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Models and simulation techniques
from stochastic geometry
Wilfrid S. Kendall
Department of Statistics, University of Warwick
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A course of 5 × 75 minute lectures at the
Madison Probability Intern Program, June-July 2003
Introduction
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Stochastic geometry is the study of random patterns, whether of points, line segments, or objects.
Related reading:
Stoyan et al. (1995);
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Barndorff-Nielsen et al. (1999);
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Møller and Waagepetersen (2003) and the recent expository essays edited
by Møller (2003);
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The Stochastic Geometry and Statistical Applications section of Advances
in Applied Probability.
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Contents
1
A rapid tour of point processes
7
2
Simulation for point processes
45
3
Perfect simulation
79
4
Deformable templates and vascular trees
107
5
Multiresolution Ising models
127
A Notes on the proofs in Chapter 5
149
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Chapter 1
A rapid tour of point processes
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Contents
1.1
1.2
1.3
1.4
1.5
The Poisson point process . . . .
Boolean models . . . . . . . . . .
General definitions and concepts
Markov point processes . . . . .
Further issues . . . . . . . . . . .
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1.1. The Poisson point process
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The simplest example of a random point pattern is the Poisson point process, which
models “completely random” distribution of points in the plane R2 , or 3-space R3 ,
or other spaces. It is a basic building block in stochastic geometry and elsewhere,
with many symmetries which typically reduce calculations to computations of area
or volume.
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Figure 1.1: Which of these point clouds exhibits interaction?
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Definition 1.1 Let X be a random point process in a window W ⊆ Rn
defined as a collection of random variables X(A), where A runs through
the Borel subsets of W ⊆ Rn . The point process X is Poisson of intensity
λ > 0 if the following are true:
1. X(A) is a Poisson random variable of mean λ Leb(A);
2. X(A1 ), . . . , X(An ) are independent for disjoint A1 , . . . , An .
Examples:
stars in the night sky;
patterns used to define collections of random discs;
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. . . or space-time structures;
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Brownian excursions indexed by local time(!).
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• In dimension 1 we can construct X using a Markov chain (“number of points
in [0, t]”) and simulate it using Exponential random variables for inter-point
spacings. Higher dimensions are trickier! See Theorem 1.5 below.
• We can replace λ Leb (and W ⊆ Rn ) by a nonnegative σ-finite measure µ
on a measurable space X , in which case X is an inhomogeneous Poisson
point process with intensity measure µ. We should also require µ to be
diffuse (P [µ({x}) = 0] = 1 for every point {x}) in order to avoid the point
process faux pas of placing than one point onto a single location.
Exercise 1.2 Show that any measurable map will carry one (perhaps
inhomogeneous) Poisson point process into another, so long as
– the image of the intensity measure is a diffuse measure;
– and we note that the image Poisson point process is degenerate
if the image measure fails to be σ-finite.
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Exercise 1.3 Show that in particular we can superpose two independent Poisson point processes to obtain a third, since the sum of independent Poisson random variables is itself Poisson.
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• It is a theorem that the second, independent scattering, requirement of Definition 1.1 is redundant! In fact even the first requirement is over-elaborate:
Theorem 1.4 Suppose a random point process X satisfies
P [X(A) = 0]
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=
exp (−µ(A))
for all measurable A .
Then X is an (inhomogeneous) Poisson point process of intensity measure µ.
– This can be proved directly. I find it more satisfying to view the result as a consequence of Choquet capacity theory, since this approach
extends to cover a theory of random closed sets (Kendall 1974). Compare inclusion-exclusion arguments and “correlation functions” used
in the lattice case in statistical mechanics (Minlos 1999).
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– We don’t have to check every measurable set:1 it is enough to consider
sets A running through the finite unions of rectangles!
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– But we cannot be too constrained in our choice of various A (the family
of convex A is not enough).
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1 Certainly
we need check only the BOUNDED measurable sets.
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• An alternative approach2 lends itself to simulation methods for Poisson point
processes in bounded windows: condition on the total number of points.
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Theorem 1.5 Suppose the window W ⊆ Rn has finite area. The following constructs X as a Poisson point process of finite intensity λ:
– X(W ) is Poisson of mean λ Leb(W );
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– Conditional on X(W ) = n, the random variables X(A) are
constructed as point-counts of a pattern of n independent points
U1 , . . . , Un uniformly distributed over W . So
X(A)
=
# {i : Ui ∈ A} .
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Exercise 1.6 Generalize Theorem 1.5 to (inhomogeneous) Poisson
point processes and to the case of W of infinite area/volume.
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2 The construction described in Theorem 1.5 is the same as the one producing the link via conditioning between Poisson and multinomial contingency table models.
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Exercise 1.7 Independent thinning. Suppose each point ξ of X, produced as in Theorem 1.5, is deleted independently of all other points
with probability 1 − p(ξ) (so p(ξ) is the retention probability). Show
that the resulting thinned random point pattern is an inhomogeneous
Poisson point process with intensity measure p(u) Leb( du).
Exercise 1.8 Formulate the generalization of the above exercise
which uses position-dependent thinning on inhomogeneous Poisson
point processes.
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• The process produced by conditioning on X(W ) = n has a claim to being
even more primitive than the Poisson point process.
Definition 1.9 A Binomial point process with n points defined on a
window W is produced by scattering n points independently and uniformly over W .
However you should notice
– point counts in disjoint subsets are now (perhaps weakly) negatively
correlated: the more in one region, the fewer can fall in another;
– the total number n of points is fixed, so we cannot extend the notion of
a (homogeneous) Binomial point process to windows of infinite measure;
– if you like that sort of thing, the Binomial point process is the simplest
representative of a class of point processes (with finite total number of
points) analogous to canonical ensembles in statistical physics. The
Poisson point process relates similarly to grand canonical ensembles
(grand, because points are indistinguishable and we randomize over
the total number of particles). Of course these are trivial cases because
there is no interaction between points — but more of that later!
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Here’s a worry.3 It is easy to prove that the construction of Theorem 1.5 produces
point count random variables which satisfy Definition 1.1 and therefore delivers a
Poisson point process. But what about the other way round? Can we travel from
Poisson point counts to a random point pattern? Or is there some subtle concealed
random behaviour in Theorem 1.5 going beyond the labelling of points, which
cannot be extracted from a point count specification?
The answer is that the construction and the point count definition are equivalent.
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3 Perhaps you think this is more properly a neurosis to be dealt with by personal counselling?!
Consider that a neurotic concern here is what allows us to introduce measure-theoretic foundations
which are most valuable in further theory.
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Theorem 1.10 Any Poisson point process given by Definition 1.1 induces
a unique probability measure on the space of locally finite point patterns,
when this is furnished with the σ-algebra generated by point counts.
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Here is a sketch of the proof.
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Take W = [0, 1)2 . For each n = 1, 2, . . . , make a dyadic dissection of [0, 1)2 into
4n different sub-rectangles [k2−n , (k + 1)2−n ) × [`2−n , (` + 1)2−n ). For a given
level n, visualize the pattern Pn obtained by painting sub-rectangles black if they
are occupied by points, white otherwise. The point counts for each of these 4n
sub-rectangles are independent Poisson of mean 4−n λ: by subadditivity of probability:
P [one or more point counts > 1] ≤ 4n (1 − (1 + 4−n λ) exp −4−n λ
= 4n (4−n λ)2 θ exp −4−n λθ
for some θ ∈ (0, 1) (use the mean value theorem for x 7→ 1 − (1 + x) exp(−x)),
which is summable in n. So by Borel-Cantelli we deduce that for all large enough
n the black sub-rectangles in pattern Pn contain just one Poisson point each. Ordering these points randomly yields the construction in Theorem 1.5.
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Simulation techniques for Poisson point processes are rather direct and straightforward: here are two possibilities.
1. We can use Theorem 1.5 to reduce the problem to the algorithm
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• draw N = n from a Poisson random variable distribution;
• draw n independent values from a Uniform distribution over the required window.
def poisson process (region ← W, intensity ← alpha) :
pattern ← [ ]
for i ∈ range (poisson (alpha)) :
pattern.append (uniform (region ← W ))
return pattern
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2. Quine and Watson (1984) describe a generalization of the 1-dimensional
“Markov chain” method mentioned above. Suppose we wish to simulate a
Poisson point process of intensity λ in ball(o, R). We describe the twodimensional case; the generalization to higher dimensions is straightforward. 4
Using polar coordinates and power transform of radius, produce a
measure-preserving map
(0, R2 /2] × (0, 2π]
→
(s, θ)
7→
ball(o, R)
√
( 2s, θ)
(1.1)
√
where ( 2s, θ) is expressed in polar coordinates, and [0, R2 /2] ×
[0, 2π] and the ball(o, R) are endowed with the measure λ Leb. Being measure-preserving, the map preserves point processes. But
the following Exercise 1.11 shows a Poisson(λ) point process on
the product space [0, R2 /2] × [0, 2π] can be thought of as a 1dimensional Poisson(2λπ) point process of “half-squared radii” on
[0, R2 /2] (which may be constructed using a sequence of independent
Exponential(2λπ) random variables), each point of which is associated
with an “angle” which is uniformly distributed on [0, 2π].
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4 The Quine-Watson method is particularly suited to the construction of Poisson point processes
which are to be used to build geometric tessellations . . .
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Exercise 1.11 Suppose X is a Poisson point process of intensity λ
on the rectangle [a, b] × [0, 1]. Show that it may be realized as a
one-dimensional Poisson point process on [a, b] of intensity λ, enriched by attaching to each one-dimensional point an independent
Uniform([0, 1]) random variable to provide the second coordinate.
Simply suppose the construction is given by Theorem 1.5, show the first coordinate of each point provides the required one-dimensional Poisson point
process, note the second coordinate is uniformly distributed as required!
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Exercise 1.12 Why is it irrelevant to this proof that the map image
omits the centre o of ball(o, R)?
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Exercise 1.13 Check the details of modifying the Quine-Watson construction for higher dimensions, to show how to build a d-dimensional
Poisson point process using independent sequences of Exponential
random variables and random variables uniformly distributed on
S d−1 .
Exercise 1.14 Consider the following method of parametrizing lines:
measure s the perpendicular distance from the origin and θ the angle
made by the perpendicular to a reference line. How should you modify the Quine-Watson construction to produce a Poisson line process
which is invariant under isometries of the plane? What about invariant Poisson hyperplane processes?
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What do large cells look like? In the early 1940s D.G. Kendall conjectured a large cell should approximate a disk. We now know this
is true, and more besides! (Miles 1995; Kovalenko 1997; Kovalenko
1999; Hug et al. 2003)
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Given a Poisson point process, we may construct a locally finite point pattern
using the methods of Theorem 1.10, and enumerate the points of the pattern using
some arbitrary rule. So we may speak of summing h(ξ) over the pattern points
ξ; by Fubini the result will not depend on the arbitrary ordering so long as h is
nonnegative.
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Theorem 1.15 If X is a Poisson point process of intensity λ and if h(x, φ)
is a nonnegative measurable function of a point x and a locally finite point
pattern φ then
X
Z
E
h(ξ, X \ {ξ})
=
E [h(u, X)] λ Leb( du) ,
(1.2)
and indeed by direct generalization
X
E
h(ξ1 , . . . , ξn , X \ {ξ1 , ξ2 , . . . , ξn })
Z
···
Rd ···Rd
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=
disjoint
ξ1 ,...,ξn ∈X
Z
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Rd
ξ∈X
(1.3)
n
E [h(u1 , . . . , un , X)] λ Leb( du1 ), . . . , Leb( dun ) .
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Exercise 1.16 Use Theorem 1.5 to prove (1.2) for h(u, X) which is positive
only when u lies in a bounded window W and which depends on X only on
the intersection of the pattern X with the window W . Hence deduce the full
Theorem 1.15.
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This delivers a variation on the independent scattering property of the Poisson
point process.
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Corollary 1.17 If g is a nonnegative measurable function of a locally finite
point pattern φ then
E [g(X \ {u})|u ∈ X]
=
E [g(X)]
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hP
i
Use Theorem 1.15 to calculate E
ξ∈X I [ξ ∈ ball(u, ε)]g(X \ {ξ}) . Interpret
R
the expectation as ball(u,ε) E [g(X \ {u})|u ∈ X] λ Leb( du)
We say the Palm distribution for a Poisson point process is the Poisson point process itself with the conditioning point added in!
1.2. Boolean models
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In stochastic geometry the counterpart to the Poisson point process is the Boolean
model; the set-union of (possibly random) geometric figures or grains located
one at each point or germ of an underlying Poisson point process. Much of the
amenability of the Poisson point process is inherited by the Boolean model, though
nevertheless its analysis can lead to very substantial calculations and theory.
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Boolean models have the advantage of being simple enough that one can carry
explicit computation. They have a long application history, for example being
used by Robbins (1944,1945) to study bombing problems (how heavily to bomb a
beach in order to ensure a high probability of landmine-clearance . . . ).
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Bombing problems and coverage: refer to Hall (1988, §3.5), Molchanov
(1996a); also note application of Exercise 1.14.
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Definition 1.18 A Boolean model Ξ is defined in terms of a germ process
which is a Poisson point process X of intensity λ, and a grain distribution
which defines a random set5 . It is a random set which is formed by the union
[
(U (ξ) + ξ) ,
ξ∈X
where the U (ξ) are independent realizations of the random set, one for each
germ or point ξ of the germ process. (Notice: the grains should also be
independent of the germs!)
The classic computation for a Boolean model Ξ is to determine the probability
P [o ∈ Ξ] that the resulting random set covers a specified point (here, the origin
o). This reduces to a matter of geometry: if o is covered then it must be covered
by at least one grain, so just think about where the corresponding germ might lie.
If U is non-random then the germ will have to lie in the reflection Û of U in the
origin, so restrict attention to germs lying in Û . If U is random then
h generalize
i
this: thin each germ ξ of the germ process using the probability P ξ ∈ Û that
5 The random set is often not random at all (a disk of fixed radius, say) or is random in a simple
parametrized manner (a disk of random radius, or a random convex polygon defined as the intersection of a sequence of random half-spaces); so we will follow Robbins (1944,1945) in treating this
informally, rather than developing the later and profound theory of random sets (Matheron 1975)!
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the corresponding grain coversho. Theiresult is an inhomogeneous Poisson point
process of intensity measure P u ∈ Û Leb( du): calculate the probability of this
process containing no points at all to produce the following:
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Theorem 1.19 Suppose Ξ is a Boolean model with germ intensity λ and
typical random grain U . Then the area / volume fraction is
P [o ∈ Ξ]
=
1 − exp (−λ E [Leb(U )]) .
(1.4)
A similar argument produces the generalization
Theorem 1.20 Suppose Ξ is a Boolean model with germ intensity λ and
typical random grain U . Let K be a compact region. Then the probability
of Ξ hitting K is
h
i
P [Ξ ∩ K 6= ∅] = 1 − exp −λ E Leb(Û ⊕ K) ,
(1.5)
where Û ⊕ K is a Minkowski sum:
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K 1 ⊕ K2
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=
{x1 + x2 : x1 ∈ K1 , x2 ∈ K2 } .
(1.6)
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Remember, from the discussion prior to Theorem 1.15, that we may view the
germs as a sequence of points (using some arbitrary method of ordering); so simulation of Ξ need only depend on generating a suitable sequence of independent
random sets U1 , U2 , . . . .
Harder analytical questions include:
• how to estimate the probability of complete coverage?
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• in a high-intensity Boolean model, what is the approximate statistical shape
of a vacant region?
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• as intensity increases, how to describe the transition to infinite connected
regions of coverage?
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Useful further reading for these questions: Hall (1988), Meester and Roy (1996),
Molchanov (1996b).
1.3. General definitions and concepts
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1.3.1. General Point Processes
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Other approaches to point processes include:
• random measures (nonnegative integer-valued, charge singletons with 0, 1);
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• measures on exponential measure spaces;
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reflected in notation (a point process X can have points belonging to it (ξ ∈ X),
can be used for point counts (X(E) when E is a subset of the ground space, and
can be used to provide a reference measure so as to define other point processes
via densities). Carter and Prenter (1972) establish essential equivalences.
We can define general point processes as probability measures on the space of locally finite point patterns, following Theorem 1.10. This allows us immediately to
define stationary point processes (the distribution respects translation symmetries,
at least locally) and isotropic point processes (the same, but for rotations). The
distribution of a general point process is characterized by its void probabilities as
in Theorem 1.4. To each point process X (whether stationary, isotropic, or not)
there is an intensity measure given by Λ(E) = E [X(E)]. (For stationary point
processes this has to be a multiple of Lebesgue measure!)
We can also define the Palm distribution for X:
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Definition 1.21 E [g(X)|u ∈ X] is required to solve
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
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E

X
f (ξ)g(X \ {ξ})
=
ξ∈X∩ball(x,ε)
Z
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f (u)E [g(X)|u ∈ X] Λ( du) . (1.7)
ball(x,ε)
A somewhat dual notion is the conditional intensity: informally, if ξ is a location and x is a locally finite point pattern not including the location ξ then the
conditional intensity λ∗ (x; ξ) is the infinitesimal probability of a point pattern X
containing ξ if X \ {ξ} = x:
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λ∗ (φ; ξ)
=
P [ξ ∈ X|X \ {ξ} = x]
.
P [X \ {ξ} = x]
(1.8)
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We shall not define this in rigorous generality, as its definition will be immediate
for the Markov point processes to which we will turn below.
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These considerations provide a general context for figuring out constructions which
move away from the “no interactions” property of Poisson point processes.
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1. Poisson cluster processes (can be viewed as Boolean models in which the
grains are simply finite random clusters of points). Neyman-Scott processes
arise from clusters of independent points.
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2. Randomize intensity measure: “doubly stochastic” or Cox process.
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Exercise 1.22 If the number of points in a Neyman-Scott cluster is
Poisson, then the Neyman-Scott point process is also a Cox process.
Line processes present an interesting issue here! Davidson conjectured that a second-order stationary line process with no parallel lines
is Cox. Kallenberg (1977) counterexample:
Stoyan et al. (1995, Figure 8.3) provides an illustration.
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3D variants with applications in Parkhouse and Kelly (1998).
3. Or apply rejection sampling to a Poisson point process, rejection probability
depending on statistics computed from the point pattern. Compare Gibbs
measure formalism of statistical mechanics.
What sort of statistics might we use?
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• estimation of intensity λ by λ̂(X) = X(W )/ Leb(W ) (using sufficient
statistic X(W ));
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• measuring how close the point pattern is to a specific location using the
spherical contact distribution function
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Hs (r)
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P [dist(o, X) ≤ r] ,
(1.9)
estimated via a Boolean model Ξr based on balls U of fixed radius r by
[
Leb(Ξr ∩ W )/ Leb(W ) = Leb
(U + ξ) ∩ W / Leb(W ) ;
ξ∈X
(1.10)
• measuring how close pattern points come to each other using the nearest
neighbour distance function
D(r)
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=
=
P [dist(o, X) ≤ r|o ∈ X] .
(1.11)
Interpret conditioning on null event “o ∈ X” using Palm distributions;
• The related reduced second moment measure
1
K(r) =
E [#{ξ ∈ X : dist(o, ξ) ≤ r} − 1|o ∈ X] ,
(1.12)
λ
estimating λK(r) by the number sr (X) of (ordered) neighbour pairs (ξ1 , ξ2 )
in X for which dist(ξ1 , ξ2 ) ≤ r (use Slivynak-Mecke Theorem 1.15).
We ignore the important issue of edge effects!
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Exercise 1.23 Use Theorem 1.15 to prove that for a homogeneous Poisson
point process D(r) = Hs (r) and K(r) = Leb(ball(o, r)).
Here is a prototype of use of these statistics in rejection algorithms:
Exercise 1.24 Consider the following rejection simulation algorithm, noting that λ̂ = X(W )/ Leb(W ) to connect with previous discussion:
def reject () :
X ← poisson process (region ← W, intensity ← lamda)
while uniform (0, 1) > theta ∗ ∗(number (X)) :
X ← poisson process (region ← W, intensity ← lamda)
return X
Show that the resulting point process is Poisson of intensity λ + θ.
Use Theorem 1.5 to reduce this to a question about rejection sampling of Poisson
random variables.
1.3.2. Marked Point Processes
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We can think of Boolean models as derived from marked Poisson point processes;
to each germ point assign a random mark encoding the geometric data (such as
disk radius) of the grain. In effect the Poisson point process now lives on Rd × M
where M is the mark space.
• Poisson cluster processes as mentioned above: the grains are finite random
clusters;
• segment and disk processes;
• line and (hard) rod processes;
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• fibre and surface processes.
Compare Exercise 1.11; the intensity measure will factorize as µ × ν so long as the
marks are independent, where µ is the intensity of the basic Poisson point process
and the measure ν is the distribution of the mark.
But beware! it can be meaningful to consider improper mark distributions.
Examples include:
Decomposition of Brownian motion into excursions;
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Marked planar Poisson point process for which the marks are disks of
random radius R, density proportional to 1/r if r ≤ threshold.
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1.4. Markov point processes
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Consider interacting point processes generating by rejection sampling (Exercise
1.24 is simple example!).
For the sake of simplicity, we suppose here that all our processes are two-dimensional.
The general procedure is to use a statistic t(X) in the algorithm
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def reject () :
X ← poisson process (region ← W, intensity ← lamda)
while uniform (0, 1) > exp (theta ∗ t (X)) :
X ← poisson process (region ← W, intensity ← lamda)
return X
We can now use standard arguments from rejection simulation. The resulting point
process distribution has a density
constantθ × exp(θt(X))
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(1.13)
with respect to the Poisson point process X of intensity λ, for normalizing constant
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constantθ
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=
E [exp(θt(X))|X is Poisson(λ)] .
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If θt(X) > 0 is possible then the above algorithm runs into trouble (how do you
reject a point pattern with probability greater than 1?!) However the probability
density at Equation (1.13) can still make sense. Importance sampling could of
course implement this. We can do better if the following holds:
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Definition 1.25 A point pattern density f (X) satisfies Ruelle stability if
f (X)
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≤
A exp(B#(X))
(1.14)
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for constants A > 0, B; where #(X) is the number of points in X.
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Exercise 1.26 6 Show that if the density exp(θt(X)) is Ruelle stable with
constants as in 1.25 then the rejection algorithm above will produce a
point pattern with the correct density if applied to a Poisson point process of density λeB instead of λ, and rejection probability is changed to
exp(θt(X) − B#(X))/A instead of exp(θt(X)).
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Use t(X) = sr (X), for γ = exp(θ) ∈ (0, 1), the number of ordered pairs separated by at least r, to obtain the celebrated Strauss point process (Strauss 1975).
6 Thanks
for the comment, Tom!
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The statistic t(X) = Leb(Ξr ∩ W ), for γ = exp(−θ) results in a process known
as the area-interaction point process (Baddeley and van Lieshout 1995), also
known in statistical physics as the Widom-Rowlinson interpenetrating spheres
model (Widom and Rowlinson 1970) in the case θ > 1.
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The (Papangelou) conditional intensity is easily defined for this kind of weighted
Poisson point process, which (borrowing statistical mechanics terminology) is
commonly known as a Gibbs point process: if the density of the Gibbs process
with respect to a unit rate Poisson point process is given by f (X) then the conditional intensity at ξ may be defined as
λ∗ (x; ξ)
=
f (x ∪ {ξ}
.
f (x)
(1.15)
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Exercise 1.27 Check the Nguyen-Zessin formula (compare Theorem 1.15):
for nonnegative h,


Z
X
E [h(u, x)λ∗ (x, u)] Leb( du) (1.16)
E
h(ξ, x \ {ξ}) =
ξ∈x
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Exercise 1.28 Compute the conditional intensity for the Poisson point process of intensity λ (use its density with respect to a unit-intensity Poisson
point process; see Exercise 1.24).
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Exercise 1.29 Compute the conditional intensity for the Strauss point process.
Page 36 of 167
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Exercise 1.30 Compute the conditional intensity for the area-interaction
point process.
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All these conditional intensities λ∗ (x; ξ) share a striking property: they do not
depend on the point pattern x outside of some neighborhood of ξ. This is an
appealing property for a point process intended to model interactions: it is captured
in Ripley and Kelly’s (1977) definition of a Markov point process:
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Definition 1.31 A Gibbs point process is a Markov point process with respect to a neighbourhood relation ∼ if its density is everywhere positive7 ,
and its conditional intensity λ∗ (x; ξ) depends only on ξ and {η ∈ φ : η ∼
ξ}.
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The celebrated Hammersley-Clifford theorem tells us that the probability density
for a Markov point process can always be factorized as a product of interactions,
functions of ∼-cliques of points of the process. Ripley and Kelly (1977) presents
a proof for the point process case, while Baddeley and Møller (1989) provides a
significant extension. A recent treatment of Markov point processes in book form
is given by Van Lieshout (2000).
However care must be taken to ensure they are well-defined: the danger of simply
writing down a statistic t(X) and defining the density by Equation (1.13) is that
the total integral
E [exp(θt(X)|X is Poisson(λ)]
may diverge so that the normalizing constant is not well-defined. (This is the case
for the Strauss point process when γ > 1, as shown by Kelly and Ripley 1976. This
particular point process cannot therefore be used to model clustering — a major
motivation for considering the more amenable area-interaction point process.)
7 The positivity requirement ensures that the ratio expression for λ∗ (x; ξ) in Equation (1.15) is
well-defined! It can be relaxed to a hereditary property.
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Exercise 1.32 Show that E γ sr (X) diverges for γ > 1 if the interaction
radius r exceeds the maximum separation of potential points in the window
W.
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Exercise 1.33 Bound exp(−θ Leb(Ξr ∩ W )) above in terms of the number
of points in W (show the interaction satisfies Ruelle-stability), and hence
deduce that the area-interaction point process is defined for all real θ.
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Page 39 of 167
Can we bias using perimeter instead of area of Ξr in area-interaction, or even
“holiness” (Euler characteristic: number-of-components minus number-of-holes)?
Like area, these characteristics are local (lead to formal densities whose conditional intensities satisfy the local requirement of Theorem 1.31). The Hadwiger
characterization theorem (Hadwiger 1957) more-or-less says that linear combinations of these three measures exhaust all local geometric possibilities! Can the
result be normalized?
Exercise 1.34 Use the argument of Exercise 1.33 to show that
exp(θ perimeter(Ξr ∩ W )) can be normalized for all θ.
Exercise 1.35 Argue similarly to show that exp(θ euler(Ξr ∩ W )) can be
normalized for all θ > 0.
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There can be no more components than disks/grains of the Boolean model Ξr .
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What about exp(θ euler(Ξr ∩ W )) for θ < 0? Key remark is Theorem 4.3 of Kendall et al. (1999): a union of
N closed disks in the plane (of whatever radii, not necessarily the same for each disk) can have at most 2N − 5
holes.
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Separate, trickier argument: disks can be replaced by
polygons satisfying a “uniform wedge” condition.
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There are counterexamples . . . .
1.5. Further issues
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Before moving on to develop machinery for simulating realizations of Markov
point processes, we make brief mention of a couple of recent pieces of work.
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1.5.1. Poisson point processes and set-indexed martingales
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Page 41 of 167
There are striking applications of set-indexed martingales to distributional results
for the areas of adaptively random regions. Recall the notion of stopping time T
for a filtration {Ft }. Here is a variant on the standard definition: we require
{ω : ∆(ω) t} ∈ Ft
where
∆(ω) = {ω : t T (ω)} .
(1.17)
Makes sense for partially-ordered family of closed sets (using set-inclusion); hence
martingales, and Optional Sampling Theorem (Kurtz 1980). Zuyev (1999) used
this to give an elegant proof of results of Møller and Zuyev (1996):
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Theorem 1.36 Suppose ∆ is a compact stopping set for X a Poisson point
process of intensity α, such that
P [X(∆) = n]
>
0
and does not depend on α .
Then Leb(∆) given X(∆) is Gamma(n, α).
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Examples: minimal centred ball containing n points (easy: compare Exercise
1.11); regions constructed using Delaunay or Voronoi tessellations.
Proof: use likelihood ratio for α (furnishing a martingale), hence
h
i
E ez Leb(∆) | X(∆) = n, α = α
=
z Leb(∆)
P e
I [X(∆) = n]αn α0−n e−(α−α0 ) Leb(∆) | α = α0
.
P [X(∆) = n | α = α0 ]
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(a) Delaunay circumdisk
(b) Voronoi flower
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Cowan et al. (2003) use this technique to consider the extent to which these examples lead to decompositions as in the minimal centred ball.
1.5.2. Stationary random countable dense sets
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We conclude this lecture with a cautionary example! Without straying very from
the theory of random point processes one can end up in the bad part of the Measure
Theory town. Suppose for whatever reason one wishes to consider random point
sets X which are countable but are also dense (set of directions of a line process?).
We need to be precise about what we can observe, so that we genuinely treat the
different points of X on equal terms. So suppose we can observe any event in the
hit-or-miss σ-algebra generated by [X ∩ A] 6= ∅, as A runs through the Borel sets.
Suppose that X is stationary.
(We thus exclude the situations considered by Aldous and Barlow (1981), where
account is taken of how various points are constructed — in effect the dense point
process is obtained by projection from a conventional locally finite point process.)
There are two ways to define such sets:
(a) measurable subset of Ω × R;
(b) Borel-set-valued random variable, hit-or-miss measurable.
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Equivalent for case of closed sets (Himmelberg 1975, Thm. 3.5). But (warning!)
the set of countable sets is not hit-or-miss measurable, though constructive and
weak countability are equivalent (section theorem).
What can we say about stationary random countable dense sets?
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Theorem 1.37 (Kendall 2000, Thm. 4.3, specialized to Euclidean case.) If
E is any hit-or-miss measurable event then either P [E] = 0 or P [E] = 1.
Whether 0 or 1 depends on E but not on X!
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Exercise 1.38 Compare these two stationary random countable dense sets:
(a) For Y a unit-intensity Poisson point process on R × (0, ∞), the countable set obtained by projection onto the first coordinate.
(b) The set Q + Z, where Q is the set of rational numbers, and Z is a
Uniform([0, 1]) random variable.
They are very different, but appear indistinguishable . . . .
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This bad behaviour should not be surprising: for example Vitali’s non-measurable
set depends on there being no measurable random variable taking values in Q + Z.
(Contrast our use of enumeration for locally finite point sets in Theorem 1.15.)
Careless thought betrays intuition!
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Chapter 2
Simulation for point processes
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Contents
2.1
2.2
2.3
2.4
2.5
2.6
General theory . . . . . . . . . . .
Measure-theoretic approach . . . .
MCMC for Markov point processes
Uniqueness and convergence . . . .
The MCMC zoo . . . . . . . . . . .
Speed of convergence . . . . . . . .
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48
51
57
63
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70
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Computations with Poisson process, Boolean model, cluster processes can be very
direct, and simulation is usually direct too. But what about Markov point processes?
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• Simple-minded rejection procedures (§1.4) are usually infeasible.
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• What about accept-reject procedures sequentially in small sub-windows,
thus constructing a point-pattern-valued Markov chain whose invariant distribution is the desired point process?
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• Or proceed to limit: add/delete single points?
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Markov chain Monte Carlo (MCMC) Gibbs’ sampler, Metropolis-Hastings!
Poisson point process as invariant distribution of spatial immigration-death
process after Preston (1977), also Ripley (1977, 1979).
To deal with more general Markov point processes it is convenient first to work
through the general theory of detailed balance equations for Markov chains on
general state-spaces. We will show how to use this to derive simulation algorithms
for our example point processes above, and discuss the more general MetropolisHastings (MH) formulation after Metropolis et al. (1953) and Hastings (1970),
which allows us to view MCMC as a sequence of proposals of random moves
which may or may not be rejected.
Simple example of detailed balance
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Underlying the Poisson point process example: the process X of total number of
points alters as follows
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• birth in time interval [0, ∆t) at rate α∆t;
• death in time interval [0, ∆t) at rate X∆t.
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Then we can solve
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P [X(t) = n − 1 and birth in [t, t + ∆t)] ≈ π(n − 1) × α∆t
= π(n) × n∆t ≈ P [X(t) = n and death in [t, t + ∆t)]
by
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π(n)
=
α
π(n − 1)
n
= ... =
αn e−α
.
n!
2.1. General theory
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• Ergodic theory for Markov chains: invariant distribution
X
π(y) =
π(x)p(x, y)
x
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or (continuous time)
X
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π(x)q(x, y) = 0 .
x
• Detailed balance;
– reversibility π(x)p(x, y) = π(y)p(y, x) ;
– dynamic reversibility π(x)p(x, y) = π(ỹ)p(ỹ, x̃) ;
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Exercise 2.1 Show detailed balance (whether reversibility or dynamical reversibility!) implies invariant distribution.
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P
• improper distributions: does x π(x) converge?
R
Pn
• estimate θ(y) dπ(y) using limn→∞ n1 m=1 θ(Ym ) ;
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Definition 2.2 Geometric ergodicity: there is R(x) > 0 with
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kπ − p(n) (x, ·)ktv ≤ R(x)ρn
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for all n, all starting points x.
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Definition 2.3 Uniform ergodicity: there are ρ ∈ (0, 1), R > 0 not
depending on the starting point x such that
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kπ − p(n) (x, ·)ktv ≤ Rρn
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for all n, uniformly for all starting points x;
2.1.1. History
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• Metropolis et al. (1953)
• Hastings (1970)
• Simulation tests
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• Geman and Geman (1984)
• Gelfand and Smith (1990)
• Geyer and Møller (1994), Green (1995)
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• ...
Definition 2.4 Generic Metropolis algorithm:
• Propose: move x → y with probability r(x, y);
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• Accept: move with probability α(x, y).
Required: balance π(x)r(x, y)α(x, y) = π(y)r(y, x)α(y, x).
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Definition 2.5 Hastings ratio: set α(x, y) = min{Λ(x, y), 1} where
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Λ(x, y)
=
π(y)r(y, x)
.
π(x)r(x, y)
2.2. Measure-theoretic approach
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We follow Tierney (1998) closely in presenting a treatment of the Hastings ratio for
general discrete-time Markov chains, since in stochastic geometry one frequently
deals with chains on quite complicated state spaces.
Here is the key detailed balance calculation in general form: compare the “fluxes”
π( dx)p(x, dy) and π( dy)p(y, dx). Both are absolutely continuous with respect
to their sum: set
Λ(y, x)
Λ(y, x) + 1
=
π( dx)p(x, dy)
π( dx)p(x, dy) + π( dy)p(y, dx)
defining a possibly infinite Λ(y, x) up to nullsets of π( dx)p(x, dy)+π( dy)p(y, dx).
Now Λ(y, x) is positive off a null-set N1 of π( dx)p(x, dy) and finite off a null-set
N2 of π( dy)p(y, dx). Moreover N1 and N2 can be chosen as transposes of each
other under the symmetry (x, y) ↔ (y, x), because they are defined in terms of
whether Λ(y, x)/(Λ(y, x) + 1) is 0 or 1, and so we may arrange1 for
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Λ(y, x)
Λ(x, y)
+
Λ(y, x) + 1 Λ(x, y) + 1
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=
1.
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(2.1)
1 working
to null-sets of π( dx)p(x, dy) + π( dy)p(y, dx) throughout!
Applying simple algebra, we deduce Λ(y, x)Λ(x, y) = 1 off N1 ∪ N2 . We know
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0 < Λ(y, x), Λ(x, y) < ∞
so
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Λ(y, x)
Λ(y, x) + 1
Λ(y, x)
Λ(x, y)
×
Λ(y, x) + 1 Λ(x, y)
Λ(x, y)
min {Λ(y, x), Λ(y, x)Λ(x, y)}
(Λ(y, x) + 1)Λ(x, y)
Λ(x, y)
= min {Λ(y, x), 1}
(2.2)
Λ(x, y) + 1
min {1, Λ(x, y)}
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off N1 ∪ N2 ,
=
min {1, Λ(x, y)}
and therefore
min {1, Λ(x, y)} π( dx)p(x, dy) = min {1, Λ(y, x)} π( dy)p(y, dx) .
The kernel min{1, Λ(x, y)}p(x, dy) may have a deficit (integral over y less than
one): compensate by adding an atom
Z
1 − min{1, Λ(x, u)}p(x, du) δx ( dy)
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to obtain a reversible kernel (actually a Metropolis-Hastings kernel!)
Z
1 − min{1, Λ(x, u)}p(x, du) δx ( dy) + min{1, Λ(x, y)}p(x, dy)
2.2.1. Projection arguments
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In case π( dx)p(x, dy) and π( dy)p(y, dx) are mutually absolutely continuous
Λ(x, y)
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π( dy)p(y, dx)
π( dx)p(x, dy)
(2.3)
is exactly the Hastings ratio. Otherwise it arises via the Metropolis map L1 projection arguments of Billera and Diaconis (2001), measurable case.
Sketch: Let K be family of all kernels and π( dx) be target probability measure.
R
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=
=
{q(x, dy) ∈ K : π( dx)q(x, dy) = π( dy)q(y, dx)}
and introduce distance (via Hahn-Jordan decomposition of signed measures)
Z Z
d(p, q) =
|π( dx) (q(x, dy) − p(x, dy))|
x6=y
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between p ∈ K and q ∈ R.
Remembering q ∈ R, introduce a deficit signed kernel (x, dy) by
π( dy)q(y, dx)
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=
π( dx)p(x, dy) − π( dx)ε(x, dy)
and define
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H+
H−
= {(x, y) : Λ(x, y) > Λ(y, x)} ,
= {(x, y) : Λ(x, y) < Λ(y, x)}
(2.4)
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where Λ(x, y) is defined in terms of p using Equation (2.3).
Notice H− = {(x, y) : (y, x) ∈ H+ }, and (x, x) 6∈ H− ∪ H+ for all x.
Furthermore, set p̃(x, dy) = α(x, y)p(x, dy) with
Λ(y, x)
α(x, y) = min 1,
Λ(x, y)
(use Λ(x, y) to compute acceptance probabilities to produce p̃ ∈ R). Now
Z Z
d(p, p̃) =
|π( dx) (p̃(x, dy) − p(x, dy))|
x6=y
Z Z
=
|π( dx) (α(x, y)p(x, dy) − p(x, dy))|
x6=y
Z Z
Λ(y, x)
=
π( dx) p(x, dy) − min 1,
p(x, dy)
Λ(x, y)
x6=y
Z Z
Λ(y, x)
=
π( dx) p(x, dy) −
p(x, dy)
Λ(x, y)
H
Z Z +
=
(π( dx)p(x, dy) − π( dy)p(y, dx))
H+
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using 0 ≤ α(x, y) ≤ 1 and definition of Λ(x, y) via a density in Equation (2.1).
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Note also that replacing p̃(x, dy) by α0 (x, y)p(x, dy), for α0 (x, y) ∈ [0, α(x, y)]
increases the distance by
Z Z
π( dx) (α(x, y) − α0 (x, y)) p(x, dy) .
(2.5)
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x6=y
Now compute
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Z Z
d(p, q)
|π( dx) (q(x, dy) − p(x, dy))|
=
x6=y
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Z Z
≥
|π( dx) (q(x, dy) − p(x, dy))| +
H+
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+ |π( dy) (q(y, dx) − p(y, dx))|
Z Z
|π( dx)ε(x, dy)| +
=
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H+
+ |π( dx)p(x, dy) − π( dx)ε(x, dy) − π( dy)p(y, dx)|
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Z Z
≥
π( dx)p(x, dy) − π( dy)p(y, dx)
=
d(p, p̃) .
H+
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Thus p̃(x, dy), defined by p̃(x, dy) = α(x, y)p(x, dy), is the kernel in R which is
closest to the original kernel p(x, dy) in this distance d(p, p̃). Moreover equality in
the last step can hold only when π( dx)ε(x, dy) restricted to H+ is a non-negative
measure dominated by π( dx)p(x, dy), which is to say when
π( dx)ε(x, dy)
=
(1 − α0 (x, y)) π( dx)p(x, dy) on H+
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Page 56 of 167
for some α0 (x, y) ∈ [0, 1]. Exactly the same arguments applied to the transposed
measure π( dy)p(y, dx), and Equation (2.4), lead us to deduce that if q(d, dy) is a
kernel in R which minimizes d(q, p) then
q(x, y)
=
α0 (x, y)p(x, dy) .
We deduce now from the discussion before Equation (2.5) that the distance is
minimized only when
α0 (x, y)
=
α(x, y) = min{1,
Λ(y, x)
}
Λ(x, y)
in L1 (π( dx)p(x, dy)). In this sense the Metropolis-Hastings sampler
p̃(x, dy)
=
α(x, y)p(x, dy)
is an L1 projection of p(x, dy) onto the class of reversible kernels.
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We note in passing that the Hastings ratio (as opposed to other possible accept/reject
rules) yields the largest spectral gap and minimizes the asymptotic variance of additive functionals (Peskun 1973; Tierney 1998).
2.3. MCMC for Markov point processes
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We now apply the above to simulation of a point process in a bounded window,
whose distribution2 has density f (ξ) with respect to a Poisson point process measure π1 ( dξ) of unit intensity (Geyer and Møller 1994; Green 1995). We need
to design a point-pattern-valued Markov chain whose limiting distribution will be
f (ξ)π1 ( dξ), and we will constrain the Markov chain to evolve solely by adding
and deleting points.
Let λ∗ (x; ξ) = f (x ∪ {ξ})/f (x) be the Papangelou conditional intensity for the
target point process, as defined in Equation (1.15). Suppose the point process
satisfies some condition such as local stability, namely
(λ∗ (x; ξ)
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γ
(2.6)
for some positive constant γ. We can weaken this: we need require only that
Z
W
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≤
(λ∗ (x; u) du
Z
≤
γ(u) d Leb(u)
<
∞.
X
However the stronger condition is useful later, in CFTP, and implies Ruelle stability.
2 Here we use the measurable space of locally finite point patterns, introduced in Theorem 1.10.
Notice that we are dealing with a bounded window, so in fact the total number of points in the pattern
will always be finite.
2.3.1. Discrete time case
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Consider a Markov chain X which evolves as follows. Fix αn so that α0 = 0 and
Z
αn + αn+1
λ∗ (x; u) du ≤ 1 .
(2.7)
W
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• suppose the current value of X is X = x with #(x) = n. Choose from:
– Death: with probability αn−1 , choose a point η of x at random and
delete it from x.
– Birth: generate a new point ξ with sub-probability density
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αn λ∗ (x; ·)/(n + 1)
and add it to x (in case of probability deficit, do nothing!);
Arrange matters so that birth and death are exclusive (feasible by (2.7) above).
It is tedious to write down the transition kernel for X: however the message of
Λ(x, y) (as given in (2.3)) is that equality of probability fluxes suffices. Point
pattern configurations are tricky (patterns are not ordered); however an equivalent
task is to arrange for equality of birth and death fluxes when the target distribution
is given by rejection sampling based on the construction of Theorem 1.5, recalling
that f is a density with respect to a unit Poisson point process. Set Leb(W ) = A.
We consider the exchange between x̃ and x where x̃ = x ∪ {ξ}, x = x̃ \ {ξ}.
Suppose #(x) = n, so #(x̃) = n + 1.
Death: the flux from x̃ to x is
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f (x̃)
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e−A dx1 . . . dxn dξ
αn
×(n + 1)!×
(n + 1)!
n+1
=
αn f (x̃)
e−A dx1 . . . dxn dξ
.
n+1
The factor (n + 1)! in the numerator of the left-hand side allows for the
(n+1)! ways of permuting x1 , . . . , xn , ξ to represent the same configuration
x̃. The death probability αn /(n + 1) allows for the requirement that the
point chosen to die should be ξ and not one of x1 , . . . , xn .
Birth: the flux from x to x̃ is
f (x)
αn ∗
e−A dx1 . . . dxn
× n! ×
λ (x; ξ) dξ
n!
n+1
e−A dx1 . . . dxn
= αn f (x)λ∗ (x; ξ)
dξ . (2.8)
n+1
The factor n! in the numerator of the left-hand side allows for the n! ways
of permuting x1 , . . . , xn to represent the same configuration x.
Using the formula (1.15) for the Papangelou conditional intensity, we deduce
equality between these two fluxes. It follows that X has the desired target distribution as an invariant distribution.
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Exercise 2.6 Consider the special case λ∗ (x; ξ) = β, choosing say
αn
=
1/(1 + β Leb(W ))
for all n > 0. Write down the simulation algorithm. Check that the resulting kernel is reversible with respect to πβ , the distribution measure of the
Poisson point process of rate β.
There is a very easy approach: recognize that there can be no spatial interaction, so
it suffices to set up detailed balance equations for the Markov chain which counts the
total number of points . . . .
2.3.2. Continuous Time case
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In the treatment above it is tempting to make αn as large as possible, to maximize
the rate of change per time-step. However we can also consider the limiting case
of a continuous-time model and reformulate the simulation in event-based terms,
thus producing a spatial birth-and-death process. For small δ, set αn = (n + 1)δ
for n to maintain Equation (2.7). (For larger n use smaller αn . . . .) Then
• suppose the current value of X is X = x. While #(x) is not too large,
choose from:
– Death: with probability #(x)δ, choose a point η of x at random and
delete it from x.
– Birth: generate a new point ξ with sub-probability density λ∗ (x; ξ)δ,
and add it to x (in case of probability deficit, do nothing!);
Taking the limit as δ → 0, and reformulating as a continuous time Markov chain:
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• suppose the current value of X is X = x:
R
• wait an Exponential(#(x) + W λ∗ (x; ξ) dξ) random time then
R
– Death: with probability #(x)/(#(x) + W λ∗ (x; ξ) dξ), choose a
point η of x at random and delete it from x;
– Birth: otherwise generate a new point ξ with probability density proportional to λ∗ (x; ξ), and add it to x.
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Exercise 2.7 Verify that the above algorithm produces an invariant distribution with Papangelou conditional intensity λ∗ (x; ξ).
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A coupling argument shows this is geometrically ergodic – but we will do better!
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Exercise 2.8 Determine a spatial birth-and-death process with invariant
distribution which is a Strauss point process.
Example of Strauss process target.
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Exercise 2.9 Determine a spatial birth-and-death process with invariant
distribution which is an area-interaction point process.
Note that the technique underlying these exercises suggests the following: a local
stability bound λ∗ (x; ξ) ≤ M implies that the point process can be expressed as
some complicated and dependent thinning of a Poisson point process of intensity
M . We shall substantiate this in Chapter 3, when we come to discuss perfect
simulation for point processes.
2.4. Uniqueness and convergence
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We have been careful to describe the target distributions as invariant distributions
rather than equilibrium distributions, because we have not yet introduced useful
methods to determine whether our chains actually converge to their invariant distributions! As we have already hinted in Chapter 1, this can be a problem; indeed
one can write down a Markov chain which purports to have the attractive case
Strauss point process as invariant distribution, even though this point process is
not well-defined!
The full story here involves discussions of φ-recurrence and Harris recurrence,
which we reluctantly omit for reasons of time. Fortunately there is a relatively
simple technique which often works to ensure uniqueness of and convergence to
an invariant distribution, once existence is established. Suppose we can ensure that
the chain X visits the empty-pattern state infinitely often, and with finite mean
recurrence time (the empty-pattern state ∅ is an ergodic atom). Then the coupling
approach to renewal theory (Lindvall 1982; Lindvall 2002) can be applied to show
that the distribution of Xn converges in total variation to its invariant distribution,
which therefore must be unique.
This pleasantly low-tech approach foreshadows the discussion of small sets later
in this chapter, and also anticipates important techniques in perfect simulation.
2.5. The MCMC zoo
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The attraction of MCMC is that it allows us to produce simulation algorithms for
all sorts of point processes and geometric processes. It is helpful however to have
in mind a general categorization: a kind of “zoo” of different MCMC algorithms.
• Here are some important special cases of the MH sampler classified by type
of proposal distribution. Let the target distribution be π (could be Bayesian
posterior . . . ).
– Independence sampler: Choose q(x, y) = q(y) (independent of current location x) which you hope is “well-matched” to π;
– Random walk sampler: Choose q(x, y) = q(y − x) to be a random
walk increment based on current location x. Tails of target distribution
π must not be too heavy and (eg) density contours not too sharply
curved (Roberts and Tweedie 1996b);
– Langevin algorithms (Also called, “Metropolis-adjusted Langevin”):
Choose q(x, y) = q(y − σ 2 g(x)/2) where g(x) is formed using x
and the “logarithmic gradient” of the density of π( dx) at x (possibly
adjusted) (σ 2 is the jump variance). Behaves badly if logarithmic gradient tends to zero at infinity; a condition requiring something of the
order “tails must not be too light” (Roberts and Tweedie 1996a).
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The Gibbs’ sampler corresponds to the case when proposal distributions are chosen to be conditional distributions: in a simple example, if state is described by X1 , X2 then propose new X1 according to L(X1 |X2 ) (under equilibrium distribution) and vice versa.
The continuous-time algorithms for simulation of point patterns using
birth-and-death processes and conditional intensities are examples of
this.
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• Slice sampler: task is to draw from density f (x). Here is a one-dimensional
example! (But note, this is only interesting because it can be made to work in
many dimensions . . . ) Suppose f unimodal. Define g0(y), g1(y) implicitly
by the requirement that [g0(y), g1(y)] is the superlevel set {x : f (x) ≥ y}.
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def mcmc slice move (x) :
y ← uniform (0, f (x))
return uniform (g0 (y), g1 (y))
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Figure 2.1: Slice sampler
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Roberts and Rosenthal (1999, Theorem 12) show rapid convergence (order
of 530 iterations!) under a specific variation of log-concavity. Relates to
notion of “auxiliary variables”.
• Simulated tempering:
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– Embed target distribution in a family of distributions indexed by some
“temperature” parameter. For example, embed density f (x) (defined
over bounded region for x) in family of normalized
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R
(f (x))1/τ
;
(f (u))1/τ du
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– Choose weights β0 , β1 , . . . , βk for selected temperatures τ0 = 1 <
τ1 < . . . τk ;
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– Now do Metropolis-Hastings for target density
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βo (f (x))1/τ0 = β0 f (x)
Z
β1 (f (x))1/τ1 / (f (u))1/τ1 du
at level 0
at level 1
...
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1/τk
βk (f (x))
Z
/
1/τk
(f (u))
du
at level k
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where moves either propose changes in x, or propose changes in level
i; (NB: choice of βi ?)
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– Finally, sample chain only when it is at level 0.
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Page 68 of 167
• Markov-deterministic hybrid: Need to sample exp(−E(q))/Z. Introduce
“momenta” p and aim to draw from
1
π(p, q) = exp −E(q) − |p|2 /Z 0 .
2
Fact: for Hamiltonian H(p, q) = E(q) + 12 |p|2 , the dynamical system
dq
dp
= (∂H/∂p) dt = p
= −(∂H/∂q) dt = −(∂E/∂q)
has π(p, q) as invariant distribution (Liouville’s theorem). So alternate between (a) evolving using dynamics and (b) drawing from Gaussian marginal
for p. Discretization means (a) is not exact; can fix this by use of MH rejection techniques.
Cf : dynamic reversibility.
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• Dynamic importance weighting (Liu, Liang, and Wong 2001): augment
Markov chain X by adding an
R importance parameter W , replace search
for equilibrium (E [θ(Xn )] → θ(x)π( dx)) by requirement
Z
lim E [Wn θ(Xn , Wn )] ∝ θ(x)π( dx) .
n→∞
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• Evolutionary Monte Carlo Liang and Wong (2000, 2001): use ideas of
genetic algorithms (multiple samples, “exchanging” sub-components) and
tempering.
• and many more ideas besides . . . .
2.6. Speed of convergence
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There is of course an issue as to how fast is the convergence to equilibrium. This
can be analytically challenging (Diaconis 1988)!
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2.6.1. Convergence and symmetry: card shuffling
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Example: transposition random walk X on permutation group of n cards. Equilibrium occurs at strong stationary times T (Aldous and Diaconis 1987; Diaconis
and Fill 1990):
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Definition 2.10 The random stopping time T is a strong stationary time for the
Markov chain X (whose equilibrium distribution is π) if
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P [T = k, Xk = s]
=
πs × P [T = k] .
Broder: notion of checked cards. Transpose by choosing 2 cards at random; LH,
RH. Check card pointed to by LH if
either LH, RH are the same unchecked card;
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or LH is unchecked, RH is checked.
Inductive claim: given number of checked cards, positions in pack of checked
cards, list of values of cards; the map determining which checked card has which
value is uniformly random.
Pn−1 Set Tm to be the time till m cards checked. The
independent sum T = m=0 Tm is a strong stationary time by induction.
How big is T ?
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Now Tm+1 −Tm is Geometrically distributed with success probability (n−m)(m+
1)/n2 . So mean of T is
" n−1
# n−1
X
X n2 1
1
Tm =
E
+
≈ 2n log n .
n+1 n−m m+1
m=0
m=0
Pn−1
MOREOVER: T = m=0 Tm is a discrete version of time to complete infection
for simple epidemic (without recovery) in n individuals, starting with 1 infective,
individuals contacting at rate n2 . A classic calculation tells us T /(2n log n) has a
limiting distribution.
NOTE: group representation theory: correct asymptotic is 12 n log n.
The MCMC approach to substitution decoding also produces a Markov chain moving by transpositions on a permutation group of around n = 60 symbols. However
it takes much longer to reach equilibrium than order of 60 log(60) ≈ 250 transpositions. Conditioning on data can have a drastic effect on convergence rates!
2.6.2. Convergence and small sets
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The most accessible theory (as expounded for example in Meyn and Tweedie
1993) uses the notion of small sets for Markov chains. In effect, small set theory allows us to argue as if (suitably regular) Markov chains are actually discrete,
even if defined on very general state-spaces. In fact this discreteness can even be
formalized as the existence of a latent discrete structure.
Definition 2.11 C is a small set (of order k) if there is a positive integer k,
positive c and a probability measure µ such that for all x ∈ X and E ⊆ X
p(k) (x, E)
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≥
c × I [x ∈ C] µ(E) .
Markov chains can be viewed as regenerating at small sets. The notion is useful
for recurrence conditions of Lyapunov type: compare low-tech “empty pattern”
method.
Small sets play a crucial rôle in notions of periodicity, recurrence, positive
recurrence, geometric ergodicity, making link to discrete state-space theory
(Meyn and Tweedie 1993).
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Foster-Lyapunov recurrence on small sets (Mykland, Tierney, and Yu 1995;
Roberts and Tweedie 1996a; Roberts and Tweedie 1996b; Roberts and
Rosenthal 2000; Roberts and Rosenthal 1999; Roberts and Polson 1994)
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Theorem 2.12 (Meyn and Tweedie 1993) Positive recurrence holds if one
can find a small set C, a constant β > 0, and a non-negative function V
bounded on C such that for all n > 0
E [V (Xn+1 )|Xn ]
V (Xn ) − 1 + β I [Xn ∈ C] .
(2.9)
Proof:
Let N be the random time at which X first (re-)visits C. It suffices3 to show
E [N |X0 ] < V (X0 ) + constant < ∞ (then use small-set regeneration).
By iteration of (2.9), we deduce E [V (Xn )|X0 ] < ∞ for all n.
If X0 6∈ C then (2.9)
tells us n 7→ V (Xn∧N ) + n ∧ N defines a nonnegative
supermartingale (I X(n∧N ) ∈ C = 0 if n < N ) . Consequently
E [N |X0 ]
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≤
3 This
≤
E [V (XN ) + N |X0 ]
≤
V (X0 ) .
martingale approach can be reformulated as an application of Dynkin’s formula.
If X0 ∈ C then the above can be used to show
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E [N |X0 ]
= E [1 × I [X1 ∈ C]|X0 ] + E [E [N |X1 ] I [X1 6∈ C]|X0 ]
≤ P [X1 ∈ C|X0 ] + E [1 + V (X1 )|X0 ]
≤ P [X1 ∈ C|X0 ] + V (X0 ) + β
where the last step uses Inequality (2.9) applied when I [Xn−1 ∈ C] = 1.
This idea can be extended to give bounds on convergence rates.
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Theorem 2.13 (Meyn and Tweedie 1993) Geometric ergodicity holds if one
can find a small set C, positive constants λ < 1, β, and a function V ≥ 1
bounded on C such that
E [V (Xn+1 )|Xn ]
≤
λV (Xn ) + β I [Xn ∈ C] .
(2.10)
Proof:
Define N as in Theorem 2.12.
Iterating (2.10), we deduce E [V (Xn )|X0 ] < ∞ and more specifically
n
7→
V (Xn∧N )/λn∧N
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is a nonnegative supermartingale. Consequently
E V (XN )/λN |X0
≤
V (X0 ) .
Using the facts that V ≥ 1, λ ∈ (0, 1) and Markov’s inequality we deduce
P [N > n|X0 ]
≤
λn V (X0 ) ,
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which delivers the required geometric ergodicity.
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Bounds may not be tight, but there are many small sets for most Markov chains
(not just the trivial singleton sets, either!). In fact small sets can be used to show:
Theorem 2.14 (Kendall and Montana 2002) If the Markov chain has a measurable transition density p(x, y) then the two-step density p(2) (x, y) can be
expressed (non-uniquely) as a non-negative countable sum
X
p(2) (x, y) =
fi (x)gi (y) .
II
Hence in such cases small sets of order 2 abound!
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This relates to the classical theory of small sets and would not surprise its originators; see for example Nummelin 1984.
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Proof:
Key Lemma, variation of Egoroff’s Theorem:
Let p(x, y) be an integrable function on [0, 1]2 . Then we can find subsets Aε ⊂
[0, 1], increasing as ε decreases, such that
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(a) for any fixed Aε the “L1 -valued function” px is uniformly continuous on
Aε : for any η > 0 we can find δ > 0 such that |x − x0 | < δ and x, x0 ∈ Aε
implies
Z 1
|px (z) − px0 (z)| dz < η ;
0
(b) every point x in Aε is of full relative density: as u, v → 0 so
Leb([x − u, x + v] ∩ Aε )
u+v
→
1.
Moreover one can discover latent discrete Markov chains representing all
such Markov chains at period 2.
However one cannot get below order 2:
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Theorem 2.15 (Kendall and Montana 2002) There are Markov chains
which possess no small sets of order 1.
Proof:
Key Theorem:
There exist Borel measurable subsets E ⊂ [0, 1]2 of positive area which are
rectangle-free, so that if A × B ⊆ E then Leb(A × B) = 0.
(Stirling’s formula, discrete approximations, Borel-Cantelli.)
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Figure 2.2: No small sets!
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Chapter 3
Perfect simulation
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3.1
3.2
3.3
CFTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fill’s (1998) method . . . . . . . . . . . . . . . . . . . . .
Sundry other matters . . . . . . . . . . . . . . . . . . . .
81
97
99
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Establishing bounds on convergence to equilibrium is important if we are to get
a sense of how to do MCMC. It would be very useful if one could finesse the
whole issue by modifying the MCMC algorithm so as to produce an exact draw
from equilibrium, and this is precisely the idea of perfect simulation. A graphic
illustration of this in the context of stochastic geometry (specifically, the dead
leaves model) is to be found at
http://www.warwick.ac.uk/statsdept/staff/WSK/dead.html.
3.1. CFTP
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There are two main approaches to perfect simulation. One is Coupling from the
Past (CFTP), introduced by Propp and Wilson (1996). Many variations have
arisen: we will describe the original Classic CFTP, and two variants Small-set
CFTP and Dominated CFTP. Part of the fascination of this area is the interplay
between theory and actual implementation: we will present illustrative simulations.
3.1.1. Classic CFTP
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Consider reflecting random walk run From The Past (Propp and Wilson 1996):
• no fixed start: run all starts simultaneously (Coupling From The Past);
• extend past till all starts coalesce by t = 0;
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• 3-line proof: perfect draw from equilibrium.
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Figure 3.1: Classic CFTP.
Implementation:
- “re-use” randomness;
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- sample at t = 0 not at coalescence;
- control coupling efficiently using monotonicity, boundedness;
- uniformly ergodic.
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Here is the 3-line proof, with a couple of extra lines to help explain!
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Theorem 3.1 If coalescence is almost sure then CFTP delivers a sample
from the equilibrium distribution of the Markov chain X corresponding to
the random input-output maps F(−u,v] .
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Proof:
For each time-range [−n, ∞) use input-output maps F(−n,t] to define
Xt−n
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=
F(−n,t] (0)
for − n ≤ t .
We have assumed finite coalescence time −T for F . Then
X0−n
L(X0−n )
=
=
X0−T
L(Xn0 )
whenever − n ≤ −T ;
(3.1)
(3.2)
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If X converges to an equilibrium π then
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dist(L(X0−T ), π)
lim dist(L(X0−n ), π) = lim dist(L(Xn0 ), π) = 0
n
n
(3.3)
(dist is total variation) hence giving the required result.
=
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Here is a simple version of perfect simulation applied to the Ising model,
following Propp and Wilson (1996) (but they do something much cleverer!).
Here the probability of a given ±1-valued configuration S is proportional to
XX
exp β
Sx Sy
x∼y
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and our recipe is, pick a site x, compute conditional probability of Sx = +1 given
rest of configuration S,
P
X
exp(β y:y∼x (-1) × Sy )
P
ρ =
= exp(−2β
Sy ) ,
exp(β y:y∼x (1) × Sy )
y:y∼x
and set Sx = +1 with probability ρ, else set Sx = −1.
Boundary conditions are “free” in the simulation: for small β we can make a
perfect draw from Ising model over infinite lattice, by coupling in space as well as
time (Kendall 1997b; Häggström and Steif 2000).
We can do better for Ising: Huber (2003) shows how to convert Swendsen-Wang
methods to CFTP.
3.1.2. A little history
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Kolmogorov (1936): 1 Consider an inhomogeneous (finite state) Markov transition
kernel Pij (s, t). Can it be represented as the kernel of a Markov chain begun in
the indefinite past?
Yes!
Apply diagonalization arguments to select a convergent subsequence from the
probability systems arising from progressively earlier and earlier starts:
P [X(0) = k]
P [X(−1) = j, X(0) = k]
P [X(−2) = i, X(−1) = j, X(0) = k]
=
=
=
...
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1 Thanks
to Thorisson (2000) for this reference . . . .
Qk (0)
Qj (−1)Pjk (−1, 0)
Qi (−2)Pij (−2, −1)Pjk (−1, 0)
3.1.3. Dead Leaves
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Nearly the simplest possible example of perfect simulation.
• Dead leaves falling in the forests at Fontainebleau;
• Equilibrium distribution models complex images;
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• Computationally amenable even in classic MCMC framework;
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• But can be sampled perfectly with no extra effort! (And uniformly ergodic)
How can this be done? see Kendall and Thönnes (1999)
[HINT: try to think non-anthropomorphically . . . .]
Occlusion CFTP. Related work: Jeulin (1997), Lee, Mumford, and Huang (2001).
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Figure 3.2: Falling leaves of perfection.
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3.1.4. Small-set CFTP
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Does CFTP depend on monotonicity? No: we can use small sets. Consider a
simple Markov chain on [0, 1] with triangular kernel.
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Figure 3.3: Small-set CFTP.
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• Note “fixed” small triangle;
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• this can couple Markov chain steps;
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• basis of small-set CFTP (Murdoch and Green 1998).
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Extends to more general cases (“partitioned multi-gamma sampler”).
Uniformly ergodic.
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Exercise 3.2 Write out the details of small-set CFTP. Deduce how to represent the output of simple small-set CFTP (uses just one small set) in terms of
a Geometric random variable and draws from a single kernel.
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Exercise 3.3 One small set is usually not enough - coalescence takes much
too long! Work out how to use several small sets to do partitioned small-set
CFTP (“partitioned multi-gamma sampler”) (Murdoch and Green 1998).
3.1.5. Point patterns
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Recall Widom-Rowlinson model (Widom and Rowlinson 1970) from physics,
“area-interaction” model in statistics (Baddeley and van Lieshout 1995).
The point pattern X has density (with respect to Poisson process):
exp (− log(γ) × (area of union of discs centred on X))
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Figure 3.4: Perfect area-interaction.
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• Represent as two-type pattern (red crosses, blue disks);
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• inspect structure to detect monotonicity;
• attractive case γ > 1: CFTP uses “bounded” state-space (Häggström et al.
1999) in sense of being uniformly ergodic.
All cases can be dealt with using Dominated CFTP (Kendall 1998).
3.1.6. Dominated CFTP
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Foss and Tweedie (Foss and Tweedie 1998) show classic CFTP is essentially
equivalent to uniform ergodicity (rate of convergence to equilibrium does not depend on start-point). For classic CFTP needs vertical coalescence: every possible
start from time −T leads to same result at time 0.
Can we lift this uniform ergodicity requirement?
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Figure 3.5: Horizontal CFTP.
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CFTP almost works with horizontal coalescence: all sufficiently early starts from
a specific location * lead to same result at time 0. But how to identify when this
has happened?
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Recall Lindley’s equation for the GI/G/1 queue identity
Wn+1
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=
I
Page 91 of 167
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=
max{0, Wn + ηn }
and its development
Wn+1
max{0, ηn , ηn + ηn−1 , . . . , ηn + ηn−1 + . . . + η1 }
=
leading by time-reversal to
Wn+1
J
max{0, Wn + Sn − Xn+1 }
=
max{0, η1 , η1 + η2 , . . . , η1 + η2 + . . . + ηn }
and thus the steady-state expression
W∞
=
max{0, η1 , η1 + η2 , . . .} .
Loynes (1962) discovered a coupling application to queues with general inputs, which pre-figures CFTP.
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Dominated CFTP idea: Generate target chains using a dominating process
for which equilibrium is known. Domination allows us to identify horizontal coalescence by checking starts from maxima given by the dominating
process.
Thus we can apply CFTP to Markov chains which are merely geometrically ergodic (Kendall 1998; Kendall and Møller 2000; Kendall and Thönnes 1999; Cai
and Kendall 2002) or worse (geometric ergodicity 6= Dominated CFTP!).
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(a) Perfect Strauss point process. (b) Perfect conditioned Boolean
model.
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Figure 3.6: Two examples of non-uniformly ergodic CFTP.
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Description2 of dominated CFTP (Kendall 1998; Kendall and Møller 2000) for
birth-death algorithm for point process satisfying the local stability condition
λ∗ (ξ; x)
≤
γ.
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Page 93 of 167
Original Algorithm:
points η die at unit death rate,
R
R are born in pattern x
at total birth rate W λ∗ (x; u) du, density λ∗ (x; ·)/ W λ∗ (x; u) du, so local
birth rate λ∗ (x; ξ).
Perfect modification: underlying supply of randomness is birth-and-death
process with unit death rate, uniform birth rate γ per unit area. Mark points
independently with (example) Uniform(0, 1) marks. Define lower and upper processes L and U as exhibiting minimal and maximal birth rates available in range of configurations between L and U :
minimal birth rate at u
maximal birth rate at u
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= min{λ∗ (x; u) : L ⊆ ξ ⊆ U } ,
= max{λ∗ (x; u) : L ⊆ ξ ⊆ U } .
Use marks to do CFTP till current iteration delivers L(0) = U (0), then
return the agreed configuration! Example uses Poisson point patterns as
marks for area-interaction, following Kendall (1997a).
2 Recall
remark about dependent thinning!
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“It’s a thinning procedure, Jim, but not as we know it.”
Dr McCoy, Startrek, stardate unknown.
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All this suggests a general formulation:
Let X be a Markov chain on X which exhibits equilibrium behaviour.
Embed the state-space X in a partially ordered space (Y, ) so that X is at bottom
of Y, in the sense that for any y ∈ Y, x ∈ X ,
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yx
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implies
y = x.
We may then use Theorem 3.1 to show:
Theorem 3.4 Define a Markov chain Y on Y such that Y evolves as X after
it hits X ; let Y (−u, t) be the value at t a version of Y begun at time −u,
(a) of fixed initial distribution L(Y (−T, −T )) = L(Y (0, 0)), and
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(b) obeying funnelling: if −v ≤ −u ≤ t then Y (−v, t) Y (−u, t).
Suppose coalescence occurs: P [Y (−T, 0) ∈ X ] → 1 as T → ∞. Then
lim Y (−T, 0) can be used for a CFTP draw from the equilibrium of X.
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So how far can we go? It looks nearly possible to define the dominating process
using the “Liapunov function” V of Theorem 2.13. There is an interesting link –
Rosenthal (2002) draws it even closer – nevertheless:
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Existence of Liapunov function doesn’t ensure dominated CFTP!
The relevant Equation (2.10) is:
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E [V (Xn+1 )|Xn ]
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≤
λV (Xn ) + β I [Xn ∈ C] .
However there are perverse examples satisfying the supermartingale inequality, but
failing the the stochastic dominance required to use V (X) for dominated CFTP
. . . :-(.3
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3 Later:
I have discovered how to fix this using sub-sampling . . .
3.2. Fill’s (1998) method
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The alternative to CFTP is Fill’s algorithm (Fill 1998; Thönnes 1999), at first sight
quite different, based on the notion of a strong stationary time T . Fill et al. (2000)
show a profound link, best explained using “blocks” as input-output maps for a
chain.
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Figure 3.7: The “block” model for input-output maps.
First recall that CFTP can be viewed in a curiously redundant fashion as follows:
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• Draw from equilibrium X(−T ) and run forwards;
• continue to increase T until X(0) is coalesced;
• return X(0).
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Figure 3.8: Conditioning the blocks on the reversed chain.
Key observation: By construction, X(−T ) is independent of X(0) and T so . . .
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• Condition on a convenient X(0);
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• Run X backwards to a fixed time −T ;
• Draw blocks conditioned on the X transitions;
• If coalescence then return X(−T ) else repeat.
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Exercise 3.5 Can you figure out how to do a dominated version of Fill’s
method?
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Exercise 3.6 Produce a perfect version of the slice sampler! (Mira, Møller,
and Roberts 2001)
3.3. Sundry other matters
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Layered Multishift: Wilson (2000b) and further work by Corcoran and others.
Basic idea: how to draw simultaneously from Uniform(x, x + 1) for all
x ∈ R, and to try to couple the draws? Answer: draw a uniformly random
unit span integer lattice, . . . . 4
Exercise 3.7 Figure out how to replace “Uniform(x, x + 1)” by any
other location-shifted family of continuous densities. (Hint: mixtures
of uniforms . . . .)
Montana used these ideas to do perfect simulation of nonlinear AR(1) in his
PhD thesis.
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4 Uniformly
random lattices in dimension 2 play a part in line process theory.
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Read-once randomness: Wilson (2000a). The clue is in:
Exercise 3.8 Figure out how to do CFTP, using the “input-output”
blocks picture of Figure 3.7, but under the constraint that you aren’t
allowed to save any blocks!
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Page 100 of 167
Perfect simulation for Peirls’ contours (Ferrari et al. 2002). The Ising model
can be reformulated in an important way by looking only at the contours
(lines separating ±1 values). In fact these form a “non-interacting hard-core
gas”, thus permitting (at least in theory) Ferrari et al. (2002) to apply their
variant of perfect simulation (Backwards-Forwards Algorithm)! Compare
also Propp and Wilson (1996).
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Perfect simulated tempering (Møller and Nicholls 1999; Brooks et al. 2002).
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3.3.1. Efficiency
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Burdzy and Kendall (2000): Coupling of couplings:. . .
|pt (x1 , y) − pt (x2 , y)| ≤
≤ |P [X1 (t) = y|X1 (0) = x1 ] − P [X2 (t) = y|X2 (0) = x2 ]|
≤ |P [X1 (t) = y|τ > t, X1 (0) = x1 ] − P [X2 (t) = y|τ > t, X2 (0) = x2 ]| ×
×P [τ > t|X(0) = (x1 , x2 )]
Suppose |pt (x1 , y) − pt (x2 , y)| ≈ c exp(−µ2 t)
while P [τ > t|X(0) = (x1 , x2 )] ≈ c exp(−µt)
Let X ∗ be a coupled copy of X but begun at (x2 , x1 ):
|P [X1 (t) = y|τ > t, X1 (0) = x1 ] − P [X2 (t) = y|τ > t, X2 (0) = x2 ] |
= |P [X1 (t) = y|τ > t, X(0) = (x1 , x2 )] −
P [X1∗ (t) = y|τ > t, X ∗ (0) = (x2 , x1 )] |
So let σ be time when X, X ∗ couple:
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≤ P [σ > t|τ > t, X(0) = (x1 , x2 )]
(≈ c exp(−µ0 t))
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Thus µ2 ≥ µ0 + µ.
See also Kumar and Ramesh (2001).
3.3.2. Example: perfect epidemic
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Context
• many models: SIR, SEIR, Reed-Frost
• data: knowledge about removals
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• draw inferences about other aspects
– infection times
– infectivity parameter
– removal rate
– ...
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Perfect MCMC for Reed-Frost model
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• based on explicit formula for size of epidemic
• (O’Neill 2001; Clancy and O’Neill 2002)
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Question
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Can we apply perfect simulation to SIR epidemic conditioned on removals?
• IDEA
– SIR → compartment model
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– conditioning on removals is reminiscent of coverage conditioning for
Boolean model
• consider SIR-epidemic-valued process, considered as event-driven simulation, driven by birth-and-event processes producing infection and removal
events . . .
• then use reversibility, perpetuation, crossover to derive perfect simulation
algorithm for conditioned case.
• PRO: conditioning on removals appears to make structure “monotonic”
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• CON: Perpetuation and nonlinear infectivity rate combine to make funnelling difficult
• So: attach independent streams of infection proposals to each level k =
I + R to “localize” perpetuation
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• Speed-up by cycling through:
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– replace all unobserved marked removal proposals (thresholds)
– try to alter all observed removal marks (perpetuate . . . )
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– replace all infection proposals (perpetuate . . . )
(Reminiscent of H-vL-M method for perfect simulation of attractive areainteraction point process)
Consistent theoretical framework
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View Poisson processes in time as 0 : 1-valued processes over “infinitesimal pixels”:
infinitesimal
infinitesimal
P X
= 1 = λ × measure
pixel
pixel
• replace thresholds: work sequentially through “infinitesimal pixels”, reject
replacement if it violates conditioning
• replace removal marks: can incorporate into above step
• replace infection proposals: work sequentially through “infinitesimal pixels” first at level 1, then level 2, then level 3 . . . (TV scan); reject replacement
if it violates conditioning.
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Theorem 3.9 Resulting construction is monotonic (so enables CFTP)
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Proof:
Establish monotonicity for
• fastest path of infection (what is observed)
• slowest feasible path of infection (what is used for perpetuation)
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Implementation:
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It works!
• speed-up?
• scaling?
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Page 106 of 167
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• refine?
• generalize?
Applications
• “omnithermal” variant to produce estimate of likelihood surface
• generalization using crossover to produce full Bayesian analysis
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Page 107 of 167
Chapter 4
Deformable templates and
vascular trees
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4.1
4.2
4.3
MCMC and statistical image analysis . . . . . . . . . . . . 108
Deformable templates . . . . . . . . . . . . . . . . . . . . 110
Vascular systems . . . . . . . . . . . . . . . . . . . . . . . 111
4.1. MCMC and statistical image analysis
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How to use MCMC on images? Huge literature, beginning with Geman and Geman (1984).
Simple example: reconstruct white/black ±1 image: observation Xij = ±1, “true
image” Sij = ±1, X|S independent
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P [Xij = xij |Sij ]
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exp (φSij xij )
cosh(φ)
(i,j)∼(i0 ,j 0 )
Posterior for S|X proportional to

X
exp β
(i,j)∼(i0 ,j 0 )

Sij Si0 j 0 + φ
X
Xij Sij 
ij
(Ising model with inhomogeneous external magnetic field φX), and we can sample
from this using MCMC algorithms.
Easy CFTP for this example
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2
while prior for S is Ising model, probability mass function proportional to


X
exp β
Sij Si0 j 0 
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“Low-level image analysis”: contrast “high-level image analysis”.
For example, pixel array {(i, j)} representing small squares in a window [0, 1]2 ,
and replace S by a Boolean model (Baddeley and van Lieshout 1993).
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Page 109 of 167
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Figure 4.1: Scheme for high-level image analysis.
Van Lieshout and Baddeley (2001), Loizeaux and McKeague (2001) provide perfect examples.
4.2. Deformable templates
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Grenander and Miller (1994): to answer “how can knowledge in complex systems
be represented? use “deformable templates” (image algebras).
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• configuration space;
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• configurations (prior probability measure from interactions) → image;
• groups of similarities allow deformations;
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• template is representation within similarity equivalence class;
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• “jump-diffusion”: analogue of Metropolis-Hastings. Besag makes link in
discussion of Grenander and Miller (1994), suggesting discrete-time LangevinMetropolis-Hastings moves.
Expect to need long MCMC runs; Grenander and Miller (1994) use SIMD technology.
4.3. Vascular systems
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We illustrate using an application to vascular structure in medical images (Bhalerao
et al. 2001, Thönnes et al. 2002, 2003). Figures:
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Figure 4.2(a) cerebral vascular system (application: preplanning of surgical
procedures).
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Figure 4.2(b) retinal vascular system (application: diabetic retinopathy).
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(a) AVI movie of cerebral vascular
system.
(b) Retina vascular system:
4002 pixel image.
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Figure 4.2: Examples of vascular structures.
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a
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Plan: use random forests of planar (or spatial!) trees as configurations, since the
underlying vascular system is nearly (but not exactly!) tree-like.
Note: our tree edges are graph vertices in the Grenander-Miller formalism.
Vary tree number, topologies and geometries using Metropolis-Hastings moves.
Hence sample from posterior so as to partly reconstruct the vascular system.
Issues:
• figuring out effective reduction of the very large datasets which can be involved, how to link the model to the data,
• careful formulation of the prior stochastic geometry model,
• choice of the moves so as not to slow down the MCMC algorithm,
• and use of multiscale information.
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4.3.1. Data
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For computational feasibility, use multiresolution Fourier Transform (Wilson, Calway, and Pearson 1992) based on dyadic recursive partitioning using quadtree.
(Compare wavelets.)
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Page 113 of 167
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Figure 4.3: Multiresolution summary (planar case!).
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Partitioning: based on mean-squared error criterion and the extent to which summarizing feature is elongated: alternative approach uses Akaike’s information criterion (Li and Bhalerao 2003). Figure 4.4(a) shows a typical feature, which in
the planar case is a weighted bivariate Gaussian kernel, while Figure 4.4(b) shows
the result of summarizing a 400 × 400 pixel retinal image using 1000 weighted
bivariate Gaussian kernels.
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Page 114 of 167
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(a) A typical feature
for the MFT.
(b) 1000 kernels for a retina.
Figure 4.4: Decomposition into features using quadtree.
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This procedure can be viewed as a spatial variant of a treed model (Chipman et al.
2002), a variation on CART (Breiman et al. 1984) in which different models are
fitted to divisions of the data into disjoint subsets, with further divisions being
made recursively until satisfactory models are obtained.
Thus subsequent Bayesian inference can be viewed as conditional on the treed
model inference.
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Page 115 of 167
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Figure 4.5: Final result of Multiresolution Fourier Transform
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4.3.2. Likelihood
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Link summarized data (mixture of weighted bivariate Gaussian kernels k(x, y, e)
to configuration, assuming additive Gaussian noise.
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f (x, y) =
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XX
→
k(x, y, e) + noise
f˜(x, y)
=
τ ∈F e∈τ
X
k(x, y, e) .
e∈E
Split likelihood by rectangles Ri .
`(F|f˜)
=
N
X
`i (F|f˜) + constant
i=1
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`i (F|f˜)
= −
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≈
1
2σ 2
1
− 2
2σ
h
X
f˜(x, y) −
h
f˜(x, y) −
Ri
i2
k(x, y, e)
τ ∈F e∈τ
(x,y)∈Ri
ZZ
XX
XX
i2
k(x, y, e) hi (x, y) dx dy
τ ∈F e∈τ
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Extend from Ri to all R2 , enabling closed form computation.
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≈
−
1
2σ 2
ZZ
R2
h
f˜(x, y) −
XX
τ ∈F e∈τ
i2
k(x, y, e) gi (x, y) dx dy (4.1)
4.3.3. Prior
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Prior using AR(1) processes for location, width, extending along forests of trees
τ produced by subcritical binary Galton-Watson processes.
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lengthe
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=
α × lengthe0 + Gaussian innovation
for autoregressive parameter α ∈ (0, 1).
(a) sub-critical binary (b) AR(1) process on (c) each edge reprebranching process.
edges.
sented by a weighted
Gaussian kernel.
Figure 4.6: Components of the prior.
(4.2)
4.3.4. MCMC implementation
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We have now formulated likelihood and model. It remains to decide on a suitable
MCMC process.
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Figure 4.7: Refined scheme for high-level image analysis.
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Changes are local (hence slow). Compute MH accept-reject probabilities using
AR(1) location distributions and
Y
Y
P [τ ] =
P [ node has n children ]
(4.3)
n=0,1,2 v∈τ :v has n children
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Figure 4.8: Reversible jump MCMC proposals.
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Computations for topological moves. Suppose family-size distribution is
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(1 − p)2 , 2p(1 − p) , p2
Hastings ratio for adding a specified leaf ` ∈ ∂ext (τ ):
H(τ ; `)
=
P [τ ∪ {`}] /#(∂int (τ ∪ {`}) ∪ ∂ext (τ ∪ {`}))
.
P [τ ∪ {`}] /#(∂int (τ ) ∪ ∂ext (τ ))
Use induction on τ : P [τ ∪ {`}] = P [τ ] p(1 − p), . . .
H(τ ; `)
=
p(1 − p)
1 + #(V (τ )) + #(∂ext (τ ))
2 + #(V (τ )) + #(∂ext (τ ))+1
where +1 is added to the denominator if ` is added to a vertex which already has
one daughter.
It is clear that this Hastings ratio is always smaller than p(1 − p), and is approximately equal to p(1 − p) for large trees τ . Consequently removals of leaves will
always be accepted, and additions will usually be accepted, but there is a computational slowdown as the tree size increases.
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Geometric ergodicity is impossible: consider second eigenvalue λ2 of the Markov
kernel by Rayleigh-Ritz:
P P
2
x
y |φ(x) − φ(y)| p(x, y)π(y)
1 − λ2 = inf P P
2
φ6=0
x
y |φ(x) − φ(y)| π(x)π(y)
P
P
x∈S
y∈S c p(x, y)π(y)
≤ 2
π(S)
whenever π(S) < 1/2. Taking S to be the set of all trees of depth n or more, and
using the fact that such trees must have at least n sites, it follows
1 − λ2
≤
2
1
p(1 − p) n
→
0.
Experiments:
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MCMC for prior model
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First we implement only the moves relating to a single
tree, and only to that tree’s topology. We change the tree
just one leaf at a time. Convergence is clearly very slow!
moreover the animation clearly indicates the problem is
to do with incipient criticality phenomena.
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MCMC for prior model with more global changes
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Moves dealing with sub-trees will speed things up
greatly! We won’t use these as such – however we will
be working with a whole forest of such trees, and using
more “global” moves of join and divide.
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MCMC for occupancy conditioned prior model
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Conditioning on data will also help! Here is a crude parody of conditioning (pixels are black/white depending on
whether or not they intersect the planar tree).
4.3.5. Results
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Figure 4.9 presents snapshots (over about 50k moves) of a first try, using accept/reject probabilities for a Metropolis-Hastings algorithm based on the above
prior and likelihood.
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Figure 4.9: AVI movie of fixed-width MCMC.
We improve the algorithm by adding
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(1) a further AR(1) process to model the width of blood vessels;
(2) a Langevin-related component to the Metropolis-Hasting proposal distribution, biasing the proposal according to local modes of the posterior density
(compare Grenander and Miller 1994 and especially Besag’s remarks!);
(3) the possibility of proposing addition (and deletion) of features of length 2;
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(4) gradual refinement of feature proposals from coarse to fine levels.
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Figure 4.10: AVI movies of improved MCMC.
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4.3.6. Refining the starting state
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Page 125 of 167
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We improve matters by choosing a better starting state for the algorithm, using a
variant of Kruskal’s algorithm.
This generates a minimal spanning tree of a graph as follows: Let G = (V, E) be
a graph with edges weighted by cost. A spanning tree is a subgraph of G which is
(a) a tree and (b) spans all vertices. We wish to find a spanning tree τ of minimal
cost:
def kruskal (edges) :
edgelist ← edges.sort () # sort so cheapest edge is first
tree ← [ ]
for e ∈ edgelist :
if cyclefree (tree, e) : # check no cycle is created
tree.append (e)
return tree
(Key remark: at end, swapping in new edge e would create a circuit which must
be broken by removal of e0 . By construction cost(e) ≥ cost(e0 ).)
Actually we need a binary tree (so amend cyclefree accordingly!) and the cost
depends on the edges to which it is proposed to join e, but a heuristic modification
of Kruskal’s algorithm produces a binary forest which is a good candidate for a
starting position for our algorithm. Figure 4.11 shows a result; notice how now
the algorithm begins with many small trees, which are progressively dropped or
joined to other trees.
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Figure 4.11: AVI movie of MCMC using starting position provided by Kruskal’s
algorithm.
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Chapter 5
Multiresolution Ising models
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Contents
5.1
5.2
5.3
5.4
5.5
5.6
Generalized quad-trees
Random clusters . . . .
Percolation . . . . . . .
Comparison . . . . . . .
Simulation . . . . . . .
Future work . . . . . .
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145
147
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The vascular problem discussed above involves multi-resolution image analysis
described, for example, in Wilson and Li (2002). This represents the image on
different scales: the simplest example is that of a quad-tree (a tree graph in which
each node has four daughters and just one parent), for which each node carries
a random value 0 (coding the colour white) and 1 (coding the colour black), and
the random values interact in the manner of a Markov point process with their
neighbours. In the vascular case the 0/1 value is replaced by a line segment summarizing the linear structure which best explains the local data at the appropriate
resolution level, but for now we prefer to analyze the simpler 0/1 case.
Conventional multi-resolution models have interactions
only running up and down the tree; however in our case
we clearly need interactions between neighbours at the
same resolution level as well. This complicates the analysis!
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A major question is to determine the extent to which values at high resolution
levels are correlated with values at lower resolution levels (Kendall and Wilson
2003). This is related to much recent interest in the question of phase transitions
for Ising models on non-Euclidean random graphs. We shall show how geometric
arguments lead to a schematic phase portrait.
5.1. Generalized quad-trees
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Define Qd as graph whose vertices are cells of all dyadic tessellations of Rd , with
edges connecting each cell u to its 2d neighbours, and also its parent M(u) (covering cell in tessellation at next resolution down) and its 2d daughters (cells which
it covers in next resolution up).
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Case d = 1:
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Neighbours at same level also are connected.
Remark: No spatial symmetry!
However: Qd can be constructed using an iterated function system generated by
elements of a skew-product group R+ ⊗s Rd .
Further define:
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• Qd;r as subgraph of Qd at resolution
levels of r or higher;
• Qd (o) as subgraph formed by o and
all its descendants.
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• Remark: there are many graph-isomorphisms Su;v between Qd;r and Qd;s ,
with natural Zd -action;
• Remark: there are graph homomorphisms injecting Q(o) into itself, sending o to x ∈ Q(o) (semi-transitivity).
Simplistic analysis
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Define Jλ to be strength of neighbour interaction, Jτ to be strength of parent interaction. If Sx = ±1 then probability of configuration is proportional to exp(−H)
where
X
1
H=−
Jhx,yi (Sx Sy − 1) ,
(5.1)
2
hx,yi∈E(G)
for Jhx,yi = Jλ , Jτ as appropriate.
If Jλ = 0 then the free Ising model on Qd (o) is a branching process (Preston
1977; Spitzer 1975); if Jτ = 0 then the Ising model on Qd (o) decomposes into
sequence of d-dimensional classical (finite) Ising models. So we know there is
a phase change at (Jλ , Jτ ) = (0, ln(5/3))
(results from branching processes),
√
and expect one at (Jλ , Jτ ) = (ln(1 + 2), 0+) (results from 2-dimensional Ising
model).
But is this all that there is to say?
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There has been a lot of work on percolation and Ising models on non-Euclidean
graphs (Grimmett and Newman 1990; Newman and Wu 1990; Benjamini and
Schramm 1996; Benjamini et al. 1999; Häggström and Peres 1999; Häggström
et al. 1999; Lyons 2000; Häggström et al. 2002). Mostly this concerns graphs
admitting substantial symmetry (unimodular, quasi-transitive), so that one can use
Häggström’s remarkable Mass Transport method:
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Theorem 5.1 Let Γ ⊂ Aut(G) be unimodular and quasi-transitive. Given
Rmass transport m(x, y, ω) ≥ 0 “diagonally invariant”, set M (x, y) =
m(x, y, ω) dω. Then the expected total transport out of any x equals
Ω
the expected total transport into x:
X
X
M (x, y) =
M (y, x) .
y∈Γx
y∈Γx
Sadly our graphs do not have quasi-transitive symmetry :-(.
Furthermore the literature mostly deals with phase transition results “for sufficiently large values of the parameter”. This is an invaluable guide, but image
analysis needs actual estimates!
5.2. Random clusters
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A similar problem, concerning Ising models on products of trees with Euclidean
lattices, is treated by Newman and Wu (1990). We follow them by exploiting the
celebrated Fortuin-Kasteleyn random cluster representation (Fortuin and Kasteleyn 1972; Fortuin 1972a; Fortuin 1972b):
The Ising model is the marginal site process at q = 2 of a site/bond process derived from a dependent bond percolation model with configuration probability Pq,p
proportional to
qC ×
Y
(phx,yi )bhx,yi × (1 − phx,yi )1−bhx,yi .
hx,yi∈E(G)
(where bhx,yi indicates whether or not hx, yi is closed, and C is the number of
connected clusters of vertices). Site spins are chosen to be the same in each cluster
independently of other clusters with equal probabilities for ±1. We find
phx,yi = 1 − exp(−Jhx,yi ) ;
(5.2)
Argue for general q (Potts model): spot the Gibbs’ sampler!
(I) given bonds, chose connected component colours independently and uniformly
from 1, . . . , q;
(II) given colouring, open bonds hx, yi independently, probability phx,yi .
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Exercise 5.2 Check step (II) gives correct conditional probabilities!
Now compute conditional probability that site o gets colour 1 given neighbours
split into components C1 , . . . , Cr with colour 1, and other neighbours in set C0 .
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Exercise 5.3 Use inclusion-exclusion to show it is this:
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!
Y
1
P [colour(o) = 1|neighbours] ∝
×q
(1 − phx,yi ) +
q
y


!
Y
Y


+
(1 − phx,yi )
1−
1 − phx,zi
y∈C0
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z∈C1 ∪...∪Cr
=
Y
(1 − phx,yi ) .
y:colour(y)6=1
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Deduce interaction strength is Jhx,yi = − log(1 − phx,yi ) as stated in Equation
(5.2) above.
FK-comparison inequalities
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If q ≥ 1 and A is an increasing event then
Pq,p (A) ≤ P1,p (A)
Pq,p (A) ≥ P1,p0 (A)
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where
p0hx,yi =
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(5.3)
(5.4)
phx,yi
phx,yi
=
.
phx,yi + (1 − phx,yi )q
q − (q − 1)phx,yi
Since P1,p is bond percolation (bonds open or not independently of each other),
we can find out about phase transitions by studying independent bond percolation.
5.3. Percolation
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Independent bond percolation on products of trees with Euclidean lattices have
been studied by Grimmett and Newman (1990), and these results were used in the
Newman and Wu work on the Ising model. So we can make good progress by
studying independent bond percolation on Qd , using pτ for parental bonds, pλ for
neighbour bonds.
Theorem 5.4 There is almost surely no infinite cluster in Qd;0 (and consequently in Qd (o)) if
q
2d τ Xλ 1 + 1 − Xλ−1
< 1,
where Xλ is the mean size of the percolation cluster at the origin for λpercolation in Zd .
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Modelled on Grimmett and Newman (1990, §3 and §5). Get
from matrix spectral asymptotics.
1+
q
1 − Xλ−1
1
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λ
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?
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Infinite clusters here
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0
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No
infinite
clusters
here
0
(may or may not be unique)
2
−d
1
τ
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The story so far: small λ, small to moderate τ .
Case of small τ
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Need d = 2 for mathematical convenience. Use Borel-Cantelli argument and
planar duality to show, for supercritical λ > 1/2 (that is, supercritical with respect
to planar bond percolation!), all but finitely many of the resolution layers Ln =
[1, 2n ] × [1, 2n ] of Q2 (o) have just one large cluster each of diameter larger than
constant × n.
Hence . . .
Theorem 5.5 When λ > 1/2 and τ is positive there is one and only one
infinite cluster in Q2 (o).
1
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Just one unique infinite cluster here
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λ
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1/2
?
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0
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No
infinite
clusters
here
0
Infinite clusters here
(may or may not be unique)
1/4
τ
The story so far: adds small τ for case d = 2.
1
Uniqueness of infinite clusters
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The Grimmett and Newman (1990) work was remarkable in pointing out that as τ
increases so there is a further phase change, from many to just one infinite cluster
for λ > 0. The work of Grimmett and Newman carries
√ through for Qd (o). However the relevant bound is improved by a factor of 2 if we take into account the
hyperbolic structure of Qd (o)!
Theorem 5.6 If τ < 2−(d−1)/2 and λ > 0 then there cannot be just one
infinite cluster in Qd;0 .
Method: sum weights of “up-paths” in Qd;0 starting,
ending at level 0. For fixed s and start point there are
infinitely many such up-paths containing s λ-bonds;
but no more than (1 + 2d + 2d )s which cannot be reduced by “shrinking” excursions. Hence control the
mean number of open up-paths stretching more than a
given distance at level 0.
Contribution to upper bound on second phase transition:
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p
Theorem 5.7 If τ > 2/3 then the infinite cluster of Q2:0 is almost surely
unique for all positive λ.
Method: prune bonds, branching processes, 2-dim comparison . . .
1
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Just one unique infinite cluster here
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λ
1/2
I
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0
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Infinite clusters here
?
0
(may or may not be unique)
No
infinite
clusters
here
Many infinite clusters
1
1/4
τ
0.707
0.816
The story so far: includes uniqueness transition for case d = 2.
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5.4. Comparison
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We need to apply the Fortuin-Kasteleyn comparison inequalities (5.3) and (5.4).
The event “just one infinite cluster” is not increasing, so we need more. Newman
and Wu (1990) show it suffices to establish a finite island property for the site
percolation derived under adjacency when all infinite clusters are removed. Thus:
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1
λ
finite islands
1/2
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0
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0
1
1/4
τ
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0.707
0.816
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Comparison arguments then show the following schematic phase diagram for the
Ising model on Q2 (o):
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All nodes substantially
Free boundary condition
correlated with each
is mixture of the two extreme
other in case of free
Gibbs states (spin 1 at
boundary conditions
boundary, spin −1 at boundary)
1.099
0.693
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Unique
J
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λ
Gibbs
state
J
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τ
Root influenced
by wired boundary
0.288
0.511
0.762
1.228
1.190
2.292
5.5. Simulation
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Approximate simulations confirm the general story (compare our original animation, which had Jτ = 2, Jλ = 1/4):
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http://www.dcs.warwick.ac.uk/˜rgw/sira/sim.html
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(1) Only 200 resolution levels;
(2) At each level, 1000 sweeps in scan order;
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(3) At each level, simulate square sub-region of 128 × 128 pixels conditioned
by mother 64 × 64 pixel region;
(4) Impose periodic boundary conditions on 128 × 128 square region;
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(5) At the coarsest resolution, all pixels set white. At subsequent resolutions,
‘all black’ initial state.
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(a) Jλ = 1, Jτ = 0.5
(b) Jλ = 1, Jτ = 1
(c) Jλ = 1, Jτ = 2
(d) Jλ = 0.5, Jτ = 0.5
(e) Jλ = 0.5, Jτ = 1
(f) Jλ = 0.5, Jτ = 2
(g) Jλ = 0.25, Jτ = 0.5
(h) Jλ = 0.25, Jτ = 1
(i) Jλ = 0.25, Jτ = 2
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5.6. Future work
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This is about the free Ising model on Q2 (o). Image analysis more naturally concerns the case of prescribed boundary conditions (say, image at finest resolution
level . . . ).
Question: will boundary conditions at “infinite fineness” propagate back to finite
resolution?
Series and Sinaı̆ (1990) show answer is yes for analogous problem on hyperbolic
disk (2-dim, all bond probabilities the same).
Gielis and Grimmett (2002) point out (eg, in Z3 case) these boundary conditions
translate to a conditioning for random cluster model, and investigate using large
deviations.
Project: do same for Q2 (o) . . . and get quantitative bounds?
Project: generalize from Qd to graphs with appropriate hyperbolic structure.
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Appendix A
Notes on the proofs in Chapter
5
A.1. Notes on proof of Theorem 5.4
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Mean size of cluster at o bounded above by
∞
X
X
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n−T (t)
Xλ τ n (Xλ − 1)T (t) Xλ
n=0 t:|t|=n
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≤
JJ
Xλ (τ Xλ )n
n=0
II
≤
J
∞
X
I
∞
X
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(1 − Xλ−1 )T (t)
t:|t|=n
Xλ (2d τ Xλ )n
n=0
≈
X
X
(1 − Xλ−1 )T (j)
j:|j|=n
∞
X
n
q
Xλ (2d τ Xλ )n 1 + 1 − Xλ−1
.
n=0
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For last step, use spectral analysis of matrix representation
n X
1
1
1
−1 T (j)
1 1
(1 − Xλ )
=
.
1
1 − Xλ−1 1
j:|j|=n
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A.2. Notes on proof of Theorem 5.5
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Uniqueness: For negative exponent ξ(1 − λ) of dual connectivity function, set
`n
=
(n log 4 + (2 + ) log n) ξ(1 − λ) .
More than one “`n -large” cluster in Ln forces existence of open path in dual lattice
longer than `n . Now use Borel-Cantelli . . . .
On the other hand super-criticality will mean some distant points in Ln are interconnected.
Existence: consider 4n−[n/2] points in Ln−1 and specified daughters in Ln . Study
probability that
(a) parent percolates more than `n−1 ,
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(b) parent and child are connected,
(c) child percolates more than `n .
Now use Borel-Cantelli again . . . .
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A.3. Notes on proof of Theorem 5.6
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Two relevant lemmata:
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Lemma A.1 Consider u ∈ Ls+1 ⊂ Qd and v = M(u) ∈ Ls ⊂ Qd . There
are exactly 2d solutions in Ls+1 of
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M(x)
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=
Su;v (x) .
One is x = u. Others are the remaining 2d − 1 vertices y such that closure of
cell representing y intersects vertex shared by closures of cells representing
u and M(u). Finally, if x ∈ Ls+1 does not solve M(x) = Su;v (x) then
kSu;v (x) − Su;v (u)ks,∞
>
kM(x) − M(u)ks,∞ .
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Lemma A.2 Given distinct v and y in the same resolution level. Count pairs
of vertices u, x in the resolution level one step higher, such that
(a) M(u) = v; (b) M(x) = y; (c) Su;v (x) = y.
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There are at most 2d−1 such vertices.
A.4. Notes on proof of Theorem 5.7
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Prune! Then a direct connection is certainly established across the boundary between the cells corresponding to two neighbouring vertices u, v in L0 if
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(a) the τ -bond leading from u to the relevant boundary is open;
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(b) a τ -branching process (formed by using τ -bonds mirrored across the boundary) survives indefinitely, where this branching process has family-size distribution Binomial(2, τ 2 );
(c) the τ -bond leading from v to the relevant boundary is open.
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Then there are infinitely many
chances of making a connection
across the cell boundary.
A.5. Notes on proof of infinite island property
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Notion of “cone boundary” ∂c (S) of finite subset S of vertices: collection of
daughters v of S such that Qd (v) ∩ S = ∅.
Use induction on S, building it layer Ln on layer Ln−1 to obtain an isoperimetric
bound: #(∂c (S)) ≥ (2d − 1)#(S). Hence deduce
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P [S in island at u]
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≤
(2d −1)n
(1 − pτ (1 − η))
where #(S) = n and η = P [u not in infinite cluster of Qd (u)].
Upper bound on number N (n) of self-avoiding paths S of length n beginning at
u0 :
N (n) ≤ (1 + 2d + 2d )(2d + 2d )n .
Hence upper bound on the mean size of the island:
∞
X
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n(1−2−d )
(1 + 2d + 2d )(2d + 2d )n ηbr
,
n=0
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where ηbr is extinction probability for branching process based on Binomial(2d , pτ )
family distribution.
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