Designing “Particle Filters”
for Particle MCMC
Adam M. Johansen
a.m.johansen@warwick.ac.uk
go.warwick.ac.uk/amjohansen/talks/
Oxford, March 4th, 2016
Outline
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Background: SMC and PMCMC
Beyond simple filters:
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Iterative Lookahead Methods
Tempering Blocks in SMC
Hierarchical Particle Filters
Conclusions
Discrete Time Filtering
Online inference for Hidden Markov Models:
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
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Given initial distribution µθ (x1 ), transition fθ (xn−1 , xn ),
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and likelihood gθ (xn , yn ),
use pθ (xn |y1:n ) to characterize latent state, but,
R
pθ (xn−1 |y1:n−1 )fθ (xn−1 , xn )dxn−1 gθ (xn , yn )
pθ (xn |y1:n ) = R R
pθ (xn−1 |y1:n−1 )fθ (xn−1 , xn0 )dxn−1 gθ (xn0 , yn )dxn0
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and other densities of interest are typically intractable.
Particle Filtering
A Simple “Bootstrap” Particle Filter (Gordon et al. [10])
At n = 1:
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Sample X11 , . . . , X1N ∼ µθ .
For n > 1:
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Sample
j
j=1 gθ (Xn−1 , yn−1 )δj (·)
Pn
k
k=1 gθ (Xn−1 , yn−1 )
PN
A1n , . . . , AN
n
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I
∼
Ai
n
For i in 1 : N sample Xni ∼ fθ (Xn−1
, ·).
Approximate pθ (dxn |y1:n ), pθ (y1:n ) with
PN
j=1
pbθ (·|y1:n ) = Pn
gθ (Xnj , yn )δXnj
k=1
gθ (Xnk , yn )
,
N
1X
pbθ (y1:n )
gθ (Xnk , yn )
=
pbθ (y1:n−1 )
n
j=1
Iteration 2
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2
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0
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Iteration 3
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Iteration 4
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Iteration 5
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Iteration 6
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Iteration 7
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Iteration 8
8
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−1
−2
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Iteration 9
8
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−2
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Iteration 10
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2
1
0
−1
−2
1
2
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5
6
7
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9
10
Various Improvements to Particle Filters
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Better Proposal Distributions:
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Locally optimal (cf. Doucet et al. [6]) case
q(xn |xn−1 , yn ) ∝ fθ (xn−1 , xn )g (xn , yn ).
Incremental importance weights
Gn (xn−1 , xn ) = f (xn−1 , xn )g (xn , yn )/q(xn−1 , xn ).
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Auxiliary Particle Filters (Pitt and Shephard [15])
(see also Johansen and Doucet [13])
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Better Resampling Schemes (cf. Douc and Cappé [5])
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“Adaptive” Resampling (cf. Del Moral et al. [4])
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Resample-Move Schemes (MCMC; Gilks and Berzuini [8])
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Tempered Transitions (Godsill and Clapp [9])
All of these retain the online character of particle filtering.
Key Properties of Particle Filters
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Fundamentally online: can approximate pθ (xn |y1:n ) at
iteration n at constant cost per iteration.
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Yield good approximations of the filtering distribution:
pθ (xn |y1:n )
at time n.
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Can approximate the marginal likelihood
Z
pθ (y1:n ) = pθ (x1:n , y1:n )dx1:n
unbiasedly.
Only the last of these is critical in an offline setting. . .
Online Particle Filters for Offline Estimation (via PMCMC)
Particle Markov chain Monte Carlo (Andrieu et al. [2])
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Embed SMC within MCMC,
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justified via explicit auxiliary variable construction,
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or in some cases by a pseudomarginal (Andrieu and Roberts
[1]) argument.
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Very widely applicable,
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but prone to poor mixing when SMC performs poorly for some
θ (Owen et al. [14, Section 4.2.1]).
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Is valid for very general SMC algorithms.
Iterative Lookahead Methods1 : Motivation
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Online algorithms can only perform so well:
Z
p(x1:n |y1:T ) = p(x1:T |y1:T )dxn+1:T 6= p(x1:n |y1:n )
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We’d benefit from targetting the smoothing distributions:
π̃n (x1:n ) = p(x1:n |y1:T ) ∝ p(x1:n |y1:n )p(yn+1:T |xn )
in place of the filtering distributions:
πn (x1:n ) = p(x1:n |y1:n )
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1
But this is really hard: can we approximate p(yn+1:T |xn )?
Joint work with Pieralberto Guarniero and Anthony Lee
Twisting the HMM (complements (Whiteley and Lee [16]))
Given (µ, f , g ) and y1:T , introducing
ψ := (ψ1 , ψ2 , . . . , ψT ) , ψt ∈ Cb (X, (0, ∞)) and
Z
Z
ψ̃0 :=
µ (x1 ) ψ1 (x1 ) dx1 ψ̃t (xt ) :=
f (xt , xt+1 ) ψt+1 (xt+1 ) dxt+1
X
X
ψ
ψ
(and ψ̃T ≡ 1) we obtain (µψ
1 , {ft }, {gt }), with
µψ
1 (x1 ) :=
µ(x1 )ψ1 (x1 )
,
ψ̃0
ftψ (xt−1 , xt ) :=
f (xt−1 , xt ) ψt (xt )
ψ̃t−1 (xt−1 )
and the sequence of non-negative functions
g1ψ (x1 ) := g (x1 , y1 )
ψ̃1 (x1 )
ψ̃0 ,
ψ1 (x1 )
gtψ (xt ) := g (xt , yt )
ψ̃t (xt )
.
ψt (xt )
Proposition
For any sequence of bounded, continuous and positive functions ψ,
let
Z
Zψ :=
XT
ψ
µψ
1 (x1 ) g1 (x1 )
T
Y
ftψ (xt−1 , xt ) gtψ (xt ) dx1:T .
t=2
Then, Zψ = pθ (y1:T ) for any such ψ.
The (variance) optimal choice is:


T
Y
ψt∗ (xt ) := g (xt , yt ) E 
g (Xp , yp ) {Xt = xt } ,
p=t+1
xt ∈ X,
for t ∈ {1, . . . , T − 1}. Then, ZψN∗ = p(y1:T ) with probability 1.
ψ−Auxiliary Particle Filters (Guarniero et al. [11])
ψ-Auxiliary Particle Filter
1. Sample ξ1i ∼ µψ independently for i ∈ {1, . . . , N}.
2. For t = 2, . . . , T , sample independently
ψ
j
ψ j
j=1 gt−1 (ξt−1 )ft (ξt−1 , ·)
,
PN
ψ
j
j=1 gt−1 (ξt−1 )
PN
ξti
∼
Necessary features of ψ
1. It is possible to sample from ftψ .
2. It is possible to evaluate gtψ .
3. To be useful: V(ZψN ) must be small.
i ∈ {1, . . . , N}.
A Recursive Approximtion
Proposition
The sequence ψ ∗ satisfies ψT∗ (xT ) = g (xT , yT ), xT ∈ X and
∗
ψt∗ (xt ) = g (xt , yt ) f xt , ψt+1
, xt ∈ X, t ∈ {1, . . . , T − 1}.
Algorithm 1 Recursive function approximations
For t = T , . . . , 1:
1. Set ψti ← g ξti , yt f ξti , ψt+1 for i ∈ {1, . . . , N}.
2. Choose ψt as a member of Ψ on the basis of ξt1:N and ψt1:N .
Iterated Auxiliary Particle Filters (Guarniero et al. [11])
Algorithm 2 An iterated auxiliary particle filter (param’s: N0 , k, τ )
1. Initialize: set ψt 0 = 1. l ← 0.
2. Repeat:
2.1 Run a ψ l -APF with Nl particles; set Ẑl ← ZψNll .
2.2 If l > k and sd(Ẑl−k:l )/mean(Ẑl−k:l ) < τ , go to 3.
2.3 Compute ψ l+1 using Algorithm 1.
2.4 If Nl−k = Nl and the sequence Ẑl−k:l is not monotonically
increasing, set Nl+1 ← 2Nl .
Otherwise, set Nl+1 ← Nl .
2.5 Set l ← l + 1. Go to 2a.
3. Run a ψ l -APF. Return Ẑ := ZψNl .
An Elementary Implementation
Function Approximation .
I Numerically obtain:
(mt∗ , Σ∗t , λ∗t )
= arg min(m,Σ,λ)
N
X
(N (xti , m, Σ)−λψti )2
i=1
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Set:
ψt (xt ) := N (xt ; mt∗ , Σ∗t ) + c(N, mt∗ , Σ∗t ).
Stopping Rule .
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k = 3 or k = 5 in the following examples
τ = 0.5
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Multinomial when ESS < N/2.
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Resampling .
A Linear Gaussian Model: Behaviour with Dimension
µ = N (·; 0, Id )
f (x, ·) = N (·; Ax, Id )
where Aij = 0.42|i−j|+1 ,
and g (x, ·) = N (·; x, Id )
0.5
1.0
1.5
iAPF
Bootstrap
0.0
Estimates
2.0
Box plots of Ẑ /Z for different dim(X) (1000 replicates; T = 100).
5
5
10
10
20
40
80
Linear Gaussian Model: Sensitivity to Parameters
1.5
1.0
Estimates
0.0
0.5
1.5
1.0
0.0
0.5
Estimates
2.0
2.0
Fixing d = 10: Bootstrap (N = 50, 000) / iAPF (N0 = 1, 000)
0.3
0.34
0.38
0.42
Values of α
Box plots of
Ẑ
Z
0.46
0.5
0.3
0.34
0.38
0.42
0.46
0.5
Values of α
for different values of the parameter α using 1000
replicates.
Linear Gaussian Model: PMMH Empirical Autocorrelations
0
100
200
300
400
500
600
0.8
ACF
0
Lag
d =5
µ = N (·; 0, Id )
f (x, ·) = N (·; Ax, Id )
0.4
0.0
ACF
0.4
0.0
0.4
0.0
ACF
A55
0.8
A41
0.8
A11
and g (x, ·) = N (·; x, 0.25Id )
100
200
300
400
500
600
0
100
200
Lag
In this case:

0.9
 0.3

A=
 0.1
 0.4
0.1
300
400
Lag
0
0
0 0
0.7 0
0 0
0.2 0.6 0 0
0.1 0.1 0.3 0
0.2 0.5 0.2 0



,


treated as unknown lower triangular matrix.
500
600
Stochastic Volatility
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A simple stochastic volatility model is defined by:
µ(·) =N (·; 0, σ 2 /(1 − α)2 )
f (x, ·) =N (·; αx, σ 2 )
and g (x, ·) =N (·; 0, β 2 exp(x)),
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where α ∈ (0, 1), β > 0 and σ 2 > 0 are unknown.
Considered T = 945 observations y1:T corresponding to the
mean-corrected daily returns for the GBP/USD exchange rate
from 1/10/81 to 28/6/85.
ACF
0.0
0.4
0.8
Estimated PMCMC Autocorrelation
0
20
40
60
80
Lag
ACF
0.0
0.4
0.8
a
0
20
40
60
80
60
80
Lag
ACF
0.0
0.4
0.8
σ
0
β
20
40
Lag
Boostrap N = 1, 000.
iAPF N0 = 100.
Comparable cost.
150,000 PMCMC iterations.
0.974
0.978
0.982
0.986
0.99
0.994
0.974
0.978
0.982
0.986
0.99
0.994
0.974
0.978
0.982
0.986
0.99
0.994
Estimates
−922
−923
−924
−925
−926
−927
−928
Values of a
Bootstrap : N = 1, 000 / N = 10, 000 / iAPF , N0 = 100
Estimates
−923
−924
−925
−926
0.61
0.65
0.69
0.73
0.77
0.81
0.61
0.65
0.69
0.73
0.77
0.81
0.61
0.65
0.69
0.73
0.77
0.81
−927
Values of β
Bootstrap : N = 1, 000 / N = 10, 000 / iAPF , N0 = 100
−922
Estimates
−923
−924
−925
−926
−927
0.12
0.13
0.14
0.15
0.16
0.17
0.12
0.13
0.14
0.15
0.16
0.17
0.12
0.13
0.14
0.15
0.16
0.17
−928
Values of σ
Bootstrap : N = 1, 000 / N = 10, 000 / iAPF , N0 = 100
Block-Sampling and Tempering in Particle Filters
Tempered Transitions (Godsill and Clapp [9])
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Introduce each likelihood term gradually, targetting:
πn,m (x1:n ) ∝ p(x1:n |y1:n−1 )p(yn |xn )βm
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between p(x1:n−1 |y1:n−1 ) and p(x1:n |y1:n ).
Can improve performance — but to a limited extent.
Block Sampling (Doucet et al. [7])
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Essentially uses yn+1:n+L in proposing xn+1 .
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Can dramatically improve performance,
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but requires good analytic approximation of
p(xn+1:n+L |xn , yn+1:n+L ).
Block Sampling: An Idealised Approach
At time n, given x1:n−1 ; discard xn−L+1:n−1 :
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Sample from q(xn−L+1:n |xn−L , yn−L+1:n ).
Weight with
W (x1:n ) =
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p(x1:n |y1:n )
p(x1:n−L |y1:n−1 )q(xn−L+1:n |xn−L , y1:n−L+1:n )
Optimally,
q(xn−L+1:n |xn−L , yn−L+1:n ) =p(xn−L+1:n |xn−L , yn−L+1:n )
p(x1:n−L |y1:n )
W (x1:n ) ∝
=p(yn |x1:n−L , yn−L+1:n−1 )
p(x1:n−L |y1:n−1 )
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Typically intractable; auxiliary variable approach in [7].
Block-Tempering (Johansen [12])
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We could combine blocking and tempering strategies.
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Run a simple SMC sampler (Del Moral et al. [3]) targetting:
1
β(t,r
)
θ
πt,r
(x1:t∧T ) =µθ (x1 )
T
∧t
Y
s=2
1
γ(t,r
)
gθ (y1 |x1 )
fθ (xs |xs−1 )
s
β(t,r
)
·
s
s
where {β(t,r
) } and {γ(t,r ) } are [0, 1]-valued
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I
I
s
γ(t,r
)
gθ (ys |xs )
,
for s ∈ [[1, T ]], r ∈ [[1, R]] and t ∈ [[1, T 0 ]], with T 0 = T + L
for some R, L ∈ N.
Can be validly embedded within PMCMC:
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Terminal likelihood estimate is unbiased.
Explicit auxiliary variable construction is possible.
(1)
Two Simple Block-Tempering Strategies
Tempering both likelihood and transition probabilities
s
s
β(t,r
) = γ(t,r ) =
1∧
R(t − s) + r
RL
∨0
(2)
Tempering only the observation density
s
β(t,r
)
=I{s ≤ t}
s
γ(t,r
)
R(t − s) + r
= 1∧
∨ 0,
RL
(3)
Tempering Only the Observation Density
s
s
β(t,R)
= β(t,0)
1
0
t
s
s
γ(t,R)
1
1
L
s
γ(t,0)
0
t−L+1
t
s
Illustrative Example
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Univariate Linear Gaussian SSM:
transition f (x 0 |x) =N (x 0 ; x, 1)
likelihood
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g (y |x) =N (y ; x, 1)
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Artificial jump in observation
seqeuence at time 75.
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Cartoon of model misspecification
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— a key difficulty with PMCMC.
Observations, y[t]
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Temper only likelihood.
−15
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Use single-site Metropolis-Within
Gibbs (standard normal proposal)
MCMC moves.
0
25
50
Time, t
75
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100
True Filtering and Smoothing Distributions
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Algorithm
smoother
● filter
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25
50
Time, t
75
100
b
Estimating the Normalizing Constant, Z
−88
L
log(Estimated Z)
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−90
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1
2
3
5
●
R
● 0
1
5
10
−92
3
4
5
log(Cost = N(1+RL))
6
b Against Computational Effort
Relative Error in Z
2
L
log(Relative Error in Z)
1
●
1
●
2
●
3
●
5
Performance:
log(N)
●
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3
●
4
●
5
●6
R
●
0
0
1
5
10
4
5
log(Cost = N(1+RL))
6
(R = 1, L = 1)
≺(R > 1, L = 1)
≺(R > 1, L > 1)
Hierarchical Particle Filters2
I
(Optimal) block sampler requires samples from
(approximately)
p(xn−L+1:n |xn−L , yn−L+1:n )
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and evaluations of
p(yn |xn−L , yn−L+1:n−1 ) =
I
p(yn−L+1:n |xn−L )
.
p(yn−L+1:n−1 |xn−L )
Particle filters can provide sample approximations of
p(xn−L+1:n |xn−L , yn−L+1:n )
I
and of
p(yn−L+1:n |xn−L ).
I
2
Can we use particle filters hierarchically?
Joint work with Arnaud Doucet. . .
SMC Distributions
Formally gives rise to the SMC Distribution:
M
ψn,L
an−L+2:n , xn−L+1:n , k; xn−L
#
"
"M
#
n
M
Y
Y
Y
i
a
p
q( x in−L+1 x n−L )
r (ap |wp−1 )
q x ip |x p−1
r (k|wn )
=
i=1
p=n−L+2
i=1
and the conditional SMC Distribution:
ek
k
M
k
e
e
ψen,L
ak
,
x
;
x
xn−L+1:n
n−L+2:n n−L+1:n n−L bn−L+1:n−1 , k, e
M
ψn,L
(e
an−L+2:n , e
xn−L+1:n , k; xn−L )
"
#
= ebk ebn n
k
e
Q
bn,n−L+1
n,p
n,p−1
k
e
e p−1 q e
e n)
q e
xn−L+1 |xn−L
r bn,p |w
xp |e
xp−1
r (k|w
p=n−L+2
Local Particle Filtering: Current Trajectories
Starting Point
4
2
0
−2
−4
0
2
4
6
8
n
10
12
14
16
Local Particle Filtering: PF Proposal
PF Step
4
2
0
−2
−4
0
2
4
6
8
n
10
12
14
16
Local Particle Filtering: CPF Auxiliary Proposal
CPF Step
4
2
0
−2
−4
0
2
4
6
8
n
10
12
14
16
Local SMC: Version 1
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Not just a Random Weight Particle Filter.
Propose from:
⊗n−1
M
U1:M
(b1:n−2 , k)p(x1:n−1 |y1:n−1 )ψn,L
(an−L+2:n , xn−L+1:n , k; xn−L )
k
k
M
e
ψn−1,L−1 e
an−L+2:n−1 , e
xn−L+1:n−1 ; xn−L ||bn−L+2:n−1 , xn−L+1:n−1
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Target:
b
k̄
n,n−L+1:n
⊗n
k̄
U1:M
(b1:n−L , b̄n,n−L+1:n−1
, k̄)p(x1:n−L , x n−L+1:n
|y1:n )
k̄
k̄
k
M
b̄n,n−L+1:n , x bn,n−L+1:n
,
;
x
ψen,L
ak
x
n−L
n−L+2:n n−L+1:n
n−L+1:n
M
ψn−1,L−1
(e
an−L+2:n−1 , e
xn−L+1:n−1 , k; xn−L ) .
I
en−L+1:n−1 .
Weight: Z̄n−L+1:n /Z
Local SMC: Version 2
Problems with this PF+CPF scheme:
I
Expensive to run 2 filters per proposal. . .
I
and large M is required. . .
I
can’t we do better?
Using a non-standard CPF/PF proposal is preferable:
I
Running a CPF from time n − L + 1 to n,
I
and extending it’s paths with a PF step to n + 1,
I
invoking a careful auxiliary variable construction,
I
we reduce computational cost and variance.
I
Leads to importance weights:
bk
∝
n−1,n−L
p̂(yn−L+1:n |xn−L
)
bk
n−1,n−L
p̂(yn−L+1:n−1 |xn−L
)
Hierarchical Particle Filtering: Current Trajectories
Starting Point
7
6
5
4
3
2
1
0
-1
-2
0
2
4
6
8
n
10
12
14
16
Hierarchical Particle Filtering: CPF Proposal Component
CPF Step
7
6
5
4
3
2
1
0
-1
-2
0
2
4
6
8
n
10
12
14
16
Hierarchical Particle Filtering: PF Proposal Component
PF Step
7
6
5
4
3
2
1
0
-1
-2
0
2
4
6
8
n
10
12
14
16
A Toy Model: Linear Gaussian HMM
I
Linear, Gaussian state transition:
f (xt |xt−1 ) = N (xt ; xt−1 , 1)
I
and likelihood
g (yt |xt ) = N (yt ; xt , 1)
I
Analytically: Kalman filter/smoother/etc.
I
Simple bootstrap HPF:
I
Local proposal:
q(xt |xt−1 , yt ) = f (xt |xt−1 )
I
Weighting:
W (xt−1 , xt ) ∝ g (yt |xt )
●
3
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
N
●
●
●
●
●
100
●
●
●
●
●
1000
●
10000
●
●
1e+05
2
●
●
M
●
●
1
●
rZ
●
●
● ●
●
●
●
●
●
●
●
●
●
●
●
●
1
●
●
● ●
● ●
●
2
●
●
●
●
●
●
4
8
16
●
●
● ●
●
32
64
●
●
128
●
●
256
●
●
●
●
●
512
●
1024
●
Bo
ots
4
tra
p
0
L
0
00
10
00
0
12
80
0
16
00
0
25
60
0
32
00
0
51
20
0
64
00
0
1e
+0
5
10
24
00
12
80
00
20
48
00
25
60
00
40
96
00
51
20
00
10
24
00
0
20
48
00
0
40
96
00
0
80
00
00
64
32
00
16
80
log(rZ)
2
●
N
100
●
●
●
1000
C
●
●
●
●
●
●
●
●
10000
0
●
●
−4
M
1e+05
●
1
●
2
●
4
8
−2
16
32
64
128
256
512
L
1024
4
Bootstrap
Conclusions
I
I
I
To fully realise the potential of PMCMC we should exploit its
flexibility.
Even very simple variants on the standard particle filter can
significantly improve performance.
Three particular possibilities were discussed here:
I
iAPF
I
I
I
I
Block-Tempering
I
I
I
The iAPF can improve performance substantially in some
settings.
Extending the extent of its applicability is ongoing work.
In principle any function approximation scheme can be
employed: provided that ftψ can be sampled from and gtψ
evaluated.
Can significantly improve performance with misspecified
models.
Requires MCMC kernels and introduces 2 tuning parameters
Hierarchical Particle Filters
I
I
Dramatically reduce the need for resampling.
Computationally rather costly.
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