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 0 Monte Carlo methods Markov chain Monte Carlo Sequential Monte Carlo 1/6 π q ⋆

Approximate Bayesian inference VSMC Message passing f x Laplace’s method Variational inference VMCMC Deterministic methods q 0 Monte Carlo methods Markov chain Monte Carlo Sequential Monte Carlo 1/6 π · · · q ⋆

Probabilistic graphical models f 2 • Compute the normalizing constant Z . Task: f 4 f 3 x 3 x 2 x 1 We consider inference in factor graphs with f 1 Z joint distribution 2/6 ∏ π ( x 1 : T ) = 1 f j ( x I j ) . j ∈F • Compute expectations w.r.t. π ( x 1 : T ) .

2 x 1 2 3 x 1 3 t x 1 t t f j x 1 x 1 Sequential Monte Carlo x 1 f 3 f 4 Iteration t 2 f 1 x 2 f 2 x 2 x 3 f 3 f 4 Iteration t 3 x 1 3 x 3 x 1 f 2 x 1 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: j j f 1 f 2 Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via x 2 x 3 f 3 f 4 Iteration t 1 f 1 3/6

2 x 1 2 3 x 1 3 Sequential Monte Carlo f 2 f 4 Iteration t 2 f 1 x 1 x 2 x 3 x 3 f 3 f 4 Iteration t 3 x 1 3 f 3 x 2 Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via f 2 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. f 1 x 1 f 2 x 2 x 3 f 3 f 4 f 1 x 1 3/6 Define intermediate SMC targets: γ t ( x 1 : t ) = ∏ j ∈F t f j ( x I j ) . Iteration t = 1 γ 1 ( x 1 )

3 x 1 3 Sequential Monte Carlo f 2 x 3 f 3 f 4 f 1 x 1 x 3 x 2 Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via f 3 f 4 Iteration t 3 x 1 3 x 2 f 2 x 1 f 1 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. f 1 x 1 f 2 x 2 x 3 f 3 f 4 3/6 Define intermediate SMC targets: γ t ( x 1 : t ) = ∏ j ∈F t f j ( x I j ) . Iteration t = 1 Iteration t = 2 γ 1 ( x 1 ) γ 2 ( x 1 : 2 )

Sequential Monte Carlo f 1 f 4 f 3 x 3 x 2 f 2 x 1 f 1 f 4 f 3 x 3 x 2 f 2 Sequential Monte Carlo (SMC) can be used for probabilistic graphical model inference via x 1 3/6 f 1 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. x 1 f 2 x 2 x 3 f 3 f 4 Define intermediate SMC targets: γ t ( x 1 : t ) = ∏ j ∈F t f j ( x I j ) . Iteration t = 1 Iteration t = 2 Iteration t = 3 γ 1 ( x 1 ) γ 2 ( x 1 : 2 ) γ 3 ( x 1 : 3 ) ∝ π ( x 1 : 3 )

2 x 1 2 3 x 1 3 1 x 1 Twisted SMC x 1 f 3 f 4 Iteration t 2 f 1 f 2 x 2 x 2 x 3 f 3 f 4 Iteration t 3 x 1 3 x 3 f 2 Dependencies on “future variables” are not taken into account! x 2 Twisted intermediate targets: f 1 x 1 x 1 f 2 x 3 f 3 f 4 Iteration t 1 f 1 4/6 ∏ γ ψ t ( x 1 : t ) := ψ t ( x 1 : t ) γ t ( x 1 : t ) = ψ t ( x 1 : t ) f j ( x I j ) . j ∈F t

2 x 1 2 3 x 1 3 Twisted SMC x 1 f 3 f 4 Iteration t 2 f 1 x 2 f 2 x 2 x 3 f 3 f 4 Iteration t 3 x 1 3 x 3 f 2 Dependencies on “future variables” are not taken into account! x 3 Twisted intermediate targets: f 1 x 1 x 1 x 2 f 2 f 3 f 4 f 1 4/6 ∏ γ ψ t ( x 1 : t ) := ψ t ( x 1 : t ) γ t ( x 1 : t ) = ψ t ( x 1 : t ) f j ( x I j ) . j ∈F t Iteration t = 1 γ ψ 1 ( x 1 )

3 x 1 3 Twisted SMC f 2 x 2 x 3 f 3 f 4 f 1 x 1 x 2 x 1 x 3 f 3 f 4 Iteration t 3 x 1 3 Dependencies on “future variables” are not taken into account! f 2 f 1 x 3 Twisted intermediate targets: f 1 x 1 f 2 x 2 f 3 f 4 4/6 ∏ γ ψ t ( x 1 : t ) := ψ t ( x 1 : t ) γ t ( x 1 : t ) = ψ t ( x 1 : t ) f j ( x I j ) . j ∈F t Iteration t = 1 Iteration t = 2 γ ψ γ ψ 1 ( x 1 ) 2 ( x 1 : 2 )

Twisted SMC f 1 f 4 f 3 x 3 x 2 f 2 x 1 f 1 f 4 f 3 x 3 x 2 f 2 Dependencies on “future variables” are not taken into account! x 1 4/6 x 2 f 1 Twisted intermediate targets: f 2 x 1 x 3 f 3 f 4 ∏ γ ψ t ( x 1 : t ) := ψ t ( x 1 : t ) γ t ( x 1 : t ) = ψ t ( x 1 : t ) f j ( x I j ) . j ∈F t Iteration t = 1 Iteration t = 2 Iteration t = 3 γ ψ γ ψ γ ψ 1 ( x 1 ) 2 ( x 1 : 2 ) 3 ( x 1 : 3 ) ∝ π ( x 1 : 3 )

t can be computed by various deterministic inference methods How do we choose the twisting functions? Proposition (Optimal twisting). With • Can be seen as a bias post-correction • Sub-optimality only affects efficiency, not consistency or unbiasedness t • Optimal twisting functions are intractable, but: 5/6 ∫ ∏ ψ ∗ t ( x 1 : t ) = f j ( x I j ) dx t + 1 : T , j ∈F\F t the SMC algorithm outputs i.i.d. draws from π and the normalizing constant estimate is exact; � Z = Z w.p.1.

How do we choose the twisting functions? Proposition (Optimal twisting). With • Can be seen as a bias post-correction • Sub-optimality only affects efficiency, not consistency or unbiasedness Optimal twisting functions are intractable, but: 5/6 ∫ ∏ ψ ∗ t ( x 1 : t ) = f j ( x I j ) dx t + 1 : T , j ∈F\F t the SMC algorithm outputs i.i.d. draws from π and the normalizing constant estimate is exact; � Z = Z w.p.1. • ψ t ≈ ψ ∗ t can be computed by various deterministic inference methods

Twisting functions via deterministic approximations x 1 dom field ex) Gaussian Markov ran- Laplace Approximation w T x T w 1 hood evaluation Loopy Belief Propagation likeli- model Topic ex) Expectation Propagation model ex) Square lattice Ising 6/6

Twisting functions via deterministic approximations Loopy Belief Propagation dom field ex) Gaussian Markov ran- Laplace Approximation w T x T w 1 x 1 6/6 hood evaluation likeli- model Topic ex) Expectation Propagation model ex) Square lattice Ising θ · · · · · ·

Twisting functions via deterministic approximations Loopy Belief Propagation dom field ex) Gaussian Markov ran- Laplace Approximation w T x T w 1 x 1 6/6 hood evaluation likeli- model Topic ex) Expectation Propagation model ex) Square lattice Ising θ · · · · · ·

Thank you for listening! Come see the poster: #51 Code available at: - github.com/freli005/smc-pgm-twist - github.com/helske/particlefield 6/6