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

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 𝑦 1. Prior distribution 𝜌(𝑦) 𝑝 # 𝑝 $ …… 𝑝 % …… 2. Likelihood function 𝑞(𝑝|𝑦) … 𝑞 𝑦 𝑝 # 𝑞 𝑦 𝑝 #:$ 𝑞 𝑦 𝑝 #:% prior 𝜌(𝑦) 3. Observations 𝑝 # , 𝑝 $ , … , 𝑝 % 𝑝 # 𝑝 $ 𝑝 % arrive sequentially 𝑞 𝑦 𝑝 #:% ∝ 𝑞 𝑦 𝑝 #:%,# 𝑞 𝑝 % 𝑦 Need efficient online update! updated posterior current posterior Likelihood Or prior Sequential Monte Carlo: 4 from prior 𝜌(𝑦) # , … , 𝑦 2 • 𝑂 particles 𝒴 2 = 𝑦 2 Reweight the particles using likelihood • Particle degeneracy problem • (Doucet et al., 2001)

Our Approach: Particle Flow # , … , 𝑦 2 4 } , from prior 𝝆(𝒚) • 𝑶 particles 𝒴 2 = {𝑦 2 • Move particles through an ordinary differential equation (ODE) 𝑒𝑦 ; 𝑦 0 = 𝑦 2 and 𝑒𝑢 = 𝑔 𝒴 2 , 𝑝 # , 𝑦 𝑢 , 𝑢 ; = 𝑦(𝑈) ⟹ solution 𝑦 # 𝑦 𝑝 # 𝑝 $ 𝑝 % …… Does a unified flow velocity 𝑔 exist? 𝑞 𝑦 𝑝 #:$ 𝑞 𝑦 𝑝 # prior 𝜌(𝑦) 𝑞 𝑦 𝑝 #:% Does P article F low B ayes’ R ule (PFBR) exist? … 𝑝 # 𝑝 $ 𝑝 % # , … , 𝑦 2 4 } 𝒴 2 = {𝑦 2 # , … , 𝑦 # 4 } 𝒴 # = {𝑦 # # , … , 𝑦 $ 4 } 𝒴 $ = {𝑦 $ B B B A 𝑔 𝒴 # , 𝑝 $ , 𝑦(𝑢) 𝑒𝑢 A 𝑔 𝒴 2 , 𝑝 # , 𝑦(𝑢) 𝑒𝑢 A 𝑔 𝒴 $ , 𝑝 C , 𝑦(𝑢) 𝑒𝑢 2 2 2

Our Approach: Particle Flow • Move particles to next posterior through an ordinary differential equation (ODE) 𝑒𝑦 ; 𝑦 0 = 𝑦 2 and 𝑒𝑢 = 𝑔 𝒴 2 , 𝑝 # , 𝑦 𝑢 , 𝑢 ; = 𝑦(𝑈) ⟹ solution 𝑦 # 𝑦 𝑝 # 𝑝 $ 𝑝 % …… Does a unified flow velocity 𝑔 exist? 𝑞 𝑦 𝑝 #:$ 𝑞 𝑦 𝑝 # prior 𝜌(𝑦) 𝑞 𝑦 𝑝 #:% Does P article F low B ayes’ R ule (PFBR) exist? … 𝑝 # 𝑝 $ 𝑝 % # , … , 𝑦 2 4 } 𝒴 2 = {𝑦 2 Yes!!! # , … , 𝑦 # 4 } 𝒴 # = {𝑦 # # , … , 𝑦 $ 4 } 𝒴 $ = {𝑦 $ E log 𝜌 𝑦 𝑞(𝑝|𝑦) − 𝑥 ∗ 𝜌 𝑦 , 𝑞 𝑝|𝑦 , 𝑦, 𝑢 𝑔: = 𝛼 B B B A 𝑔 𝒴 # , 𝑝 $ , 𝑦(𝑢) 𝑒𝑢 A 𝑔 𝒴 2 , 𝑝 # , 𝑦(𝑢) 𝑒𝑢 A 𝑔 𝒴 $ , 𝑝 C , 𝑦(𝑢) 𝑒𝑢 2 2 2

Existence of Particle Flow Bayes’ Rule Langevin dynamics ✓ density 𝒓 𝒚, 𝒖 converges to posterior 𝒒 𝒚|𝒑 ✘ stochastic flow 𝑒𝑦 𝑢 = 𝛼 E log 𝜌 𝑦 𝑞(𝑝|𝑦) 𝑒𝑢 + 2 𝒆𝒙 𝒖 Fokker-Planck Equation + Continuity Equation ✓ density 𝒓 𝒚, 𝒖 converges to posterior 𝒒 𝒚|𝒑 deterministic, closed-loop ✓ deterministic flow 𝑒𝑦 𝑢 = 𝛼 E log 𝜌 𝑦 𝑞(𝑝|𝑦) − ∇ E log 𝒓(𝒚, 𝒖) 𝑒𝑢 ✘ closed-loop flow: depends on 𝒓(𝒚, 𝒖) Optimal control theory: closed-loop to open loop ✓ density 𝒓 𝒚, 𝒖 converges to posterior 𝒒 𝒚|𝒑 deterministic, open-loop ✓ deterministic flow E log 𝜌 𝑦 𝑞(𝑝|𝑦) − 𝑥 ∗ 𝜌 𝑦 , 𝑞 𝑝|𝑦 , 𝑦, 𝑢 𝑒𝑢 𝑒𝑦 𝑢 = 𝛼 ✓ open-loop flow

Parameterization The unified flow velocity is in form of: E log 𝜌 𝑦 𝑞(𝑝|𝑦) − 𝑥 ∗ 𝜌 𝑦 , 𝑞 𝑝|𝑦 , 𝑦, 𝑢 𝒈(𝝆 𝒚 , 𝒒 𝒑 𝒚 , 𝒚, 𝒖): = 𝛼 𝑶 𝒆𝒚 𝟐 𝝔 𝒚 𝒐 , 𝒑, 𝒚 𝒖 , 𝒖 𝒆𝒖 = 𝒈 𝓨, 𝒑, 𝒚 𝒖 , 𝒖 ≔ 𝒊 𝑶 Z 𝒐\𝟐 Deep set 𝒊 neural networks

Experiment 1: Multimodal Posterior Gaussian Mixture Model • prior 𝑦 # , 𝑦 $ ∼ 𝒪 0,1 observations o|𝑦 # , 𝑦 $ ∼ # $ 𝒪 𝑦 # , 1 + # $ 𝒪(𝑦 # + 𝑦 $ , 1) • • With 𝑦 # , 𝑦 $ = (1, −2) , the resulting posterior 𝒒(𝒚|𝒑 𝟐 , … , 𝒑 𝒏 ) will have two modes: − 3 − 3 − 3 − 4 x 10 x 10 x 10 x 10 5 14 3 0.018 3 10 3 2.2 3 3 4 2 9 0.016 2 12 2 2 2 3 1.8 8 0.014 2 1.6 10 1 7 1 1 1 0.012 1 1.4 2 6 0 0 8 0 1.2 0 0.01 0 5 − 1 1 0.008 6 − 1 4 − 1 − 1 − 1 − 2 0.8 0.006 3 − 3 0.6 1 4 − 2 − 2 − 2 − 2 0.004 2 − 4 0.4 − 5 2 − 3 − 3 0.2 − 3 1 − 3 0.002 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 (a) True posterior (d) Gibbs Sampling (e) One-pass SMC (b) Stochastic Variational (c) Stochastic Gradient Inference Langevin Dynamics

Experiment 1: Multimodal Posterior PFBR vs one-pass SMC Visualization of the evolution of posterior density from left to right.

Experiment 2: Efficiency in #Particles Our Approach 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).

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