Exact simulation of the sample paths of a diffusion
with a finite entrance boundary
Paul A. Jenkins
Departments of Statistics
University of Warwick
Dynstoch, University of Warwick
11 Sep 2014
Joint work with Dario Spanò
0 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline
1
Introduction
2
Overview of the exact algorithm
3
Bessel-EA
4
Wright-Fisher diffusion
5
Summary
0 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Diffusion model
The time-evolution of a genetic variant, or allele, is well
approximated by a diffusion process on the interval [0, 1].
1
Allele
frequency
0
Time
2 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Diffusion model
The time-evolution of a genetic variant, or allele, is well
approximated by a diffusion process on the interval [0, 1].
1
Allele
frequency
0
Time
Wright-Fisher SDE
dXt = µθ (Xt )dt +
p
Xt (1 − Xt )dWt ,
X0 = x,
t ≥ 0.
The infinitesimal drift, µθ (x), encapsulates directional forces such as
natural selection, migration, mutation, . . .
2 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Population genetic Motivation I: Demographic inference
Given a sample of DNA sequences obtained in the present-day,
what can we infer about the demographic history of the population?
Example (Gutenkunst et al., 2009)
Expansion out-of-Africa
Demography from Multid
Settlement of the New World
3 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Population genetic Motivation II: Time-series analysis of selection
Given a sample of genetic data obtained over several generations,
what can we infer about the strength of natural selection?
Example (Biston betulaeria; Mathieson & McVean, 2013)
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from 1953 to 2002
the y-axis in each de
are observations, c
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4 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Inference from diffusion processes
Like many interesting diffusions, the transition function of the
Wright-Fisher diffusion is unknown.
5 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Inference from diffusion processes
Like many interesting diffusions, the transition function of the
Wright-Fisher diffusion is unknown.
Inference typically proceeds by
1
Model-discretization such as an Euler approximation:
Xt+dt | (Xt = z) ∼ N (µθ (z)dt, σ 2 (z)dt),
2
. . . followed by (sequential) Monte Carlo simulation (or numerical
solution of Kolmogorov PDEs, or spectral expansions, . . . )
5 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Inference from diffusion processes
Like many interesting diffusions, the transition function of the
Wright-Fisher diffusion is unknown.
Inference typically proceeds by
1
Model-discretization such as an Euler approximation:
Xt+dt | (Xt = z) ∼ N (µθ (z)dt, σ 2 (z)dt),
2
. . . followed by (sequential) Monte Carlo simulation (or numerical
solution of Kolmogorov PDEs, or spectral expansions, . . . )
But—discretization introduces a bias we would like to remove.
5 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Inference from diffusion processes
Like many interesting diffusions, the transition function of the
Wright-Fisher diffusion is unknown.
Inference typically proceeds by
1
Model-discretization such as an Euler approximation:
Xt+dt | (Xt = z) ∼ N (µθ (z)dt, σ 2 (z)dt),
2
. . . followed by (sequential) Monte Carlo simulation (or numerical
solution of Kolmogorov PDEs, or spectral expansions, . . . )
But—discretization introduces a bias we would like to remove.
5 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Inference from diffusion processes
Like many interesting diffusions, the transition function of the
Wright-Fisher diffusion is unknown.
Inference typically proceeds by
1
Model-discretization such as an Euler approximation:
Xt+dt | (Xt = z) ∼ N (µθ (z)dt, σ 2 (z)dt),
2
. . . followed by (sequential) Monte Carlo simulation (or numerical
solution of Kolmogorov PDEs, or spectral expansions, . . . )
But—discretization introduces a bias we would like to remove.
Three sentence summary
There exist so-called exact algorithms for simulating diffusions
without discretization error, even if the transition density is
unknown.
They can perform poorly when there are entrance boundaries.
I will outline how to fix these problems.
5 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline
1
Introduction
2
Overview of the exact algorithm
3
Bessel-EA
4
Wright-Fisher diffusion
5
Summary
8 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA)—one-dimensional bridge version
Goal: return exact bridge samples from the one-dimensional
diffusion X = (Xt : t ≥ 0) on R satisfying
dXt = µθ (Xt )dt + σ(Xt )dWt ,
X0 = x,
0 ≤ t ≤ T.
9 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA)—one-dimensional bridge version
Goal: return exact bridge samples from the one-dimensional
diffusion X = (Xt : t ≥ 0) on R satisfying
dXt = µθ (Xt )dt + σ(Xt )dWt ,
1
X0 = x,
0 ≤ t ≤ T.
Reduce the problem to unit diffusion coefficient via the Lamperti
transform Xt 7→ Yt :
Z Xt
1
Yt :=
du,
σ(u)
so now we work with
dYt = αθ (Yt )dt + dWt ,
Y0 = y ,
0 ≤ t ≤ T.
9 / 38
Introduction
Exact algorithm
Bessel-EA
dYt = αθ (Yt )dt + dBt ,
Wright-Fisher diffusion
Y0 = y ,
Summary
0 ≤ t ≤ T.
Exact algorithm (EA)
2
Now we can consider a rejection algorithm using Brownian
bridge paths as candidates.
If Qy is the target law (of Y ) and Wy is the law of a Brownian
motion then we need
dQy
(Y )
dWy
to provide the rejection probability
10 / 38
Introduction
Exact algorithm
Bessel-EA
dYt = αθ (Yt )dt + dBt ,
Wright-Fisher diffusion
Y0 = y ,
Summary
0 ≤ t ≤ T.
Exact algorithm (EA)
2
Now we can consider a rejection algorithm using Brownian
bridge paths as candidates.
If Qy is the target law (of Y ) and Wy is the law of a Brownian
motion then we need
(Z
)
Z
T
dQy
1 T 2
αθ (Yt )dYt −
α (Yt )dt
(Y ) = exp
2 0 θ
dWy
0
to provide the rejection probability, by the Girsanov theorem.
10 / 38
Introduction
Exact algorithm
Bessel-EA
dYt = αθ (Yt )dt + dBt ,
Wright-Fisher diffusion
Y0 = y ,
Summary
0 ≤ t ≤ T.
Exact algorithm (EA)
2
Now we can consider a rejection algorithm using Brownian
bridge paths as candidates.
If Qy is the target law (of Y ) and Wy is the law of a Brownian
motion then we need
(Z
)
Z
T
dQy
1 T 2
αθ (Yt )dYt −
α (Yt )dt
(Y ) = exp
2 0 θ
dWy
0
to provide the rejection probability, by the Girsanov theorem.
Such a rejection algorithm is impossible: it requires simulation
of complete (infinite-dimensional) Brownian sample paths!
10 / 38
Introduction
Exact algorithm
Bessel-EA
dYt = αθ (Yt )dt + dBt ,
Wright-Fisher diffusion
Y0 = y ,
Summary
0 ≤ t ≤ T.
Exact algorithm (EA)
3
Key observation: The Radon-Nikodým derivative can be put in
the form
( Z
)
T
dQy
(Y ) ∝ exp −
φ(Ys )ds ≤ 1,
dWy
0
where φ(·) := 12 [αθ2 (·) + αθ0 (·)] + C.
11 / 38
Introduction
Exact algorithm
Bessel-EA
dYt = αθ (Yt )dt + dBt ,
Wright-Fisher diffusion
Y0 = y ,
Summary
0 ≤ t ≤ T.
Exact algorithm (EA)
3
Key observation: The Radon-Nikodým derivative can be put in
the form
( Z
)
T
dQy
(Y ) ∝ exp −
φ(Ys )ds ≤ 1,
dWy
0
where φ(·) := 12 [αθ2 (·) + αθ0 (·)] + C.
Assume we can arrange for φ ≥ 0. Then the right-hand side is
the probability that a Poisson point process of unit rate on
[0, T ] × [0, ∞) has no points under the graph of t 7→ φ(Ys ).
11 / 38
Introduction
Exact algorithm
dQy
dWy (Y )
Bessel-EA
Summary
n R
o
T
∝ exp − 0 φ(Ys )ds ≤ 1.
Exact algorithm (EA)
4
Wright-Fisher diffusion
φ(Yt)
A proposed Brownian
path should be
rejected if a simulated
Poisson point process
has any points under
its graph.
0
0
t
T
12 / 38
Introduction
Exact algorithm
dQy
dWy (Y )
Bessel-EA
Summary
n R
o
T
∝ exp − 0 φ(Ys )ds ≤ 1.
Exact algorithm (EA)
4
Wright-Fisher diffusion
φ(Yt)
A proposed Brownian
path should be
rejected if a simulated
Poisson point process
has any points under
its graph.
0
0
t
T
12 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Brownian bridge (Yt )0≤t≤T from Y0 to YT .
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Brownian bridge (Yt )0≤t≤T from Y0 to YT .
2
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Brownian bridge (Yt )0≤t≤T from Y0 to YT .
2
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
3
Accept if all points are in the epigraph of t 7→ φ(Yt ), otherwise
return to 1.
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Brownian bridge (Yt )0≤t≤T from Y0 to YT .
2
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
3
Accept if all points are in the epigraph of t 7→ φ(Yt ), otherwise
return to 1.
Problems
1
We still need an infinite-dimensional Brownian path.
2
The Poisson point process has unbounded intensity.
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Brownian bridge (Yt )0≤t≤T from Y0 to YT .
2
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
3
Accept if all points are in the epigraph of t 7→ φ(Yt ), otherwise
return to 1.
Problems
1
We still need an infinite-dimensional Brownian path.
2
The Poisson point process has unbounded intensity.
Solutions
1
Exploit retrospective sampling; switch the order of simulation!
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Brownian bridge (Yt )0≤t≤T from Y0 to YT .
2
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
3
Accept if all points are in the epigraph of t 7→ φ(Yt ), otherwise
return to 1.
Problems
1
We still need an infinite-dimensional Brownian path.
2
The Poisson point process has unbounded intensity.
Solutions
1
Exploit retrospective sampling; switch the order of simulation!
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
2
Simulate the Brownian bridge at the times of the Poisson points.
3
Accept if all points are in the epigraph of t 7→ φ(Yt ), otherwise
return to 1.
Problems
1
We still need an infinite-dimensional Brownian path.
2
The Poisson point process has unbounded intensity.
Solutions
1
Exploit retrospective sampling; switch the order of simulation!
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Retrospective sampling
Exact algorithm (EA) for simulating a bridge from Y0 to YT
1
Simulate a Poisson point process of unit rate on [0, T ] × [0, ∞).
2
Simulate the Brownian bridge at the times of the Poisson points.
3
Accept if all points are in the epigraph of t 7→ φ(Yt ), otherwise
return to 1.
Problems
1
We still need an infinite-dimensional Brownian path.
2
The Poisson point process has unbounded intensity.
Solutions
1
Exploit retrospective sampling; switch the order of simulation!
2
Assume φ is bounded, φ ≤ K (for now), and use Poisson
thinning (“EA1”).
13 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA1); Beskos & Roberts (2005)
K
φ(Yt)
0
0
t
T
14 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA1); Beskos & Roberts (2005)
1
Simulate a Poisson point process on [0, T ] × [0, K ].
K
φ(Yt)
0
0
t
T
14 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA1); Beskos & Roberts (2005)
1
Simulate a Poisson point process on [0, T ] × [0, K ].
2
Simulate the Brownian bridge at the times of the Poisson points.
K
φ(Yt)
0
0
t
T
14 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA1); Beskos & Roberts (2005)
1
Simulate a Poisson point process on [0, T ] × [0, K ].
2
Simulate the Brownian bridge at the times of the Poisson points.
3
If any of the former are beneath any of the latter, return to 1.
K
φ(Yt)
0
0
t
T
14 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA)
Output of the algorithm is a set of skeleton points of the bridge.
Any further points can be filled in by further draws from the
Brownian bridge—no further reference to the target law, Qy , is
necessary!
15 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA)
Output of the algorithm is a set of skeleton points of the bridge.
Any further points can be filled in by further draws from the
Brownian bridge—no further reference to the target law, Qy , is
necessary!
φ(·) =
1 2
[α (·) + αθ0 (·)] + C.
2 θ
This function φ is important.
The assumption φ ≤ K is restrictive, but it can in fact be relaxed
(“EA2”, Beskos et al., 2006).
15 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA)
Output of the algorithm is a set of skeleton points of the bridge.
Any further points can be filled in by further draws from the
Brownian bridge—no further reference to the target law, Qy , is
necessary!
φ(·) =
1 2
[α (·) + αθ0 (·)] + C.
2 θ
This function φ is important.
The assumption φ ≤ K is restrictive, but it can in fact be relaxed
(“EA2”, Beskos et al., 2006).
15 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Exact algorithm (EA)
Output of the algorithm is a set of skeleton points of the bridge.
Any further points can be filled in by further draws from the
Brownian bridge—no further reference to the target law, Qy , is
necessary!
φ(·) =
1 2
[α (·) + αθ0 (·)] + C.
2 θ
This function φ is important.
The assumption φ ≤ K is restrictive, but it can in fact be relaxed
(“EA2”, Beskos et al., 2006).
There have been many further refinements to this algorithm
(multidimensions, jumps, killing, reflection, . . . ):
Beskos et al. (2006, 2008, 2012), Casella & Roberts (2008, 2011),
Chen & Huang (2013), Étoré & Martinez (2013), Giesecke & Smelov
(2013), Gonçalves & Roberts (2013), Mousavi & Glynn (2013),
Blanchet & Murthy (2014), Pollock et al. (2014).
15 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Efficiency
The exact algorithm will be less efficient wherever φ(Xt ) is very
large—unavoidable when the diffusion travels through a region
where the drift (or its derivative) is very large.
18 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Efficiency
The exact algorithm will be less efficient wherever φ(Xt ) is very
large—unavoidable when the diffusion travels through a region
where the drift (or its derivative) is very large.
Example: Entrance boundary at 0
“A diffusion at x will almost
surely not hit 0 before hitting
any b > x. A diffusion started
at 0 will enter (0, ∞) in finite
time.”
If σ 2 (x) = 1, then φ explodes
at the boundary.
18 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Efficiency
The exact algorithm will be less efficient wherever φ(Xt ) is very
large—unavoidable when the diffusion travels through a region
where the drift (or its derivative) is very large.
Example: Entrance boundary at 0
“A diffusion at x will almost
surely not hit 0 before hitting
any b > x. A diffusion started
at 0 will enter (0, ∞) in finite
time.”
If σ 2 (x) = 1, then φ explodes
at the boundary.
φ(u)
0
0
1
2
u
3
4
5
18 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Efficiency
The exact algorithm will be less efficient wherever φ(Xt ) is very
large—unavoidable when the diffusion travels through a region
where the drift (or its derivative) is very large.
Example: Entrance boundary at 0
“A diffusion at x will almost
surely not hit 0 before hitting
any b > x. A diffusion started
at 0 will enter (0, ∞) in finite
time.”
If σ 2 (x) = 1, then φ explodes
at the boundary.
φ(u)
0
0
1
2
u
3
4
5
18 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Large φ is a symptom of a poor likelihood ratio, i.e. Brownian
motion is a poor mimic of the target diffusion.
19 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Large φ is a symptom of a poor likelihood ratio, i.e. Brownian
motion is a poor mimic of the target diffusion.
Idea: Replace Brownian motion with a different candidate
process—one with an entrance boundary.
19 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Large φ is a symptom of a poor likelihood ratio, i.e. Brownian
motion is a poor mimic of the target diffusion.
Idea: Replace Brownian motion with a different candidate
process—one with an entrance boundary.
But: the exact algorithms rely heavily on our knowledge about
Brownian bridges:
The distribution of bridge coordinates.
The distribution of the minimum, mT , and its time, tm .
The distribution of bridge coordinates conditioned on (mT , tm ).
The ability to sample from these distributions exactly.
19 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Large φ is a symptom of a poor likelihood ratio, i.e. Brownian
motion is a poor mimic of the target diffusion.
Idea: Replace Brownian motion with a different candidate
process—one with an entrance boundary.
But: the exact algorithms rely heavily on our knowledge about
Brownian bridges:
The distribution of bridge coordinates.
The distribution of the minimum, mT , and its time, tm .
The distribution of bridge coordinates conditioned on (mT , tm ).
The ability to sample from these distributions exactly.
19 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Large φ is a symptom of a poor likelihood ratio, i.e. Brownian
motion is a poor mimic of the target diffusion.
Idea: Replace Brownian motion with a different candidate
process—one with an entrance boundary.
But: the exact algorithms rely heavily on our knowledge about
Brownian bridges:
The distribution of bridge coordinates.
The distribution of the minimum, mT , and its time, tm .
The distribution of bridge coordinates conditioned on (mT , tm ).
The ability to sample from these distributions exactly.
Question. Does there exist a diffusion:
with infinitesimal variance equal to 1,
with an entrance boundary, and such that
the finite-dimensional distributions of its bridges are known, and
which can be simulated exactly, and
(bonus) whose extrema are well characterized?
19 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline
1
Introduction
2
Overview of the exact algorithm
3
Bessel-EA
4
Wright-Fisher diffusion
5
Summary
19 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Bessel process
Infinitesimal
variance 1?
X Drift β(y ) = (δ − 1)/(2y),
variance σ 2 (y) = 1.
Entrance
boundary?
Finitedimensional
distributions?
Exact
simulation?
Distributions
of extrema?
20 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Bessel process
Infinitesimal
variance 1?
X Drift β(y ) = (δ − 1)/(2y),
variance σ 2 (y) = 1.
Entrance
boundary?
X Zero is an entrance boundary when δ ≥ 2.
Finitedimensional
distributions?
Exact
simulation?
Distributions
of extrema?
20 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Bessel process
Infinitesimal
variance 1?
X Drift β(y ) = (δ − 1)/(2y),
variance σ 2 (y) = 1.
Entrance
boundary?
X Zero is an entrance boundary when δ ≥ 2.
Finitedimensional
distributions?
X p(y,0)→(z,T ) (x; t) =
−
T
2t(T −t) e
z(T −t)
+ 2t(TxT−t) + 2T (Tyt−t)
2tT
√
Iν (
√
xy
xz
)
)Iν (
t
(T −t)2
√
yz
Iν ( 2 )
T
,
where ν = 2(δ + 1), is the transition density
of the (squared) Bessel bridge.
Exact
simulation?
Distributions
of extrema?
20 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Bessel process
Infinitesimal
variance 1?
X Drift β(y ) = (δ − 1)/(2y),
variance σ 2 (y) = 1.
Entrance
boundary?
X Zero is an entrance boundary when δ ≥ 2.
Finitedimensional
distributions?
X p(y,0)→(z,T ) (x; t) =
−
T
2t(T −t) e
z(T −t)
+ 2t(TxT−t) + 2T (Tyt−t)
2tT
√
Iν (
√
xy
xz
)
)Iν (
t
(T −t)2
√
yz
Iν ( 2 )
T
,
where ν = 2(δ + 1), is the transition density
of the (squared) Bessel bridge.
Exact
simulation?
X δ ∈ Z≥0 : radial part of a δ-dimensional
Brownian motion.
δ ∈ R≥0 : See Makarov & Glew (2010).
Distributions
of extrema?
20 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Bessel process
Infinitesimal
variance 1?
X Drift β(y ) = (δ − 1)/(2y),
variance σ 2 (y) = 1.
Entrance
boundary?
X Zero is an entrance boundary when δ ≥ 2.
Finitedimensional
distributions?
X p(y,0)→(z,T ) (x; t) =
−
T
2t(T −t) e
z(T −t)
+ 2t(TxT−t) + 2T (Tyt−t)
2tT
√
Iν (
√
xy
xz
)
)Iν (
t
(T −t)2
√
yz
Iν ( 2 )
T
,
where ν = 2(δ + 1), is the transition density
of the (squared) Bessel bridge.
Exact
simulation?
Distributions
of extrema?
X δ ∈ Z≥0 : radial part of a δ-dimensional
Brownian motion.
δ ∈ R≥0 : See Makarov & Glew (2010).
(X) Partly.
20 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Bessel-EA
Exact simulation from a diffusion with law Qy using the Bessel
process (law Bδy Qy ) is possible by the following:
Theorem.
Under regularity conditions (similar to EA), Qy is the marginal
distribution of Y when
n
h
io
e ) ,
(Y , Φ) ∼ (Bδy ⊗ L) Φ ⊆ epigraph φ(Y
where L is the law of a Poisson point process Φ of unit rate on
[0, T ] × [0, ∞),and
1
e
φ(u)
:= [αθ2 (u) − β 2 (u) + αθ0 (u) − β 0 (u)] + C.
2
22 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline of proof.
Similar to the Brownian case: regularity conditions permit a
Girsanov transformation and rearrangement
so)that
( Z
T
dQy
e t )dt ≤ 1,
φ(Y
(Y ) ∝ exp −
dBδy
0
provides the rejection probability for sampling from the conditional
law
n
io
h
e ) .
(Bδy ⊗ L) Φ ⊆ epigraph φ(Y
23 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline of proof.
Similar to the Brownian case: regularity conditions permit a
Girsanov transformation and rearrangement
so)that
( Z
T
dQy
e t )dt ≤ 1,
φ(Y
(Y ) ∝ exp −
dBδy
0
provides the rejection probability for sampling from the conditional
law
n
io
h
e ) .
(Bδy ⊗ L) Φ ⊆ epigraph φ(Y
So what?
We have just replaced one candidate process for another, the
only substantial difference the appearance of
1
e
φ(u)
:= [αθ2 (u) − β 2 (u) + αθ0 (u) − β 0 (u)] + C.
2
instead of
1
φ(u) := [αθ2 (u) + αθ0 (u)] + C.
2
23 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
A diffusion (Xt )0≤t≤T with drift and diffusion coefficients
µ(x) = κx,
σ 2 (x) = x + ωx 2 ,
commenced from X0 = x0 and grown to XT = xT .
24 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
A diffusion (Xt )0≤t≤T with drift and diffusion coefficients
µ(x) = κx,
σ 2 (x) = x + ωx 2 ,
commenced from X0 = x0 and grown to XT = xT .
The population has not died out, so we can condition the
process on non-absorption at 0.
24 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
A diffusion (Xt )0≤t≤T with drift and diffusion coefficients
σ 2 (x) = x + ωx 2 ,
µ(x) = κx,
commenced from X0 = x0 and grown to XT = xT .
The population has not died out, so we can condition the
process on non-absorption at 0.
Conditioning and Lamperti transforming leads to new drift
κ
α(y ) = √ tanh
ω
√
√
√ ωy
ω
−
coth ωy
2
2
tanh
+
h√
ωy
2
i
ω − 2κ
h√ i,
√
ωy
−2
w 1 − cosh 4κ
ω
2
with an entrance boundary at 0.
24 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
What does the drift look like at the boundary?
3
α(y ) =
+ O(y )
as y → 0.
2y
25 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
What does the drift look like at the boundary?
3
α(y ) =
+ O(y )
as y → 0.
2y
δ−1
.
2y
So we should choose δ = 4 for our candidate process.
Compare with the Bessel process:
β(y) =
25 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
What does the drift look like at the boundary?
3
α(y ) =
+ O(y )
as y → 0.
2y
δ−1
.
2y
So we should choose δ = 4 for our candidate process.
Compare with the Bessel process:
β(y) =
φ(u)
0
0
5
u
10
25 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
What does the drift look like at the boundary?
3
α(y ) =
+ O(y )
as y → 0.
2y
δ−1
.
2y
So we should choose δ = 4 for our candidate process.
Compare with the Bessel process:
β(y) =
φ(u)
0
0
5
u
10
25 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
φe is (tightly) bounded (by K say), while φ is unbounded as
y → 0.
26 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: A population growth model.
φe is (tightly) bounded (by K say), while φ is unbounded as
y → 0.
Hence we can use the following Bessel-EA to return skeleton
bridges:
1
2
3
Simulate a Poisson point process on [0, T ] × [0, K ].
Simulate a Bessel bridge of dimension δ = 4 at the times of the
Poisson points.
If any of the former are beneath any of the latter, return to 1.
26 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Results
Bessel-EA1
κ
1.0
1.0
1.0
1.0
1.0
1.0
y
10.0
1.0
0.25
0.15
0.1
0.025
Attempts
1.1
1.0
1.0
1.0
1.1
1.0
Poisson
points
0.2
0.2
0.2
0.2
0.2
0.2
Y0 = y to Y0.15 = 1, ω = 3.
Skeleton Random Total
points variables Time (s)
0.2
1.9
0
0.2
1.9
0
0.2
2.0
0
0.2
2.0
1
0.2
2.0
1
0.2
2.0
0
Poisson
points
0.1
0.1
1288.6
7531.1
DNF
DNF
Skeleton
points
0.1
0.1
420.6
617.4
DNF
DNF
Brownian-EA (“EA2”)
κ
1.0
1.0
1.0
1.0
1.0
1.0
y
10.0
1.0
0.25
0.15
0.1
0.025
Attempts
1.0
1.1
1.2
1.4
DNF
DNF
Random
variables
7.3
7.4
3846.1
16921.4
DNF
DNF
Total
Time (s)
0
0
6
16
DNF
DNF
27 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Results
Bessel-EA1
κ
10.0
10.0
10.0
10.0
10.0
10.0
y
10.0
1.0
0.25
0.15
0.1
0.025
Attempts
5.2
3.0
2.3
2.2
2.2
2.1
Poisson
points
14.1
7.9
6.1
6.0
5.9
5.8
Y0 = y to Y0.15 = 1, ω = 3.
Skeleton Random Total
points variables Time (s)
6.8
56.4
1
4.9
36.4
1
4.4
30.8
1
4.3
30.3
0
4.4
30.4
0
4.3
29.6
1
Poisson
points
9.8
5.9
81.4
23052.1
DNF
DNF
Skeleton
points
4.8
3.6
10.7
1981.9
DNF
DNF
Brownian-EA (“EA2”)
κ
10.0
10.0
10.0
10.0
10.0
10.0
y
10.0
1.0
0.25
0.15
0.1
0.025
Attempts
5.0
2.9
2.6
2.9
DNF
DNF
Random
variables
40.9
29.8
201.9
52056.9
DNF
DNF
Total
Time (s)
0
0
0
52
DNF
DNF
28 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
When is the singularity in the drift at an entrance boundary matched
by a Bessel process?
29 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
When is the singularity in the drift at an entrance boundary matched
by a Bessel process?
Here’s a partial answer.
Theorem.
Suppose we have a diffusion Y satisfying the requirements of EA1.
Then the diffusion Y ∗ obtained by conditioning this process on
{Tb < T0 }, can be simulated via Bessel-EA1 with δ = 3.
29 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
When is the singularity in the drift at an entrance boundary matched
by a Bessel process?
Here’s a partial answer.
Theorem.
Suppose we have a diffusion Y satisfying the requirements of EA1.
Then the diffusion Y ∗ obtained by conditioning this process on
{Tb < T0 }, can be simulated via Bessel-EA1 with δ = 3.
Outline of proof.
Deduce regularity requirements for Bessel-EA1 from the
assumptions of EA1.
Compute the conditioned drift α∗ (y ) by bare hands, using a
Doob h-transform.
We find φe∗ (u) is bounded iff δ = 3 (among all possible
δ ≥ 2).
29 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline
1
Introduction
2
Overview of the exact algorithm
3
Bessel-EA
4
Wright-Fisher diffusion
5
Summary
30 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion with natural selection
The frequency Xt ∈ [0, 1] of a gene in a large population
evolves according to
p
dXt = γXt (1 − Xt )dt + Xt (1 − Xt )dWt ,
X0 = x, t ≥ 0.
31 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion with natural selection
The frequency Xt ∈ [0, 1] of a gene in a large population
evolves according to
p
dXt = γXt (1 − Xt )dt + Xt (1 − Xt )dWt ,
X0 = x, t ≥ 0.
γ parametrizes the selective advantage of this gene.
31 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion with natural selection
The frequency Xt ∈ [0, 1] of a gene in a large population
evolves according to
p
dXt = γXt (1 − Xt )dt + Xt (1 − Xt )dWt ,
X0 = x, t ≥ 0.
γ parametrizes the selective advantage of this gene.
A gene is observed to be at frequency X0 = x0 and XT = xT .
What do its sample paths look like?
31 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion with natural selection
The frequency Xt ∈ [0, 1] of a gene in a large population
evolves according to
p
dXt = γXt (1 − Xt )dt + Xt (1 − Xt )dWt ,
X0 = x, t ≥ 0.
γ parametrizes the selective advantage of this gene.
A gene is observed to be at frequency X0 = x0 and XT = xT .
What do its sample paths look like?
0 is absorbing so we should condition the diffusion not to have
hit 0 (T0 > T ).
31 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion with natural selection
The frequency Xt ∈ [0, 1] of a gene in a large population
evolves according to
p
dXt = γXt (1 − Xt )dt + Xt (1 − Xt )dWt ,
X0 = x, t ≥ 0.
γ parametrizes the selective advantage of this gene.
A gene is observed to be at frequency X0 = x0 and XT = xT .
What do its sample paths look like?
0 is absorbing so we should condition the diffusion not to have
hit 0 (T0 > T ).
Conditioning and Lamperti transforming leads to new drift
α(y ) =
h
y i
1
γ sin(y ) coth γ sin2
− cot(y).
2
2
(Schraiber et al., 2013).
31 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion.
What does the drift look like at the boundary?
3
+ O(y )
as y → 0.
α(y ) =
2y
32 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion.
What does the drift look like at the boundary?
3
+ O(y )
as y → 0.
α(y ) =
2y
So we should again choose δ = 4 for our candidate process.
32 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion.
What does the drift look like at the boundary?
3
+ O(y )
as y → 0.
α(y ) =
2y
So we should again choose δ = 4 for our candidate process.
φ(u)
0
0
u
pi
32 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion.
What does the drift look like at the boundary?
3
+ O(y )
as y → 0.
α(y ) =
2y
So we should again choose δ = 4 for our candidate process.
φ(u)
0
0
u
pi
32 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Example: The Wright-Fisher diffusion.
What does the drift look like at the boundary?
3
+ O(y )
as y → 0.
α(y ) =
2y
So we should again choose δ = 4 for our candidate process.
φ(u)
0
0
u
pi
e
Problem: φ(u)
is still unbounded as u → π. This leaves us with
only an approximately exact algorithm.
32 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Ongoing work to fix this issue
In the spirit of (Brownian)-EA2, conditioning on the maximum of a
Bessel bridge would solve the problem:
33 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Ongoing work to fix this issue
In the spirit of (Brownian)-EA2, conditioning on the maximum of a
Bessel bridge would solve the problem:
1
Simulate the maximum MT (and the time, tM , it is attained) of a
Bessel bridge from Y0 to YT .
33 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Ongoing work to fix this issue
In the spirit of (Brownian)-EA2, conditioning on the maximum of a
Bessel bridge would solve the problem:
1
Simulate the maximum MT (and the time, tM , it is attained) of a
Bessel bridge from Y0 to YT .
2
Use another version of the Wright-Fisher diffusion as our
candidate process.
33 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
The Wright-Fisher diffusion with mutation but no selection
p
dXt = [θ1 (1 − Xt ) − θ2 Xt ]dt + Xt (1 − Xt )dWt ,
X0 = x,
Summary
t ≥ 0.
The transition density has eigenfunction expansion
∞
m
X
X
f (x, y; t) =
qm (t)
Bm,x (l) · Dθ1 +l,θ2 +m−l (y ),
| {z } |
{z
}
m=0
l=0
Binomial PMF
Beta density
where qm (t) is the transition function of a certain pure death process
on N (related to Kingman’s coalescent).
34 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
The Wright-Fisher diffusion with mutation but no selection
p
dXt = [θ1 (1 − Xt ) − θ2 Xt ]dt + Xt (1 − Xt )dWt ,
X0 = x,
Summary
t ≥ 0.
The transition density has eigenfunction expansion
∞
m
X
X
f (x, y; t) =
qm (t)
Bm,x (l) · Dθ1 +l,θ2 +m−l (y ),
| {z } |
{z
}
m=0
l=0
Binomial PMF
Beta density
where qm (t) is the transition function of a certain pure death process
on N (related to Kingman’s coalescent).
This is a known infinite mixture of beta random variables.
34 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
The Wright-Fisher diffusion with mutation but no selection
p
dXt = [θ1 (1 − Xt ) − θ2 Xt ]dt + Xt (1 − Xt )dWt ,
X0 = x,
Summary
t ≥ 0.
The transition density has eigenfunction expansion
∞
m
X
X
f (x, y; t) =
qm (t)
Bm,x (l) · Dθ1 +l,θ2 +m−l (y ),
| {z } |
{z
}
m=0
l=0
Binomial PMF
Beta density
where qm (t) is the transition function of a certain pure death process
on N (related to Kingman’s coalescent).
This is a known infinite mixture of beta random variables.
Convenient for simulation!
1
Simulate M ∼ {qm (t) : m = 0, 1, . . .}.
(a realization of Kingman’s coalescent with mutation, time t).
34 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
The Wright-Fisher diffusion with mutation but no selection
p
dXt = [θ1 (1 − Xt ) − θ2 Xt ]dt + Xt (1 − Xt )dWt ,
X0 = x,
Summary
t ≥ 0.
The transition density has eigenfunction expansion
∞
m
X
X
f (x, y; t) =
qm (t)
Bm,x (l) · Dθ1 +l,θ2 +m−l (y ),
| {z } |
{z
}
m=0
l=0
Binomial PMF
Beta density
where qm (t) is the transition function of a certain pure death process
on N (related to Kingman’s coalescent).
This is a known infinite mixture of beta random variables.
Convenient for simulation!
1
Simulate M ∼ {qm (t) : m = 0, 1, . . .}.
(a realization of Kingman’s coalescent with mutation, time t).
2
Simulate L ∼ Binomial(M, x).
34 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
The Wright-Fisher diffusion with mutation but no selection
p
dXt = [θ1 (1 − Xt ) − θ2 Xt ]dt + Xt (1 − Xt )dWt ,
X0 = x,
Summary
t ≥ 0.
The transition density has eigenfunction expansion
∞
m
X
X
f (x, y; t) =
qm (t)
Bm,x (l) · Dθ1 +l,θ2 +m−l (y ),
| {z } |
{z
}
m=0
l=0
Binomial PMF
Beta density
where qm (t) is the transition function of a certain pure death process
on N (related to Kingman’s coalescent).
This is a known infinite mixture of beta random variables.
Convenient for simulation!
1
Simulate M ∼ {qm (t) : m = 0, 1, . . .}.
(a realization of Kingman’s coalescent with mutation, time t).
2
Simulate L ∼ Binomial(M, x).
3
Return Y ∼ Beta(θ1 + L, θ2 + M − L).
34 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Issues
Example
Mixture weights are known only as an infinite series:
∞
X
(θ + 2k − 1)Γ(θ + m + k − 1) −k(k+θ−1)t/2
qm (t) =
(−1)k−m
e
.
m!(k − m)!Γ(θ + m)
k=m
35 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Proposition (Jenkins & Spanò, in preparation).
The coefficients of the ancestral process of Kingman’s coalescent,
{qm (t) : m = 0, 1, . . .},
can be rearranged in such a way that this distribution can be
simulated exactly.
36 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Outline
1
Introduction
2
Overview of the exact algorithm
3
Bessel-EA
4
Wright-Fisher diffusion
5
Summary
36 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Summary
It is possible to simulate efficiently from several diffusions with a
finite entrance boundary, without discretization error.
37 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Summary
It is possible to simulate efficiently from several diffusions with a
finite entrance boundary, without discretization error.
Candidate diffusions other than Brownian motion:
Bessel process
Wright-Fisher diffusion
suggest the potential for further generalizing the exact
algorithms.
37 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Plug
Jenkins, P. A. “Exact simulation of the sample paths of a diffusion
with a finite entrance boundary.” arXiv:1311.5777.
Acknowledgements
Many helpful conversations:
Steve Evans
Murray Pollock
Gareth Roberts
Joshua Schraiber
38 / 38
Introduction
Exact algorithm
Bessel-EA
Wright-Fisher diffusion
Summary
Plug
Jenkins, P. A. “Exact simulation of the sample paths of a diffusion
with a finite entrance boundary.” arXiv:1311.5777.
Acknowledgements
Many helpful conversations:
Steve Evans
Murray Pollock
Gareth Roberts
Joshua Schraiber
Thank you for listening!
38 / 38