On Particle Gibbs Samplingmwl25/mcmski/slides/ss_mcmski14.pdf · 0.6 0.8 ACF X 0 multinomial...
Transcript of On Particle Gibbs Samplingmwl25/mcmski/slides/ss_mcmski14.pdf · 0.6 0.8 ACF X 0 multinomial...
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On Particle Gibbs Sampling
Sumeetpal S. SinghCambridge University Engineering Department
joint work: N. Chopin (CREST-ENSAE)
MCMSKICharmonix 7 January 2014
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State-space Model
Running example (Yu & Meng 2011):
X0 ∼ N(µ, σ2), Xt+1−µ = ρ(Xt−µ)+σεt , εt ∼i.i .d . N(0, 1)
Yt |Xt = xt ∼ Poisson(ext )
In general:
Xt |Xt−1 = xt−1 ∼ mθ(xt−1, xt)dxt
Yt |Xt = xt ∼ gθ(xt , yt)dyt
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State-space Model
Running example (Yu & Meng 2011):
X0 ∼ N(µ, σ2), Xt+1−µ = ρ(Xt−µ)+σεt , εt ∼i.i .d . N(0, 1)
Yt |Xt = xt ∼ Poisson(ext )
In general:
Xt |Xt−1 = xt−1 ∼ mθ(xt−1, xt)dxt
Yt |Xt = xt ∼ gθ(xt , yt)dyt
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Inference Objective
The posterior: p(θ, x0:T |y0:T ), θ = (µ, ρ, σ)
The Gibbs sampler: (θ, x0:T )→ (θ′, x ′0:T )
σ′|(x0:T , µ, ρ) ∼ Gamma (· · ·)...
µ′|(x0:T , σ′, ρ′) ∼ Normal (· · ·)
x ′0:T |(σ′, µ′, ρ′)
One at a time: xi |(x ′0:i−1, xi+1:T , σ′, µ′, ρ′)
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Inference Objective
The posterior: p(θ, x0:T |y0:T ), θ = (µ, ρ, σ)
The Gibbs sampler: (θ, x0:T )→ (θ′, x ′0:T )
σ′|(x0:T , µ, ρ) ∼ Gamma (· · ·)...
µ′|(x0:T , σ′, ρ′) ∼ Normal (· · ·)
x ′0:T |(σ′, µ′, ρ′)
One at a time: xi |(x ′0:i−1, xi+1:T , σ′, µ′, ρ′)
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Inference Objective
The posterior: p(θ, x0:T |y0:T ), θ = (µ, ρ, σ)
The Gibbs sampler: (θ, x0:T )→ (θ′, x ′0:T )
σ′|(x0:T , µ, ρ) ∼ Gamma (· · ·)...
µ′|(x0:T , σ′, ρ′) ∼ Normal (· · ·)
x ′0:T |(σ′, µ′, ρ′)
One at a time: xi |(x ′0:i−1, xi+1:T , σ′, µ′, ρ′)
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Particle Gibbs kernel (Andrieu, Doucet and Holenstein,2010):
Replace Gibbs step for x0:T with
X ′0:T |(θ′, x0:T ) ∼ Pθ′,N(x0:T , dx′0:T )
Invariant measure p(x0:T |θ′, y0:T )
The kernel is a randomly chosen output of a particle filter, Nparticles, one fixed to x0:T
Meant to emulate the Gibbs step (change the wholetrajectory)
Typically:
x ′0:T 6= x0:T but x ′i = xi for small i .
Increasing N fixes this.
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Particle Gibbs kernel (Andrieu, Doucet and Holenstein,2010):
Replace Gibbs step for x0:T with
X ′0:T |(θ′, x0:T ) ∼ Pθ′,N(x0:T , dx′0:T )
Invariant measure p(x0:T |θ′, y0:T )
The kernel is a randomly chosen output of a particle filter, Nparticles, one fixed to x0:T
Meant to emulate the Gibbs step (change the wholetrajectory)
Typically:
x ′0:T 6= x0:T but x ′i = xi for small i .
Increasing N fixes this.
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Running Particle Gibbs
Sampling p(θ, x0:399|y0:399)
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Statistic: counting proportion x ′i 6= xi for i = 0, . . . , 399
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Running Particle Gibbs
Sampling p(θ, x0:400|y0:400)
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ACF
X0
multinomialresidualsystematicmulti+BS
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X399
multinomialresidualsystematicmulti+BS
Statistic: autocorrellation X0, X399 (200 particles)
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Particle filter
Input: {x i0:t}Ni=1 ≈ p(x0:t |θ, y0:t)
Output: {x i0:t+1}Ni=1 ≈ p(x0:t+1|θ, y0:t+1)
Append:(x i0:t ,Xit+1) where X i
t+1 ∼ mθ(x it , dxt+1)
Weight: x i0:t+1 gets weight w it+1 = gθ(x it+1, yt+1)
Output: Approximation of p(x0:t+1|θ, y0:t+1) are N independentsamples from
N∑i=1
w it+1δx i0:t+1
(dx ′0:t+1)
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Particle filter
Input: {x i0:t}Ni=1 ≈ p(x0:t |θ, y0:t)
Output: {x i0:t+1}Ni=1 ≈ p(x0:t+1|θ, y0:t+1)
Append:(x i0:t ,Xit+1) where X i
t+1 ∼ mθ(x it , dxt+1)
Weight: x i0:t+1 gets weight w it+1 = gθ(x it+1, yt+1)
Output: Approximation of p(x0:t+1|θ, y0:t+1) are N independentsamples from
N∑i=1
w it+1δx i0:t+1
(dx ′0:t+1)
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Sampling from Pθ,N(x?0:T , dx0:T )
Idea: N particle system but force x10:T = x?0:T
Intermediate step tInput: {x i0:t}Ni=1 with x1
0:t = x?0:t
Output: {x i0:t+1}Ni=1 with x10:t+1 = x?0:t+1
Append particles i = 2, . . . ,N:
(x0:t ,Xit+1) where X i
t+1 ∼ mθ(x it , dxt+1)
Weight: x i0:t+1 gets weight w it+1 = gθ(x it+1, yt+1)
Output x10:t+1 = x?0:t+1 and N − 1 independent samples from
w?t+1δx?0:t+1
(dx ′0:t+1) +N∑i=2
w it+1δx i0:t+1
(dx ′0:t+1)
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Sampling from Pθ,N(x?0:T , dx0:T )
Idea: N particle system but force x10:T = x?0:T
Intermediate step tInput: {x i0:t}Ni=1 with x1
0:t = x?0:t
Output: {x i0:t+1}Ni=1 with x10:t+1 = x?0:t+1
Append particles i = 2, . . . ,N:
(x0:t ,Xit+1) where X i
t+1 ∼ mθ(x it , dxt+1)
Weight: x i0:t+1 gets weight w it+1 = gθ(x it+1, yt+1)
Output x10:t+1 = x?0:t+1 and N − 1 independent samples from
w?t+1δx?0:t+1
(dx ′0:t+1) +N∑i=2
w it+1δx i0:t+1
(dx ′0:t+1)
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Sampling from Pθ,N(x?0:T , dx0:T )
Idea: N particle system but force x10:T = x?0:T
Intermediate step tInput: {x i0:t}Ni=1 with x1
0:t = x?0:t
Output: {x i0:t+1}Ni=1 with x10:t+1 = x?0:t+1
Append particles i = 2, . . . ,N:
(x0:t ,Xit+1) where X i
t+1 ∼ mθ(x it , dxt+1)
Weight: x i0:t+1 gets weight w it+1 = gθ(x it+1, yt+1)
Output x10:t+1 = x?0:t+1 and N − 1 independent samples from
w?t+1δx?0:t+1
(dx ′0:t+1) +N∑i=2
w it+1δx i0:t+1
(dx ′0:t+1)
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Sampling from Pθ,N(x?0:T , dx0:T )
Idea: N particle system but force x10:T = x?0:T
Intermediate step tInput: {x i0:t}Ni=1 with x1
0:t = x?0:t
Output: {x i0:t+1}Ni=1 with x10:t+1 = x?0:t+1
Append particles i = 2, . . . ,N:
(x0:t ,Xit+1) where X i
t+1 ∼ mθ(x it , dxt+1)
Weight: x i0:t+1 gets weight w it+1 = gθ(x it+1, yt+1)
Output x10:t+1 = x?0:t+1 and N − 1 independent samples from
w?t+1δx?0:t+1
(dx ′0:t+1) +N∑i=2
w it+1δx i0:t+1
(dx ′0:t+1)
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cPF example
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cPF example
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cPF example
0 1 2 3 4 5 6 7 8 9 10−2
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Uniform Ergodicity
Assumption
gθ(xt , yt) ≤ Gt,θ (density upper bounded)
Predicted density lower bounded:∫mθ(dx0)gθ(x0, y0) ≥ 1
G0,θ,
∫mθ(xt−1, dxt)gθ(xt , yt) ≥
1
Gt,θ
For any θ and ε > 0, for N large enough,
|(PN,θϕ) (x0:T )− (PN,θϕ) (x̌0:T )| ≤ ε
for all x0:T , x̌0:T , and ϕ : XT+1 → [−1, 1]
⇒ Large N gives samples arbitrarily close to target p(x0:T |θ, y0:T )⇒ Uniform ergodicity
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Proof Idea
Use coupling: define (X ?0:T , X̌
?0:T ) such that
Law(X ?0:T ) = PN,θ(x0:T , dx
?0:T ) Law(X̌ ?
0:T ) = PN,θ(x̌0:T , dx̌?0:T )
IfP(X ?
0:T 6= X̌ ?0:T ) ≤ ε
then
PNT (x0:T ,A)− PN
T (x̌0:T ,A) = E{(
IA(X ?0:T )− IA(X̌ ?
0:T ))I{X0:T 6=X̌0:T }
}≤ P
(X ?
0:T 6= X̌ ?0:T
)≤ ε
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Coupling cPFs
Aim: Coupling outputs of PN,θ(x0:T , dx?0:T ) and PN,θ(x̌0:T , dx̌
?0:T )
If cPF outputs {x i0:t}Ni=1 and {x̌ i0:t}Ni=1 satisfy
x i0:t = x̌ i0:t for i ∈ Ct
where C0 = {2, . . . ,N}Same holds after append move: (x i0:t , x
it+1) = (x̌ i0:t , x̌
it+1) for
i ∈ Ct
Resampling move: select particles in Ct for survival if possible
Coupling probability determined by law of CT
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Backward Sampling
N. Whiteley (RSS discussion of PMCMC) suggested an extrabackward step for cPF that tries to modify (recursively, backwardin time) the ancestry of the selected trajectory.
There is a forward “version” by Lindsten and Schon (2012)
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Idea of Backward Sampling
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Running Backward Sampling
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upda
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Left Statistic: counting proportion x ′i 6= xi for i = 0, . . . , 399
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Backward Sampling
Success depends on state transition law mθ(xt , dxt+1)
The cPF kernel Pθ,N with a BS step dominates the no BS versionin asymptotic efficiency
i.e. gives rise to a CLT with a smaller asymptotic variance (Idea:self-adjoint operator + lag-1 domination; see Tierney (1998), Mira& Geyer (1999) )
The cPF kernel geometrically ergodic ⇒ cPF kernel with BS isgeometrically ergodic
(Idea: both kernels are positive operators)
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Final Remarks
Established uniform ergodicity of cPF using coupling
Dependance on N and T not given although in practise Nshould scale linearly with T– now proved by (Andrieu, Lee, Vihola) and (Douc, Lindsten,Moulines)
Whiteley’s BS is better: in asymptotic efficiency and inheritsgeometric ergodicity
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