Coupling Particle Systems
Pierre E. Jacob (Harvard)
joint work with
Fredrik Lindsten and Thomas Schön (Uppsala)
CRiSM Workshop on Estimating Constants
April 20, 2016
Pierre E. Jacob
Coupling Particle Systems
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Outline
1 Motivation for coupling particle filters
2 How to couple two particle filters
3 A new smoothing algorithm
Pierre E. Jacob
Coupling Particle Systems
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Hidden Markov models
X0
X1
X2
...
XT
θ
y2
y1
...
yT
Figure: Graph representation of a general hidden Markov model.
(Xt ): initial µθ , transition fθ . (Yt ) given (Xt ): measurement gθ .
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Coupling Particle Systems
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Hidden Markov models
How to estimate/predict the latent process (Xt ) given the
observations (Yt ) and a fixed parameter θ?
How to estimate the parameter θ?
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Coupling Particle Systems
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Example: Hidden Autoregressive
Hidden process Xt = AXt−1 + εt , where εt ∼ Nd (0, I),
X0 ∼ Nd (0, I).
Aij = θ|i−j|+1 for i, j ∈ 1 : d.
Observations Yt = Xt + ηt , where ηt ∼ Nd (0, I).
taken from Guarniero, Johansen & Lee, 2015.
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Example: Phytoplankton–Zooplankton
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observations
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100
200
time
300
Figure: A time series of 365 observations generated according to a
phytoplankton–zooplankton model.
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Coupling Particle Systems
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Example: Phytoplankton–Zooplankton
Hidden process (Xt ) = (αt , pt , zt ).
At each (integer) time, αt ∼ N (µα , σα2 ).
Given αt ,
dpt
= αt pt − cpt zt ,
dt
dzt
= ecpt zt − ml zt − mq zt2 .
dt
Observations: log Yt ∼ N (log pt , σy2 ).
Initial distribution: (log p0 , log z0 ) ∼ N (µ0 , σ02 ).
Unknown parameters: θ = (µ0 , σ0 , µα , σα , σy , c, e, ml , mq ).
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Particle filter
X0
X1
X2
...
XT
θ
y2
y1
Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
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θ
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y1
Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
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Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
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Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
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Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
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Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
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Pierre E. Jacob
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Coupling Particle Systems
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Particle filter
At step t = 0,
1
Sample xk0 ∼ µθ (dx0 ), for all k ∈ 1 : N .
2
Set w0k = N −1 , for all k ∈ 1 : N .
At step t ≥ 1,
1
1:N ).
Sample ancestors a1:N
∼ r(da1:N | wt−1
t
2
t
Sample xkt ∼ fθ (dxt | xt−1
), for all k ∈ 1 : N .
3
Compute wtk = gθ (yt | xkt ), for all k ∈ 1 : N .
← resampling
ak
Pierre E. Jacob
Coupling Particle Systems
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Particle filter, rewritten
At step t = 0,
1
k , compute xk = M (U k , θ), for all k ∈ 1 : N .
Sample UM
0
M
2
Set w0k = N −1 , for all k ∈ 1 : N .
At step t ≥ 1,
1
1:N ).
Sample ancestors a1:N
∼ r(da1:N | wt−1
t
2
k , compute xk = F (x t , U k , θ), for all k ∈ 1 : N .
Sample UF,t
t
t−1
F,t
3
Compute wtk = gθ (yt | xkt ), for all k ∈ 1 : N .
← resampling
ak
Pierre E. Jacob
Coupling Particle Systems
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Output
Approximation of the filtering distributions
∀t ∈ {1, . . . , T } p(dxt |y1:t , θ)
by
∀t ∈ {1, . . . , T }
pN (dxt |y1:t , θ) =
N
X
wtk δxk (dxt ).
t
k=1
Approximation of the likelihood function L(θ) = p(y1:T |θ)
by
N
T
Y
1 X
wtk .
pN (y1:T |θ) =
N
t=1
k=1
Pierre E. Jacob
Coupling Particle Systems
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Idea
Particle filters are increasingly used as parts of
encompassing algorithms.
e.g. Particle MCMC, Iterated Filtering
Some of these algorithms compare the outputs of multiple
particle filters.
Better algorithms can be obtained by correlating particle
filters.
i.e. correlation helps comparison.
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Coupling Particle Systems
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Example: approximation of the likelihood
independent
common
−190
transport
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log−likelihood
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−200
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0.2
0.3
0.4
0.5 0.2
0.3
θ
0.4
0.5 0.2
0.3
0.4
0.5
Figure: Estimates of the log-likelihood obtained by particle filters, in a
hidden auto-regressive model, T = 100 observations, N = 64 particles.
See Pitt & Malik, 2011, Lee 2008.
Pierre E. Jacob
Coupling Particle Systems
12/ 56
Example: finite difference
A simple estimator of ∇`(θ) = ∇ log L(θ) is:
c
∇`(θ)
=
log p̂N (y1:T | θ + h) − log p̂N (y1:T | θ − h)
.
2h
The two log-likelihood estimators can be obtained using
independent particle filters given θ + h and θ − h. . .
. . . but if we could positively correlate the two log-likelihood
c
estimators, the variance of ∇`(θ)
would be smaller.
Pierre E. Jacob
Coupling Particle Systems
13/ 56
Example: finite difference
standard deviation
100
98.1
19.1
18.6
9.8
10
5.1
4.6
3.8
2.9
2.7
2.4
2.2
2.3
2.2 2.4 2.2
1
0.01
0.05
0.1
h
method:
indep.
common
sorted
index−matching
transport
c
Figure: Standard deviation of ∇`(θ),
for some θ, in a hidden
auto-regressive model, T = 100 observations, N = 128 particles.
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Coupling Particle Systems
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Example: finite difference
292.7
100
99.6 95.3
standard deviation
58.1
39.8 37.6
30.5
24.6 23.3
19.5
16.9
16 15.8 14.6 14.4
10
1
0.01
0.05
0.1
h
method:
indep.
common
sorted
index−matching
transport
Figure: Same but in dimension 5.
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Coupling Particle Systems
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Example: finite difference
671.9
252.3264.2
116.3109.1
136.1
standard deviation
100
61.8 64.7
70.8
54.5 51.8
43 43.9 41.4 41.3
10
1
0.01
0.05
0.1
h
method:
indep.
common
sorted
index−matching
transport
Figure: Same but in dimension 10.
Pierre E. Jacob
Coupling Particle Systems
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Example: Metropolis-Hastings
Assume we can compute the target density π(θ|y1:T ) pointwise.
Set some θ(1) .
2: for i = 2 to M do
3:
Propose θ? ∼ q(·|θ(i−1) ).
4:
Compute the ratio:
1:
!
q(θ(i−1) |θ? )
π(θ? )p(y1:T |θ? )
.
α = min 1,
π(θ(i−1) )p(y1:T |θ(i−1) ) q(θ? |θ(i−1) )
Set θ(i) = θ? with probability α, otherwise set θ(i) = θ(i−1) .
6: end for
5:
Pierre E. Jacob
Coupling Particle Systems
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Example: particle Metropolis-Hastings
Assume we can run a particle filter to get pN (y1:T |θ).
Set some θ(1) and sample pN (y1:T |θ(1) ).
2: for i = 2 to M do
3:
Propose θ? ∼ q(·|θ(i−1) ) and sample pN (y1:T |θ? ).
4:
Compute the ratio:
1:
!
q(θ(i−1) |θ? )
π(θ? )pN (y1:T |θ? )
.
α = min 1,
π(θ(i−1) )pN (y1:T |θ(i−1) ) q(θ? |θ(i−1) )
Set θ(i) = θ? with probability α, otherwise set θ(i) = θ(i−1) .
6: end for
5:
Andrieu, Doucet & Holenstein, 2010.
Pierre E. Jacob
Coupling Particle Systems
18/ 56
Example: particle Metropolis-Hastings
The acceptance ratio involves a ratio of particle filter
estimators:
!
π(θ? )pN (y1:T |θ? )
q(θ(i−1) |θ? )
α = min 1,
.
π(θ(i−1) )pN (y1:T |θ(i−1) ) q(θ? |θ(i−1) )
If we positively correlate pN (y1:T |θ? ) and pN (y1:T |θ(i−1) ),
the ratio of estimators becomes more precise.
Deligiannidis, Doucet, Pitt & Kohn, 2015: epic improvements
for the case large T / small N .
Pierre E. Jacob
Coupling Particle Systems
19/ 56
Outline
1 Motivation for coupling particle filters
2 How to couple two particle filters
3 A new smoothing algorithm
Pierre E. Jacob
Coupling Particle Systems
19/ 56
Coupled particle filter
By coupled particle filters we mean . . .
k k N
two particle systems, (wtk , xkt )N
k=1 and (w̃t , x̃t )k=1 ,
conditioned on θ and θ̃ respectively,
using common random numbers UM and UF for the initial
and propagation steps.
One still has the freedom to choose a “coupled resampling”
scheme.
Pierre E. Jacob
Coupling Particle Systems
20/ 56
Coupled particle filter
At step t = 0,
1
k , and compute, for all k ∈ 1 : N ,
Sample UM
k
k
xk0 = M (UM
, θ) and x̃k0 = M (UM
, θ̃).
2
Set w0k = N −1 and w̃0k = N −1 , for all k ∈ 1 : N .
At step t ≥ 1,
1
Sample ancestors:
1:N
1:N
(at , ãt ) ∼ r̄(·|wt−1
, w̃t−1
).
2
← coupled resampling
k , and compute, for all k ∈ 1 : N ,
Sample UF,t
ak
ãk
k
k
t
t
xkt = F (xt−1
, UF,t
, θ) and x̃kt = F (x̃t−1
, UF,t
, θ̃).
3
Compute wtk = g(yt | xkt , θ) and w̃tk = g(yt | x̃kt , θ̃).
Pierre E. Jacob
Coupling Particle Systems
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Coupled resampling
k k N
Given two particle systems, (wk , xk )N
k=1 and (w̃ , x̃ )k=1 . . .
We want (?) to sample a1:N and ã1:N in {1, . . . , N }N such
that
∀k
∀j
P(ak = j) = wj and P(ãk = j) = w̃j .
Equivalently, we want to sample (ak , ãk )N
k=1 from a
probability matrix P such that
P1 = w
and
P T 1 = w̃.
Independent resampling corresponds to P = w w̃T . What else?
Pierre E. Jacob
Coupling Particle Systems
22/ 56
Transport resampling
Suppose that we want to sample a couple (a, ã), from some
probability matrix P , such that the resampled particles, xa
and x̃ã , are as similar as possible.
Similarity can be encoded by a distance d on the space of x.
The expected distance between xa and x̃ã , conditional
upon the particles, is given by
h
i
E d(xa , x̃ã ) =
N X
N
X
Pij d(xi , x̃j ).
i=1 j=1
Pierre E. Jacob
Coupling Particle Systems
23/ 56
Transport resampling
Introduce J (w, w̃), the set of matrices satisfying
P1 = w
and
P T 1 = w̃.
2
Compute D = (d(xi , x̃j ))N
i,j=1 , for a cost of O(N ).
Optimal transport problem: solving
P? =
inf
N X
N
X
P ∈J (w,w̃)
Pierre E. Jacob
Pij Dij .
i=1 j=1
Coupling Particle Systems
24/ 56
Transport resampling
Sampling from the optimal P ? minimizes the expected
distance between the two sets of particles, under the
marginal constraint.
(Which is not exactly the same as maximizing the
correlation between e.g. likelihood estimators).
Computing P ? requires O(N 3 ) operations, but efficient
approximations have been proposed (Cuturi 2013 and
following work) in O(N 2 ).
The cost is linear in the dimension of x, and independent of
the model.
Pierre E. Jacob
Coupling Particle Systems
25/ 56
Index-matching resampling
At the initial step,
k
k
xk0 = M (UM
, θ) and x̃k0 = M (UM
, θ̃).
so that particles with the same index are similar (if M is
continuous in θ).
At subsequent steps, the same random numbers UFk are
k
k
used to propagate xa and x̃ã .
Standard resampling breaks the correspondence between
k
k
similarity and indices: xa and x̃ã might not be similar.
Pierre E. Jacob
Coupling Particle Systems
26/ 56
Index-matching resampling
To preserve the correspondence between indices, we want
to maximize the probability of drawing couples of ancestors
such that a = ã.
. . . i.e. we want P in J (w, w̃) with maximum trace.
We can define:
P = diag(min{w, w̃}) + (1 − α)r r̃T ,
with
α=
N
X
min{wk , w̃k },
k=1
r = (w − min{w, w̃})/(1 − α),
r̃ = (w̃ − min{w, w̃})/(1 − α).
P has maximum trace in J (w, w̃): we cannot augment its
diagonal without violating the marginal constraints.
Pierre E. Jacob
Coupling Particle Systems
27/ 56
Index-matching resampling
1000
particle
750
500
250
0
0
50
100
150
time
200
# consecutive matching
0
Pierre E. Jacob
30
60
90
Coupling Particle Systems
28/ 56
Index-matching resampling
Distance between 1, 000 pairs of particles, sampled
independently and then propagated with common random
numbers.
distance
15
10
5
0
0
5
10
time
15
20
Hidden AR, θ = 0.30, θ̃ = 0.31.
Pierre E. Jacob
Coupling Particle Systems
29/ 56
Index-matching resampling
Distance between 1, 000 pairs of particles, sampled
independently and then propagated with common random
numbers.
distance
15
10
5
0
0
5
10
time
15
20
Hidden AR, θ = 0.30, θ̃ = 0.40.
Pierre E. Jacob
Coupling Particle Systems
30/ 56
Sorting and resampling
Univariate setting: x is of dimension 1.
k N
Sort the two systems (xk )N
k=1 and (x̃ )k=1 .
Perform e.g. systematic resampling on each sorted system,
using the same random numbers.
Thus if ak selects xj in the first system, ãk is likely to
select a x̃i close to xj .
This can be extended to multivariate settings by sorting the
particles according to the Hilbert space-filling curve.
See Deligiannidis, Doucet, Pitt & Kohn, 2015, and Pitt &
Malik, 2011, Gerber & Chopin, 2015.
Pierre E. Jacob
Coupling Particle Systems
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Outline
1 Motivation for coupling particle filters
2 How to couple two particle filters
3 A new smoothing algorithm
Pierre E. Jacob
Coupling Particle Systems
31/ 56
Who cares about unbiased estimators?
Coupled resampling leads to a practical unbiased estimator
Hu of the smoothing quantity
Z
h(x0:T ) p(dx0:T |y1:T , θ),
for a test function h (for fixed θ).
(1)
(R)
Computing Hu , . . . , Hu
H̄u =
in parallel, we obtain
R
1 X
H (r) ,
R r=1 u
along with a CLT-based error estimate.
By contrast, for existing smoothing techniques, parallelism
is not trivial, nor is the construction of error estimates.
Pierre E. Jacob
Coupling Particle Systems
32/ 56
Proposed estimator
Use Rhee & Glynn (2014) trick to turn a Conditional
Particle Filter kernel into an unbiased estimator of
smoothing functionals.
Coupled resampling schemes are instrumental in this
construction.
Instead of two particle systems given θ and θ̃, we consider
two particle systems with same θ but different “reference
trajectories”.
Pierre E. Jacob
Coupling Particle Systems
33/ 56
Trajectories from particle filters
Upon running a particle filter, we get trajectories x1:N
0:T with
weights wT1:N .
4
paths
2
0
−2
0
25
50
time
75
100
Figure: Hidden auto-regressive model, T = 100 observations, N = 128.
Pierre E. Jacob
Coupling Particle Systems
34/ 56
Conditional particle filter
Input: a trajectory x̌0:T .
At step t = 0,
1
Sample xk0 ∼ µθ (dx0 ), for all k ∈ 1 : N − 1, set xN
0 = x̌0 .
2
Set w0k = N −1 , for all k ∈ 1 : N .
At step t ≥ 1,
1
2
3
−1
1:N ), set aN = N .
Sample ancestors a1:N
∼ r(da1:N −1 | wt−1
t
t
ak
t
Sample xkt ∼ fθ (dxt | xt−1
), for all k ∈ 1 : N − 1, set
N
xt = x̌t .
Compute wtk = gθ (yt | xkt ), for all k ∈ 1 : N .
Output: sample a trajectory, xk0:T with probability wTk .
Pierre E. Jacob
Coupling Particle Systems
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Conditional particle filter
paths
2.5
0.0
−2.5
0
25
50
time
75
100
Figure: M = 100 paths, for the hidden auto-regressive model, T = 100
observations, N = 128.
Pierre E. Jacob
Coupling Particle Systems
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Conditional particle Filter
x0
1
0
−1
0
25
50
iteration
75
100
Figure: M = 100 samples for x0 , for the hidden auto-regressive model,
T = 100 observations, N = 128.
Pierre E. Jacob
Coupling Particle Systems
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Conditional particle Filter
x100
1
0
−1
−2
0
25
50
iteration
75
100
Figure: M = 100 samples for x100 , for the hidden auto-regressive
model, T = 100 observations, N = 128.
Pierre E. Jacob
Coupling Particle Systems
38/ 56
Unbiased estimators
Von Neumann & Ulam (∼ 1950), Kuti (∼ 1980), Rychlik
(∼ 1990), McLeish (∼ 2010), Rhee & Glynn
(2012, 2013, 2014).
Introduce
a sequence of random variables (H (n) ) with
E[H (n) ] −−−→
Z
n→∞
h(x0:T ) p(dx0:T |y1:T , θ),
e.g. H (n) = h(X (n) ), with (X (n) ) generated by CPF,
a sequence (∆(n) ) such that
E[∆(n) ] = E[H (n) − H (n−1) ],
E
"∞
X
#
|∆(n) | < ∞,
n=0
with H (−1) = 0 by convention.
Pierre E. Jacob
Coupling Particle Systems
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Unbiased estimators
Then
E
∞
X
∆(n) =
n=0
∞
X
E[∆(n) ] =
n=0
∞
X
E[H (n) − H (n−1) ]
n=0
= lim E[H
(n)
n→∞
Z
]=
h(x0:T ) p(dx0:T |y1:T , θ).
Thus, consider
Hu =
K
X
∆(n)
,
P (K ≥ n)
n=0
where K is an integer-valued random variable. Then
E[Hu ] = E[
∞
X
∆(n) 1(K ≥ n)
n=0
P (K ≥ n)
Pierre E. Jacob
Z
]=
h(x0:T ) p(dx0:T |y1:T , θ).
Coupling Particle Systems
40/ 56
Unbiased estimators
Idea from Rhee & Glynn, 2014. Write
X (n) = ϕn (X (n−1) ) = ϕn ◦ ϕn−1 ◦ . . . ◦ ϕ1 (X (0) ).
Introduce
∆
X̃ (0) = X (0) ,
X̃ (1) = ϕ2 (X̃ (0) ),
...,
X̃ (n) = ϕn+1 ◦. . .◦ϕ2 (X (0) ).
Then ∆(n) = h(X (n) ) − h(X̃ (n−1) ) is such that
E[∆(n) ] = E[H (n) − H (n−1) ]
and we might have
E
"∞
X
#
|∆
(n)
| < ∞.
n=0
Pierre E. Jacob
Coupling Particle Systems
41/ 56
Unbiased estimators based on CPF chains
Start from X (0) and X̃ (0) generated by two particle filters.
Apply one step of CPF kernel to X (0) , to get X (1) .
For n ≥ 2, apply the CPF kernel to both X (n−1) and
X̃ (n−2) , with the same random numbers, to get X (n) and
X̃ (n−1) .
We can see each step as a joint CPF acting on pairs of
trajectories, and use coupled resampling ideas.
Can we expect ∆(n) = h(X (n) ) − h(X̃ (n−1) ) to decrease to
zero in average?
Pierre E. Jacob
Coupling Particle Systems
42/ 56
Norm of ∆(n) with independent resampling
8
Norm of ∆
6
4
2
0
0
25
50
iteration
75
100
Hidden auto-regressive model, T = 20 observations, N = 32.
Pierre E. Jacob
Coupling Particle Systems
43/ 56
Norm of ∆(n) with independent resampling
Norm of ∆
10
5
0
0
25
50
iteration
75
100
T = 100 observations, N = 128.
Pierre E. Jacob
Coupling Particle Systems
44/ 56
Norm of ∆(n) with index-matching resampling
Norm of ∆
6
4
2
0
0
25
50
iteration
75
100
T = 20 observations, N = 32.
Pierre E. Jacob
Coupling Particle Systems
45/ 56
Norm of ∆(n) with index-matching resampling
Norm of ∆
12
8
4
0
0
25
50
iteration
75
100
T = 100 observations, N = 128.
Pierre E. Jacob
Coupling Particle Systems
46/ 56
Coupled conditional particle Filter
We consider coupled conditional particle filters, acting on
pairs of trajectories:
(X (n) , X̃ (n−1) ) = ϕ̄n (X (n−1) , X̃ (n−2) )
A coupled CPF kernel uses common random numbers for
both systems, and a coupled resampling scheme.
We focus on index-matching resampling.
We see that after a number of coupled CPF steps,
X (n) = X̃ (n) exactly, and thus ∆(n) = 0.
We can thus stop early in the computation of
Hu =
K
X
∆(n)
.
P (K ≥ n)
n=0
Pierre E. Jacob
Coupling Particle Systems
47/ 56
Proposed estimator
Case h = Id: we estimate the smoothing means.
Sample an integer-valued random variable K.
Sample ϕ1 , draw X (0) and set X (1) = ϕ1 (X (0) ).
Compute ∆(0) = X (0) , set Hu ← ∆(0) .
∆
Sample X̃ (0) = X (0) , compute ∆(1) = X (1) − X̃ (0) .
Set Hu ← Hu + ∆(1) /P(K ≥ 1).
For n = 2, . . . , K,
Sample ϕ̄n , set (X (n) , X̃ (n−1) ) = ϕ̄n (X (n−1) , X̃ (n−2) ).
Compute ∆(n) = X (n) − X̃ (n−1) .
Stop if ∆(n) = 0.
Set Hu ← Hu + ∆(n) /P(K ≥ n).
Return Hu .
Pierre E. Jacob
Coupling Particle Systems
48/ 56
Example: Phytoplankton–Zooplankton
count
100
50
0
1
10
iteration
Geometric
100
1000
Time to merge
Figure: Phytoplankton–Zooplankton model, T = 365, N = 1, 024,
R = 1, 000 estimators, with a Geometric truncation with mean 100.
Pierre E. Jacob
Coupling Particle Systems
49/ 56
P smoothing means
Example: Phytoplankton–Zooplankton
30
20
10
0
0
100
time
200
300
Figure: Smoothing means of P , T = 365, N = 1, 024, R = 1, 000
estimators, with a Geometric truncation with mean 100.
The bars represent ±2σ around the estimated means. The blue
line is obtained from a long CPF run.
Pierre E. Jacob
Coupling Particle Systems
50/ 56
Example: Phytoplankton–Zooplankton
Z smoothing means
15
10
5
0
0
100
time
200
300
Figure: Smoothing means of Z, T = 365, N = 1, 024, R = 1, 000
estimators, with a Geometric truncation with mean 100.
The bars represent ±2σ around the estimated means. The blue
line is obtained from a long CPF run.
Pierre E. Jacob
Coupling Particle Systems
51/ 56
Example: Phytoplankton–Zooplankton
Z smoothing means
8
6
4
2
0
0
20
time
40
60
Figure: Smoothing means of Z, first 65/365 time steps, N = 1, 024,
R = 1, 000 estimators, with a Geometric truncation with mean 100.
Pierre E. Jacob
Coupling Particle Systems
52/ 56
Example: Phytoplankton–Zooplankton
Z smoothing means
10.0
7.5
5.0
2.5
300
320
time
340
360
Figure: Smoothing means of Z, last 65/365 time steps, N = 1, 024,
R = 1, 000 estimators, with a Geometric truncation with mean 100.
Pierre E. Jacob
Coupling Particle Systems
53/ 56
trace of relative variance
Example: Phytoplankton–Zooplankton
0.100
0.010
0.001
0
100
200
time
300
Figure: Trace of relative variance of the smoothing mean estimator,
T = 365, N = 1, 024, R = 1, 000 estimators, with a Geometric
truncation with mean 100.
Pierre E. Jacob
Coupling Particle Systems
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Discussion
Coupled resampling schemes can be used to improve a
variety of particle-based algorithms.
New estimator of smoothing functionals, easy to parallelize
and with error estimates.
Benefits greatly from ancestor sampling:
variance
1.00
0.01
0
100
200
time
300
400
256 Coupling
1024 Particle
4096Systems
Pierre E. Jacob
500
55/ 56
The End
Thank you for listening!
Soon on arXiv. . .
PJ, Fredrik Lindsten, Thomas Schön, Coupling Particle Filters.
Pitt & Malik, 2011, Particle filters for continuous likelihood
evaluation and maximisation, J. of Econometrics.
Rhee & Glynn, 2014, Exact estimation for markov chain
equilibrium expectations, arXiv.
Deligiannidis, Doucet, Pitt & Kohn, 2015, The correlated
pseudo-marginal method, arXiv.
Pierre E. Jacob
Coupling Particle Systems
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