Graphical model inference: Sequential Monte Carlo meets - - PowerPoint PPT Presentation

Graphical model inference: Sequential Monte Carlo meets deterministic approximations

Fredrik Lindsten (Linköping University and Uppsala University) Jouni Helske (Linköping University) Matti Vihola (University of Jyväskylä)

Approximate Bayesian inference

Deterministic methods

Message passing

f x

Laplace’s method Variational inference

q⋆ π q0

Monte Carlo methods

Markov chain Monte Carlo Sequential Monte Carlo

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Approximate Bayesian inference

Deterministic methods

Message passing

f x

Laplace’s method Variational inference

q⋆ π q0

Monte Carlo methods

Markov chain Monte Carlo Sequential Monte Carlo

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VSMC VMCMC · · ·

Probabilistic graphical models

We consider inference in factor graphs with joint distribution π(x1:T) = 1 Z ∏

j∈F

fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Task:

• Compute expectations w.r.t. π(x1:T).
• Compute the normalizing constant Z.

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Sequential Monte Carlo

Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via sequential graph decompositions:

Christian A. Naesseth, Fredrik Lindsten and Thomas B. Schön. Sequential Monte Carlo methods for graphical

• models. Advances in Neural Information Processing Systems 27, December, 2014.

Define intermediate SMC targets:

t x1 t j

t fj x j

f1 x1 f2 x2 x3 f3 f4

Iteration t 1

1 x1

f1 x1 f2 x2 x3 f3 f4

Iteration t 2

2 x1 2

f1 x1 f2 x2 x3 f3 f4

Iteration t 3

3 x1 3

x1 3

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Sequential Monte Carlo

Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via sequential graph decompositions:

Christian A. Naesseth, Fredrik Lindsten and Thomas B. Schön. Sequential Monte Carlo methods for graphical

• models. Advances in Neural Information Processing Systems 27, December, 2014.

Define intermediate SMC targets: γt(x1:t) = ∏

j∈Ft fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t = 1

γ1(x1)

f1 x1 f2 x2 x3 f3 f4

Iteration t 2

2 x1 2

f1 x1 f2 x2 x3 f3 f4

Iteration t 3

3 x1 3

x1 3

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Sequential Monte Carlo

Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via sequential graph decompositions:

Christian A. Naesseth, Fredrik Lindsten and Thomas B. Schön. Sequential Monte Carlo methods for graphical

• models. Advances in Neural Information Processing Systems 27, December, 2014.

Define intermediate SMC targets: γt(x1:t) = ∏

j∈Ft fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t = 1

γ1(x1)

f1 x1 f2 x2 x3 f3 f4

Iteration t = 2

γ2(x1:2)

f1 x1 f2 x2 x3 f3 f4

Iteration t 3

3 x1 3

x1 3

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Sequential Monte Carlo

Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via sequential graph decompositions:

Christian A. Naesseth, Fredrik Lindsten and Thomas B. Schön. Sequential Monte Carlo methods for graphical

• models. Advances in Neural Information Processing Systems 27, December, 2014.

Define intermediate SMC targets: γt(x1:t) = ∏

j∈Ft fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t = 1

γ1(x1)

f1 x1 f2 x2 x3 f3 f4

Iteration t = 2

γ2(x1:2)

f1 x1 f2 x2 x3 f3 f4

Iteration t = 3

γ3(x1:3) ∝ π(x1:3)

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Twisted SMC

Dependencies on “future variables” are not taken into account! Twisted intermediate targets: γψ

t (x1:t) := ψt(x1:t)γt(x1:t) = ψt(x1:t)

∏

j∈Ft

fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t 1

1 x1

f1 x1 f2 x2 x3 f3 f4

Iteration t 2

2 x1 2

f1 x1 f2 x2 x3 f3 f4

Iteration t 3

3 x1 3

x1 3

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Twisted SMC

Dependencies on “future variables” are not taken into account! Twisted intermediate targets: γψ

t (x1:t) := ψt(x1:t)γt(x1:t) = ψt(x1:t)

∏

j∈Ft

fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t = 1

γψ

1 (x1)

f1 x1 f2 x2 x3 f3 f4

Iteration t 2

2 x1 2

f1 x1 f2 x2 x3 f3 f4

Iteration t 3

3 x1 3

x1 3

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Twisted SMC

Dependencies on “future variables” are not taken into account! Twisted intermediate targets: γψ

t (x1:t) := ψt(x1:t)γt(x1:t) = ψt(x1:t)

∏

j∈Ft

fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t = 1

γψ

1 (x1)

f1 x1 f2 x2 x3 f3 f4

Iteration t = 2

γψ

2 (x1:2)

f1 x1 f2 x2 x3 f3 f4

Iteration t 3

3 x1 3

x1 3

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Twisted SMC

Dependencies on “future variables” are not taken into account! Twisted intermediate targets: γψ

t (x1:t) := ψt(x1:t)γt(x1:t) = ψt(x1:t)

∏

j∈Ft

fj(xIj).

f1 x1 f2 x2 x3 f3 f4

Iteration t = 1

γψ

1 (x1)

f1 x1 f2 x2 x3 f3 f4

Iteration t = 2

γψ

2 (x1:2)

f1 x1 f2 x2 x3 f3 f4

Iteration t = 3

γψ

3 (x1:3) ∝ π(x1:3) 4/6

How do we choose the twisting functions?

Proposition (Optimal twisting). With ψ∗

t (x1:t) =

∫ ∏

j∈F\Ft

fj(xIj)dxt+1:T, the SMC algorithm outputs i.i.d. draws from π and the normalizing constant estimate is exact; Z = Z w.p.1. Optimal twisting functions are intractable, but:

• t

t can be computed by various deterministic inference methods

• Sub-optimality only affects efficiency, not consistency or unbiasedness
• Can be seen as a bias post-correction

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How do we choose the twisting functions?

Proposition (Optimal twisting). With ψ∗

t (x1:t) =

∫ ∏

j∈F\Ft

fj(xIj)dxt+1:T, the SMC algorithm outputs i.i.d. draws from π and the normalizing constant estimate is exact; Z = Z w.p.1. Optimal twisting functions are intractable, but:

• ψt ≈ ψ∗

t can be computed by various deterministic inference methods

• Sub-optimality only affects efficiency, not consistency or unbiasedness
• Can be seen as a bias post-correction

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Twisting functions via deterministic approximations

Loopy Belief Propagation ex) Square lattice Ising model Expectation Propagation ex) Topic model likeli- hood evaluation

x1 w1 xT wT

Laplace Approximation ex) Gaussian Markov ran- dom field

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Twisting functions via deterministic approximations

Loopy Belief Propagation ex) Square lattice Ising model Expectation Propagation ex) Topic model likeli- hood evaluation

θ x1 w1 xT wT

· · · · · · Laplace Approximation ex) Gaussian Markov ran- dom field

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Twisting functions via deterministic approximations

Loopy Belief Propagation ex) Square lattice Ising model Expectation Propagation ex) Topic model likeli- hood evaluation

θ x1 w1 xT wT

· · · · · · Laplace Approximation ex) Gaussian Markov ran- dom field

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Thank you for listening! Come see the poster: #51

Code available at:

• github.com/freli005/smc-pgm-twist
• github.com/helske/particlefield

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