Rejection Sampling Variational Inference Karan Grewal CSC2547 / - PowerPoint PPT Presentation

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Rejection Sampling Variational Inference Karan Grewal CSC2547 / STA4273 Overview Variational Inference Interested in computing posterior , but it is often intractable parametrize a variational family of distributions to

Rejection Sampling Variational Inference Karan Grewal CSC2547 / STA4273
Overview
Variational Inference • Interested in computing posterior , but it is often intractable • parametrize a variational family of distributions to approximate true posterior • Maximize Evidence Lower Bound (ELBO):
Rejection Sampling • Want to sample from , parametrize a proposal distribution s.t. • Accept sample with probability source: https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture17.pdf
Reparameterized Rejection Sampler • Problem: what if we want our variational family to follow a distribution that requires rejection sampling to approximate? • Rejection sampling causes discontinuities
Example: Gamma Distribution source: http://www.epixanalytics.com/ • To sample from , sample from and divide by , the acceptance probability is dependent on
Reparameterized Rejection Sampler 1. Reparameterize : 2. Find marginal distribution of accepted sample :
Reparameterized Rejection Sampler 3. Rewrite ELBO:
Related Work • Automatic Differentiation Variational Inference (ADVI) - fit with Gaussian posterior; cannot learn a Gamma or Dirichlet posterior • Black-Box Variational Inference (BBVI) - sample from to approximate gradient • Generalized Reparameterization Gradient (G-REP) - find a distribution that makes dependent on choice of variational family
Results • model = sparse gamma Deep Exponential Family source: https://arxiv.org/pdf/1610.05683.pdf
Future Work • Combining Rejection Sampling Variational Inference with Metropolis-Hastings • Metropolis-Hastings: Acquire a sequence of samples from a distribution that is difficult to sample from directly; use rejection sampling
Supplementary: Gradient Derivation

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