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

Nov 26, 2022 •488 likes •623 views

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

Recommend

Sampling Methods Oliver Schulte - CMPT 419/726 Bishop PRML Ch. 11 Sampling Rejection Sampling

Sampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo Sampling Methods Oliver Schulte - CMPT 419/726 Bishop PRML Ch. 11 Sampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo Recall Inference For

573 views • 41 slides

Variational Inference for GPs: Presenters Group1: Stochastic variational inference. Slides 2 - 28

Variational Inference for GPs: Presenters Group1: Stochastic variational inference. Slides 2 - 28 Chaoqi Wang Sana Tonekaboni Will Grathwohl Group2: Variational inference for GPs. Slides 29 - 57 Trefor Evans Kingsley Chang Shems Saleh James

1.32k views • 98 slides

Stein Variational Newton & other Sampling-Based Inference Methods Robert Scheichl

Stein Variational Newton & other Sampling-Based Inference Methods Robert Scheichl Interdisciplinary Center for Scientific Computing & Institute of Applied Mathematics Universit at Heidelberg Collaborators: G. Detommaso (Bath); T.

970 views • 78 slides

Bayes net wrapup Exact inference algorithms Use to compute P(X1, ..., Xn) or P(X1, ..., Xn

Bayes net wrapup Exact inference algorithms Use to compute P(X1, ..., Xn) or P(X1, ..., Xn | Y1, ..., Ym) Approximate inference algorithms Direct sampling Rejection sampling Likelihood weighting Direct Sampling To

567 views • 5 slides

CS480/680 Machine Learning Lecture 11: February 11 th , 2020 Variational Inference Zahra

CS480/680 Machine Learning Lecture 11: February 11 th , 2020 Variational Inference Zahra Sheikhbahaee VARIATIONAL ALGORITHMS FOR APPROXIMATE BAYESIAN INFERENCE (Beal 2003, chapter 2) Variational Inference: A Review for Statisticians (Blei et al.

713 views • 28 slides

Approximate inference: Sampling methods Probabilistic Graphical Models Sharif University of

Approximate inference: Sampling methods Probabilistic Graphical Models Sharif University of Technology Spring 2018 Soleymani Approximate inference Approximate inference techniques Deterministic approximation Variational algorithms

1.96k views • 97 slides

Sampling Methods CMSC 678 UMBC Outline Recap Monte Carlo methods Sampling Techniques Uniform

Approximate Inference: Sampling Methods CMSC 678 UMBC Outline Recap Monte Carlo methods Sampling Techniques Uniform sampling Importance Sampling Rejection Sampling Metropolis-Hastings Gibbs sampling Example: Collapsed Gibbs Sampler for

1.03k views • 71 slides

Deep Variational Inference FLARE Reading Group Presentation Wesley Tansey 9/28/2016 What is

Deep Variational Inference FLARE Reading Group Presentation Wesley Tansey 9/28/2016 What is Variational Inference? What is Variational Inference? p*(x) Want to estimate some distribution, p*(x) What is Variational Inference?

684 views • 52 slides

Sampling Methods CMSC 691 UMBC (Some) Learning Techniques MAP/MLE: Point estimation, basic EM

Approximate Inference: Sampling Methods CMSC 691 UMBC (Some) Learning Techniques MAP/MLE: Point estimation, basic EM Variational Inference: Functional Optimization Sampling/Monte Carlo today Outline Monte Carlo methods Sampling

985 views • 71 slides

Lecture Variational 13 Inference Panini Kaushal Scribes : - Margulies Smedeuranh Niklas

Lecture Variational 13 Inference Panini Kaushal Scribes : - Margulies Smedeuranh Niklas Variational Inference Approximate by posterior Idea maximizing , bound lower Ptt variational fly ply ? ) a , , ) " 9) * =

709 views • 15 slides

Variational Inference for Bayes vMF Mixture Hanxiao Liu September 23, 2014 1 / 14 Variational

Variational Inference for Bayes vMF Mixture Hanxiao Liu September 23, 2014 1 / 14 Variational Inference Review Lower bound the likelihood L ( ; X ) = E q log p ( X | ) log p ( X , Z | ) q ( Z ) = E q + E q log q (

310 views • 14 slides

Fast and Simple Natural-Gradient Variational Inference with Mixture of Exponential-family

Fast and Simple Natural-Gradient Variational Inference with Mixture of Exponential-family Approximations Wu Lin (UBC) June 11, 2019 Joint work with Mohammad Emtiyaz Khan (AIP, RIKEN) and Mark Schmidt (UBC) 1 / 7 Variational Inference (VI) VI

119 views • 7 slides

Approximate inference (Ch. 14) Likelihood Weighting P(b|a) 1 P(a) 0.5 A B P(b|a) 0.2 In

Approximate inference (Ch. 14) Likelihood Weighting P(b|a) 1 P(a) 0.5 A B P(b|a) 0.2 In LW, say we generated 2 samples: [a] : w = 1, [a], w=0.2 If we did rejection sampling, we need about 5 a to actually get a b, so 10

645 views • 33 slides

Variational Inference CMSC 691 UMBC Goal: Posterior Inference Hyperparameters Unknown

Approximate Inference: Variational Inference CMSC 691 UMBC Goal: Posterior Inference Hyperparameters Unknown parameters Data: p ( | ) Likelihood model: p( | ) (Some) Learning Techniques MAP/MLE: Point

1.09k views • 81 slides

Approximate Inference 9.520 Class 19 Ruslan Salakhutdinov BCS and CSAIL, MIT 1 Plan 1.

Approximate Inference 9.520 Class 19 Ruslan Salakhutdinov BCS and CSAIL, MIT 1 Plan 1. Introduction/Notation. 2. Examples of successful Bayesian models. 3. Laplace and Variational Inference. 4. Basic Sampling Algorithms. 5. Markov chain

833 views • 45 slides

Neural Variational Inference and Learning Andriy Mnih, Karol Gregor 22 June 2014 1 / 14

Neural Variational Inference and Learning Andriy Mnih, Karol Gregor 22 June 2014 1 / 14 Introduction Training directed latent variable models is difficult because inference in them is intractable. Both MCMC and traditional variational

322 views • 14 slides

Lecture 14: Inference in Dirichlet Processes (Blei & Jordan, Variational inference for

CS598JHM: Advanced NLP (Spring 2013) http://courses.engr.illinois.edu/cs598jhm/ Lecture 14: Inference in Dirichlet Processes (Blei & Jordan, Variational inference for Dirichlet Process Mixture models, Bayesian Analysis 2006) Julia

354 views • 21 slides

Sampling and Inference Sampling and Inference The Quality of Data and Measures 2012 1 Why do we

Sampling and Inference Sampling and Inference The Quality of Data and Measures 2012 1 Why do we sample? Cost/ benefit benefit Benefit Benefit (precision) Cost (h (hassle factor) l f t ) N 2 Effects of samples Obvious:

652 views • 46 slides

From Identification using Rejection Sampling to Signatures via the Fiat-Shamir Transform:

From Identification using Rejection Sampling to Signatures via the Fiat-Shamir Transform: Application to the BLISS Signature Pauline Bert and Adeline Roux-Langlois Journes C2 2018 Univ Rennes, CNRS, IRISA 1 Contribution Fiat-Shamir

326 views • 19 slides

Adaptive rejection Metropolis sampling Dr. Jarad Niemi STAT 615 - Iowa State University November

Adaptive rejection Metropolis sampling Dr. Jarad Niemi STAT 615 - Iowa State University November 14, 2017 Jarad Niemi (STAT615@ISU) Adaptive rejection Metropolis sampling November 14, 2017 1 / 20 (Logarithmically) Concave Univariate Function

620 views • 20 slides

Sampling-Based Inference 1 Inference by stochastic simulation Basic idea: 1) Draw N samples from

Sampling-Based Inference 1 Inference by stochastic simulation Basic idea: 1) Draw N samples from a sampling distribution S 0.5 2) Compute an approximate posterior probability P 3) Show this converges to the true probability P Coin Outline:

773 views • 31 slides

Variational Inference and Learning Michael Gutmann Probabilistic Modelling and Reasoning

Variational Inference and Learning Michael Gutmann Probabilistic Modelling and Reasoning (INFR11134) School of Informatics, University of Edinburgh Spring semester 2018 Recap Learning and inference often involves intractable integrals

786 views • 36 slides

161-17 Stochastic Variational Lecture Inference I Kaushal Panchi Scribes : Jay DeYoung

161-17 Stochastic Variational Lecture Inference I Kaushal Panchi Scribes : Jay DeYoung Recap Vaniatianal Inference : Approximate Goal Posterior : 17,13 ) pH )p( 137 Generative Model 13 ) pc plx ,z : = , Approx

347 views • 18 slides

Week 2: Inference for SLR Inference: sampling distributions, testing confidence intervals, and

BUS41100 Applied Regression Analysis Week 2: Inference for SLR Inference: sampling distributions, testing confidence intervals, and prediction intervals Max H. Farrell The University of Chicago Booth School of Business Back to House Prices

698 views • 52 slides