Particle Flow Bayes Rule Xinshi Chen 1* , Hanjun Dai 1* , Le - - PowerPoint PPT Presentation

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Nov 13, 2022 128 likes •239 views

Particle Flow Bayes Rule Xinshi Chen 1* , Hanjun Dai 1* , Le Song 1,2 1 Georgia Tech, 2 Ant Financial (*equal contribution) ICML 2019 Sequential Bayesian Inference 1. Prior distribution () # $ %

SLIDE 1

Particle Flow Bayes’ Rule

Xinshi Chen1*, Hanjun Dai1*, Le Song1,2

1Georgia Tech, 2Ant Financial

(*equal contribution) ICML 2019

SLIDE 2

Sequential Bayesian Inference

𝑦 𝑝# 𝑝$ 𝑝%

𝑞 𝑦 𝑝# 𝑞 𝑦 𝑝#:$ 𝑞 𝑦 𝑝#:% … …… …… prior 𝜌(𝑦) 𝑝# 𝑝$ 𝑝%

𝑞 𝑦 𝑝#:% ∝ 𝑞 𝑦 𝑝#:%,# 𝑞 𝑝% 𝑦

updated posterior current posterior Likelihood Or prior

1. Prior distribution 𝜌(𝑦) 2. Likelihood function 𝑞(𝑝|𝑦) 3. Observations 𝑝#, 𝑝$, … , 𝑝% arrive sequentially Need efficient online update! Sequential Monte Carlo:

𝑂 particles 𝒴2 = 𝑦2

#, … , 𝑦2 4 from prior 𝜌(𝑦)

Reweight the particles using likelihood
Particle degeneracy problem

(Doucet et al., 2001)

SLIDE 3

Our Approach: Particle Flow

𝑶 particles 𝒴2 = {𝑦2

#, … , 𝑦2 4}, from prior 𝝆(𝒚)

Move particles through an ordinary differential equation (ODE)

⟹ solution 𝑦#

; = 𝑦(𝑈)

𝑒𝑦 𝑒𝑢 = 𝑔 𝒴2, 𝑝#, 𝑦 𝑢 , 𝑢 𝑦 0 = 𝑦2

;

and

𝒴2 = {𝑦2

#, … , 𝑦2 4}

𝒴# = {𝑦#

#, … , 𝑦# 4}

𝒴$ = {𝑦$

#, … , 𝑦$ 4}

2 B

𝑔 𝒴2, 𝑝#, 𝑦(𝑢) 𝑒𝑢 A

2 B

𝑔 𝒴#, 𝑝$, 𝑦(𝑢) 𝑒𝑢 A

2 B

𝑔 𝒴$, 𝑝C, 𝑦(𝑢) 𝑒𝑢

𝑦 𝑝# 𝑝$ 𝑝%

𝑞 𝑦 𝑝# 𝑞 𝑦 𝑝#:$ 𝑞 𝑦 𝑝#:% … …… prior 𝜌(𝑦) 𝑝# 𝑝$ 𝑝% Does a unified flow velocity 𝑔 exist? Does Particle Flow Bayes’ Rule (PFBR) exist?

SLIDE 4

Our Approach: Particle Flow

Move particles to next posterior through an ordinary differential equation (ODE)

⟹ solution 𝑦#

; = 𝑦(𝑈)

𝑒𝑦 𝑒𝑢 = 𝑔 𝒴2, 𝑝#, 𝑦 𝑢 , 𝑢 𝑦 0 = 𝑦2

;

and

𝒴2 = {𝑦2

#, … , 𝑦2 4}

𝒴# = {𝑦#

#, … , 𝑦# 4}

𝒴$ = {𝑦$

#, … , 𝑦$ 4}

2 B

𝑔 𝒴2, 𝑝#, 𝑦(𝑢) 𝑒𝑢 A

2 B

𝑔 𝒴#, 𝑝$, 𝑦(𝑢) 𝑒𝑢 A

2 B

𝑔 𝒴$, 𝑝C, 𝑦(𝑢) 𝑒𝑢

𝑦 𝑝# 𝑝$ 𝑝%

𝑞 𝑦 𝑝# 𝑞 𝑦 𝑝#:$ 𝑞 𝑦 𝑝#:% … …… prior 𝜌(𝑦) 𝑝# 𝑝$ 𝑝% Does a unified flow velocity 𝑔 exist? Does Particle Flow Bayes’ Rule (PFBR) exist?

Yes!!! 𝑔: = 𝛼

E log 𝜌 𝑦 𝑞(𝑝|𝑦) − 𝑥∗ 𝜌 𝑦 , 𝑞 𝑝|𝑦 , 𝑦, 𝑢

SLIDE 5

Existence of Particle Flow Bayes’ Rule

𝑒𝑦 𝑢 = 𝛼

E log 𝜌 𝑦 𝑞(𝑝|𝑦) 𝑒𝑢 +

2 𝒆𝒙 𝒖

Langevin dynamics

✓ density 𝒓 𝒚, 𝒖 converges to posterior 𝒒 𝒚|𝒑 ✘ stochastic flow

Fokker-Planck Equation + Continuity Equation deterministic, closed-loop

𝑒𝑦 𝑢 = 𝛼

E log 𝜌 𝑦 𝑞(𝑝|𝑦) − ∇E log 𝒓(𝒚, 𝒖) 𝑒𝑢

✓ density 𝒓 𝒚, 𝒖 converges to posterior 𝒒 𝒚|𝒑 ✓ deterministic flow ✘ closed-loop flow: depends on 𝒓(𝒚, 𝒖)

Optimal control theory: closed-loop to open loop deterministic, open-loop

𝑒𝑦 𝑢 = 𝛼

E log 𝜌 𝑦 𝑞(𝑝|𝑦) − 𝑥∗ 𝜌 𝑦 , 𝑞 𝑝|𝑦 , 𝑦, 𝑢 𝑒𝑢

✓ density 𝒓 𝒚, 𝒖 converges to posterior 𝒒 𝒚|𝒑 ✓ deterministic flow ✓ open-loop flow

SLIDE 6

Parameterization

The unified flow velocity is in form of:

𝒈(𝝆 𝒚 , 𝒒 𝒑 𝒚 , 𝒚, 𝒖): = 𝛼

E log 𝜌 𝑦 𝑞(𝑝|𝑦) − 𝑥∗ 𝜌 𝑦 , 𝑞 𝑝|𝑦 , 𝑦, 𝑢

𝒆𝒚 𝒆𝒖 = 𝒈 𝓨, 𝒑, 𝒚 𝒖 , 𝒖 ≔ 𝒊 𝟐 𝑶 Z

𝒐\𝟐 𝑶

𝝔 𝒚𝒐 , 𝒑, 𝒚 𝒖 , 𝒖

Deep set 𝒊 neural networks

SLIDE 7

Experiment 1: Multimodal Posterior

Gaussian Mixture Model

prior 𝑦#, 𝑦$ ∼ 𝒪 0,1
bservations o|𝑦#, 𝑦$ ∼ #

$ 𝒪 𝑦#, 1 + # $ 𝒪(𝑦# + 𝑦$, 1)

With 𝑦#, 𝑦$ = (1, −2), the resulting posterior 𝒒(𝒚|𝒑𝟐, … , 𝒑𝒏) will have two modes:

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −3 −2 −1 1 2 3 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −3 −2 −1 1 2 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 x 10 −3 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −5 −4 −3 −2 −1 1 2 3 4 5 2 4 6 8 10 12 14 x 10 −3 −2 −1.5 −1 −0.5 0.5 1 1.5 2 −3 −2 −1 1 2 3 1 2 3 4 5 6 7 8 9 10 x 10 −3

(a) True posterior (b) Stochastic Variational Inference (c) Stochastic Gradient Langevin Dynamics (d) Gibbs Sampling (e) One-pass SMC

−2 −1.5 −1 −0.5 0.5 1 1.5 2 −3 −2 −1 1 2 3 1 2 3 x 10 −4

SLIDE 8

Experiment 1: Multimodal Posterior

PFBR vs one-pass SMC

Visualization of the evolution of posterior density from left to right.

SLIDE 9

Experiment 2: Efficiency in #Particles

Comparison to SMC and ASMC (Autoencoding SMC, Filtering Variational Objectives, and Variational SMC) (Le et al., 2018; Maddison et al., 2017; Naesseth et al., 2018). Our Approach

SLIDE 10

Particle Flow Bayes’ Rule

Sequential Bayesian Inference

Our Approach: Particle Flow

Our Approach: Particle Flow

Existence of Particle Flow Bayes’ Rule

Parameterization

𝒈(𝝆 𝒚 , 𝒒 𝒑 𝒚 , 𝒚, 𝒖): = 𝛼

𝒆𝒚 𝒆𝒖 = 𝒈 𝓨, 𝒑, 𝒚 𝒖 , 𝒖 ≔ 𝒊 𝟐 𝑶 Z

𝝔 𝒚𝒐 , 𝒑, 𝒚 𝒖 , 𝒖

Experiment 1: Multimodal Posterior

Experiment 1: Multimodal Posterior

Experiment 2: Efficiency in #Particles

Thanks! Poster: Pacific Ballroom #218, Tue, 06:30 PM Contact: xinshi.chen@gatech.edu