Statistical Machine Learning Lecture 8: Markov Chain Monte Carlo Sampling
|
|
- Madison Bailey
- 6 years ago
- Views:
Transcription
1 1 / 27 Statistical Machine Learning Lecture 8: Markov Chain Monte Carlo Sampling Melih Kandemir Özyeğin University, İstanbul, Turkey
2 2 / 27 Monte Carlo Integration The big question : Evaluate E p(z) [f(z)] = f(z)p(z)dz Examples Bayesian prediction: p(z new z, D) = p(z new θ)p(θ D)dθ = E p(θ D) [p(z new θ)] Difficult variational updates: log q(z 1 ) E p(z2 )[log p(z 1, z 2 )] Difficult E-step in EM: Q(θ, θ old ) = E p(z D,θold )[log p(z, D θ)]
3 3 / 27 Approximating the integral by samples E p(z) [f(z)] = f(z)p(z)dz L f(z (l) ) l=1 where z (l) are samples drawn from p(z (l) ). As long as iid samples are drawn from the true p(z (l) ), 20 samples are sufficient for a good approximation.
4 Sampling from inverse CDF 1 Draw u Uniform(0, 1) Calculate y = h 1 (u) Because: P r(h 1 (u) y) = P r(u h(y)) = h(y) Problem: How do we compute h 1 (u) for an arbitrary distribution? 1 Bishop, PRML, / 27
5 2 5 / 27 Rejection Sampling 2 Target distribution p(z), and envelop distribution q(z) Procedure: z (t) q(z) u (t) Uniform(0, kq(z (t) )) Accept sample if u (t) p(z) p(accept) = p(z) kq(z) q(z)dz = 1 p(z)dz k
6 Adaptive Rejection Sampling 3 Envelope function is a set of piecewise exponential functions: q(z) = k i λ i exp{ λ i (z z i 1 )} z i 1 z z i Each rejected sample is added as a grid point. Acceptance rate decays exponentially wrt dimensionality! 3 Bishop, PRML, / 27
7 7 / 27 Importance Sampling (1) E p(z) [f(z)] = = f(z)p(z)dz f(z) p(z) q(z) q(z)dz Draw l samples from q(z). Then, E p(z) [f(z)] 1 L L l=1 f(z (l) p(z (l) ) ) q(z (l) ) }{{} importance weight
8 8 / 27 Importance Sampling (2) (+) All samples are retained. (-) Too much dependent on how similar q(z) is to p(z). (-) No diagnostic measures available!
9 9 / 27 Markov Chain Monte Carlo Robust to high dimensionalities Samples form a Markov chain with a transition function T (z z ) Samples are drawn from the target distribution p(z) if, p(z) is invariant wrt T (z z ), p(z) = p(z )T (z z )dz. the Markov chain governed by T (z z ) is ergodic. Invariance : Ensured by detailed balance: p(z)t (z z) = p(z )T (z z ) Ergodicity : More tricky. Imposed by sampling algorithms.
10 10 / 27 Metropolis-Hastings Procedure: Propose the next state by Q(z ( z), e.g. N (z, σ 2 Accept with probability min 1, p(z )Q(z z ) ) p(z)q(z z) Stay at the current state (add another copy of it to the samples list) otherwise The proposal variance σ 2 is very influential. Determines step size If large, low acceptance rate If small, slow convergence
11 11 / 27 Metropolis-Hastings (2) Detailed balance is provided: ( p(z)t (z z) = p(z)q(z z) min 1, p(z )Q(z z ) ) p(z)q(z z) = min ( p(z)q(z z), p(z )Q(z z ) ) ( p(z)q(z = p(z )Q(z z ) z) ) min p(z )Q(z z ), 1 = p(z )T (z z )
12 Metropolis-Hastings (3) 4 1-D Demo: 4 Murray,MLSS, / 27
13 13 / 27 Gibbs Sampling Procedure: Initialize z (1) 1, z(1) 2, z(1) 3 For l = 1 to L 1 z (l+1) 1 p(z 1 z (l) 2, z(l) 3 ) z (l+1) 2 p(z 2 z (l+1) 1, z (l) 3 ) z (l+1) 3 p(z 3 z (l+1) 1, z (l+1) 2 )
14 14 / 27 Gibbs Sampling (2) Invariance: All conditioned variates are constant by definition, and the remaining variable is sampled from the true distribution. Ergodicity: Guaranteed if all conditional probabilities are non-zero in their entire domain. Gibbs sampling is a special case of Metropolis-Hastings with q k (z z) = p(z k z \k ), thus A(z z) = p(z k z \k)p(z \k)p(z k z \k) p(z k z \k )p(z \k )p(z k z \k ) = 1 Hence, all samples are accepted.
15 Gibbs Sampling (3) 5 Step size is governed by covariances of conditional distributions. Iterative conditional modes: Instead of sampling, update wrt a point estimate (e.g. mean, mode). 5 Bishop, PRML, / 27
16 16 / 27 Collapsed Gibbs Sampling Integrating out some of the variables may yield others to appear conditionally-independent, which entails faster convergence. Rao-Blackwell Theorem: Let z and θ be dependent variables, and f(z, θ) be some scalar function. Then, var z,θ [f(z, θ)] var z [E θ [f(z, θ) z]].
17 Example: Gaussian Mixture Model 6 Employ conjugate priors to: cluster means cluster covariances mixture probabilities Then integrate them out! 6 Murphy, Mach. Learn., / 27
18 18 / 27 Implementation tricks Thinning : Take every Kth sample to decorrelate Burn-in : Discard first (e.g. half) of the samples which were prior to mixing Multiple runs : To neutralize the effect of initialization
19 19 / 27 Diagnosing Convergence 1: Traceplots
20 20 / 27 Diagnosing Convergence 2: Running mean plots
21 21 / 27 Diagnosing Conv. 3: Rubin-Gelman Metric Calculate within-chain variance W and between-chain variance B Calculate estimated variance ˆ V ar(θ) = (1 1/n)W + (1/n)B Calculate and monitor Potential Scale Reduction Factor (PSRF) Vˆar(θ) ˆR = W ˆR should get smaller until convergence.
22 22 / 27 Diagnosing Convergence 4: Other metrics Geweke diagnostic: Take first x and last y samples in the chain and test if they come from the same distribution. Raftery and Lewis diagnostic: Calculate nr of iterations until a desired level of accuracy is reached for a posterior quantile. Heidelberg and Welch diagnostic: Repeated significance testing (stationary vs null)
23 23 / 27 Example: Bayesian logistic regression p(f i w, x i ) = N (f i w T x i, σ 2 ), i = 1,, N 1 p(y i f i ) = 1 + e f, iy i i = 1,, N p(w d α d ) = N (w d 0, α 1 d ), d = 1,, D p(α d ) = G(α d a, b), d = 1,, D
24 24 / 27 Let s aim for a Gibbs samples We require the following conditional distributions: p(w f, α, X, y), (1) p(α w, f, X, y), (2) p(f w, α, X, y) (3)
25 The log joint N N log p(w, f, α, X, y) = log p(f i w, x i ) + log p(y i f i ) i=1 i=1 D D + log p(w i α i ) + log p(α i ) d=1 d=1 = 1 2 log σ2 I 1 2σ 2 (f T w T X T )(f Xw) N log(1 + e y if i ) + 1 D log α d wt Aw + i=1 D (a 1) log α d d=1 d=1 d=1 D bα d + const where A dd = α d and A ij = 0, i j 25 / 27
26 26 / 27 The conditionals p(α d α d, w, f, X, y) = G(α d a + 1 2, b w2 d ) ( ( p(w f, α, X, y) = N w X T X + A ) 1 X T f, ( X T X + A ) ) 1 p(f w, α, X, y) = Metropolis with q(f i ) = N (f i w T x i, σ 2 )
27 27 / 27 Useful references Robert and Casella, Monte Carlo Statistical Methods, 2004 Bishop, Pattern Recognition & Mach. Learning, 2006, Ch. 11 Murphy, Machine Learning: A Probabilistic Perspective, 2012 Gelman et al., Bayesian Data Analysis, 2013 Murray, Markov Chain Monte Carlo, MLSS, 2009
Pattern Recognition and Machine Learning. Bishop Chapter 11: Sampling Methods
Pattern Recognition and Machine Learning Chapter 11: Sampling Methods Elise Arnaud Jakob Verbeek May 22, 2008 Outline of the chapter 11.1 Basic Sampling Algorithms 11.2 Markov Chain Monte Carlo 11.3 Gibbs
More informationApproximate Inference using MCMC
Approximate Inference using MCMC 9.520 Class 22 Ruslan Salakhutdinov BCS and CSAIL, MIT 1 Plan 1. Introduction/Notation. 2. Examples of successful Bayesian models. 3. Basic Sampling Algorithms. 4. Markov
More informationSTAT 425: Introduction to Bayesian Analysis
STAT 425: Introduction to Bayesian Analysis Marina Vannucci Rice University, USA Fall 2017 Marina Vannucci (Rice University, USA) Bayesian Analysis (Part 2) Fall 2017 1 / 19 Part 2: Markov chain Monte
More informationMarkov Chain Monte Carlo
1 Motivation 1.1 Bayesian Learning Markov Chain Monte Carlo Yale Chang In Bayesian learning, given data X, we make assumptions on the generative process of X by introducing hidden variables Z: p(z): prior
More informationSampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo. Sampling Methods. Oliver Schulte - CMPT 419/726. Bishop PRML Ch.
Sampling Methods Oliver Schulte - CMP 419/726 Bishop PRML Ch. 11 Recall Inference or General Graphs Junction tree algorithm is an exact inference method for arbitrary graphs A particular tree structure
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationMarkov chain Monte Carlo
Markov chain Monte Carlo Markov chain Monte Carlo (MCMC) Gibbs and Metropolis Hastings Slice sampling Practical details Iain Murray http://iainmurray.net/ Reminder Need to sample large, non-standard distributions:
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 7 Approximate
More informationProbabilistic Graphical Models Lecture 17: Markov chain Monte Carlo
Probabilistic Graphical Models Lecture 17: Markov chain Monte Carlo Andrew Gordon Wilson www.cs.cmu.edu/~andrewgw Carnegie Mellon University March 18, 2015 1 / 45 Resources and Attribution Image credits,
More informationBayesian Methods for Machine Learning
Bayesian Methods for Machine Learning CS 584: Big Data Analytics Material adapted from Radford Neal s tutorial (http://ftp.cs.utoronto.ca/pub/radford/bayes-tut.pdf), Zoubin Ghahramni (http://hunch.net/~coms-4771/zoubin_ghahramani_bayesian_learning.pdf),
More informationIntroduction to Machine Learning
Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 25: Markov Chain Monte Carlo (MCMC) Course Review and Advanced Topics Many figures courtesy Kevin
More informationSampling Methods. Bishop PRML Ch. 11. Alireza Ghane. Sampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo
Sampling Methods Bishop PRML h. 11 Alireza Ghane Sampling Methods A. Ghane /. Möller / G. Mori 1 Recall Inference or General Graphs Junction tree algorithm is an exact inference method for arbitrary graphs
More informationSampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo. Sampling Methods. Machine Learning. Torsten Möller.
Sampling Methods Machine Learning orsten Möller Möller/Mori 1 Recall Inference or General Graphs Junction tree algorithm is an exact inference method for arbitrary graphs A particular tree structure defined
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and Non-Parametric Bayes Mark Schmidt University of British Columbia Winter 2016 Admin I went through project proposals: Some of you got a message on Piazza. No news is
More informationBasic Sampling Methods
Basic Sampling Methods Sargur Srihari srihari@cedar.buffalo.edu 1 1. Motivation Topics Intractability in ML How sampling can help 2. Ancestral Sampling Using BNs 3. Transforming a Uniform Distribution
More informationMonte Carlo Inference Methods
Monte Carlo Inference Methods Iain Murray University of Edinburgh http://iainmurray.net Monte Carlo and Insomnia Enrico Fermi (1901 1954) took great delight in astonishing his colleagues with his remarkably
More information17 : Markov Chain Monte Carlo
10-708: Probabilistic Graphical Models, Spring 2015 17 : Markov Chain Monte Carlo Lecturer: Eric P. Xing Scribes: Heran Lin, Bin Deng, Yun Huang 1 Review of Monte Carlo Methods 1.1 Overview Monte Carlo
More informationBayesian Inference and MCMC
Bayesian Inference and MCMC Aryan Arbabi Partly based on MCMC slides from CSC412 Fall 2018 1 / 18 Bayesian Inference - Motivation Consider we have a data set D = {x 1,..., x n }. E.g each x i can be the
More informationLecture 7 and 8: Markov Chain Monte Carlo
Lecture 7 and 8: Markov Chain Monte Carlo 4F13: Machine Learning Zoubin Ghahramani and Carl Edward Rasmussen Department of Engineering University of Cambridge http://mlg.eng.cam.ac.uk/teaching/4f13/ Ghahramani
More informationIntroduction to Machine Learning CMU-10701
Introduction to Machine Learning CMU-10701 Markov Chain Monte Carlo Methods Barnabás Póczos & Aarti Singh Contents Markov Chain Monte Carlo Methods Goal & Motivation Sampling Rejection Importance Markov
More information16 : Markov Chain Monte Carlo (MCMC)
10-708: Probabilistic Graphical Models 10-708, Spring 2014 16 : Markov Chain Monte Carlo MCMC Lecturer: Matthew Gormley Scribes: Yining Wang, Renato Negrinho 1 Sampling from low-dimensional distributions
More informationMarkov chain Monte Carlo
Markov chain Monte Carlo Probabilistic Models of Cognition, 2011 http://www.ipam.ucla.edu/programs/gss2011/ Roadmap: Motivation Monte Carlo basics What is MCMC? Metropolis Hastings and Gibbs...more tomorrow.
More informationST 740: Markov Chain Monte Carlo
ST 740: Markov Chain Monte Carlo Alyson Wilson Department of Statistics North Carolina State University October 14, 2012 A. Wilson (NCSU Stsatistics) MCMC October 14, 2012 1 / 20 Convergence Diagnostics:
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference Sudipto Banerjee University of Minnesota July 20th, 2008 1 Bayesian Principles Classical statistics: model parameters are fixed and unknown. A Bayesian thinks of parameters
More informationMarkov Chain Monte Carlo
Markov Chain Monte Carlo Recall: To compute the expectation E ( h(y ) ) we use the approximation E(h(Y )) 1 n n h(y ) t=1 with Y (1),..., Y (n) h(y). Thus our aim is to sample Y (1),..., Y (n) from f(y).
More informationStatistical Machine Learning Lectures 4: Variational Bayes
1 / 29 Statistical Machine Learning Lectures 4: Variational Bayes Melih Kandemir Özyeğin University, İstanbul, Turkey 2 / 29 Synonyms Variational Bayes Variational Inference Variational Bayesian Inference
More informationMCMC algorithms for fitting Bayesian models
MCMC algorithms for fitting Bayesian models p. 1/1 MCMC algorithms for fitting Bayesian models Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota MCMC algorithms for fitting Bayesian models
More informationCSC 2541: Bayesian Methods for Machine Learning
CSC 2541: Bayesian Methods for Machine Learning Radford M. Neal, University of Toronto, 2011 Lecture 3 More Markov Chain Monte Carlo Methods The Metropolis algorithm isn t the only way to do MCMC. We ll
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond January 18, 2017 Contents 1 Batch and Recursive Estimation 2 Towards Bayesian Filtering 3 Kalman Filter and Bayesian Filtering and Smoothing
More informationLecture 4: Probabilistic Learning. Estimation Theory. Classification with Probability Distributions
DD2431 Autumn, 2014 1 2 3 Classification with Probability Distributions Estimation Theory Classification in the last lecture we assumed we new: P(y) Prior P(x y) Lielihood x2 x features y {ω 1,..., ω K
More informationBAYESIAN METHODS FOR VARIABLE SELECTION WITH APPLICATIONS TO HIGH-DIMENSIONAL DATA
BAYESIAN METHODS FOR VARIABLE SELECTION WITH APPLICATIONS TO HIGH-DIMENSIONAL DATA Intro: Course Outline and Brief Intro to Marina Vannucci Rice University, USA PASI-CIMAT 04/28-30/2010 Marina Vannucci
More informationDensity Estimation. Seungjin Choi
Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/
More informationan introduction to bayesian inference
with an application to network analysis http://jakehofman.com january 13, 2010 motivation would like models that: provide predictive and explanatory power are complex enough to describe observed phenomena
More informationECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering
ECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering Lecturer: Nikolay Atanasov: natanasov@ucsd.edu Teaching Assistants: Siwei Guo: s9guo@eng.ucsd.edu Anwesan Pal:
More informationComputer Vision Group Prof. Daniel Cremers. 10a. Markov Chain Monte Carlo
Group Prof. Daniel Cremers 10a. Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative is Markov Chain
More informationBayesian Learning. HT2015: SC4 Statistical Data Mining and Machine Learning. Maximum Likelihood Principle. The Bayesian Learning Framework
HT5: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford http://www.stats.ox.ac.uk/~sejdinov/sdmml.html Maximum Likelihood Principle A generative model for
More informationLECTURE 15 Markov chain Monte Carlo
LECTURE 15 Markov chain Monte Carlo There are many settings when posterior computation is a challenge in that one does not have a closed form expression for the posterior distribution. Markov chain Monte
More informationComputer Vision Group Prof. Daniel Cremers. 14. Sampling Methods
Prof. Daniel Cremers 14. Sampling Methods Sampling Methods Sampling Methods are widely used in Computer Science as an approximation of a deterministic algorithm to represent uncertainty without a parametric
More informationReminder of some Markov Chain properties:
Reminder of some Markov Chain properties: 1. a transition from one state to another occurs probabilistically 2. only state that matters is where you currently are (i.e. given present, future is independent
More informationEco517 Fall 2013 C. Sims MCMC. October 8, 2013
Eco517 Fall 2013 C. Sims MCMC October 8, 2013 c 2013 by Christopher A. Sims. This document may be reproduced for educational and research purposes, so long as the copies contain this notice and are retained
More informationMarkov Chain Monte Carlo (MCMC) and Model Evaluation. August 15, 2017
Markov Chain Monte Carlo (MCMC) and Model Evaluation August 15, 2017 Frequentist Linking Frequentist and Bayesian Statistics How can we estimate model parameters and what does it imply? Want to find the
More informationParticle-Based Approximate Inference on Graphical Model
article-based Approimate Inference on Graphical Model Reference: robabilistic Graphical Model Ch. 2 Koller & Friedman CMU, 0-708, Fall 2009 robabilistic Graphical Models Lectures 8,9 Eric ing attern Recognition
More informationThe Expectation-Maximization Algorithm
1/29 EM & Latent Variable Models Gaussian Mixture Models EM Theory The Expectation-Maximization Algorithm Mihaela van der Schaar Department of Engineering Science University of Oxford MLE for Latent Variable
More informationComputer Vision Group Prof. Daniel Cremers. 11. Sampling Methods
Prof. Daniel Cremers 11. Sampling Methods Sampling Methods Sampling Methods are widely used in Computer Science as an approximation of a deterministic algorithm to represent uncertainty without a parametric
More informationSession 3A: Markov chain Monte Carlo (MCMC)
Session 3A: Markov chain Monte Carlo (MCMC) John Geweke Bayesian Econometrics and its Applications August 15, 2012 ohn Geweke Bayesian Econometrics and its Session Applications 3A: Markov () chain Monte
More informationComputer Vision Group Prof. Daniel Cremers. 11. Sampling Methods: Markov Chain Monte Carlo
Group Prof. Daniel Cremers 11. Sampling Methods: Markov Chain Monte Carlo Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative
More informationMH I. Metropolis-Hastings (MH) algorithm is the most popular method of getting dependent samples from a probability distribution
MH I Metropolis-Hastings (MH) algorithm is the most popular method of getting dependent samples from a probability distribution a lot of Bayesian mehods rely on the use of MH algorithm and it s famous
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee September 03 05, 2017 Department of Biostatistics, Fielding School of Public Health, University of California, Los Angeles Linear Regression Linear regression is,
More informationMarkov Chain Monte Carlo (MCMC)
Markov Chain Monte Carlo (MCMC Dependent Sampling Suppose we wish to sample from a density π, and we can evaluate π as a function but have no means to directly generate a sample. Rejection sampling can
More informationMarkov Chain Monte Carlo methods
Markov Chain Monte Carlo methods By Oleg Makhnin 1 Introduction a b c M = d e f g h i 0 f(x)dx 1.1 Motivation 1.1.1 Just here Supresses numbering 1.1.2 After this 1.2 Literature 2 Method 2.1 New math As
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond Department of Biomedical Engineering and Computational Science Aalto University January 26, 2012 Contents 1 Batch and Recursive Estimation
More information16 : Approximate Inference: Markov Chain Monte Carlo
10-708: Probabilistic Graphical Models 10-708, Spring 2017 16 : Approximate Inference: Markov Chain Monte Carlo Lecturer: Eric P. Xing Scribes: Yuan Yang, Chao-Ming Yen 1 Introduction As the target distribution
More informationProbabilistic Machine Learning
Probabilistic Machine Learning Bayesian Nets, MCMC, and more Marek Petrik 4/18/2017 Based on: P. Murphy, K. (2012). Machine Learning: A Probabilistic Perspective. Chapter 10. Conditional Independence Independent
More informationCS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling
CS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling Professor Erik Sudderth Brown University Computer Science October 27, 2016 Some figures and materials courtesy
More informationCS281A/Stat241A Lecture 22
CS281A/Stat241A Lecture 22 p. 1/4 CS281A/Stat241A Lecture 22 Monte Carlo Methods Peter Bartlett CS281A/Stat241A Lecture 22 p. 2/4 Key ideas of this lecture Sampling in Bayesian methods: Predictive distribution
More informationMCMC and Gibbs Sampling. Sargur Srihari
MCMC and Gibbs Sampling Sargur srihari@cedar.buffalo.edu 1 Topics 1. Markov Chain Monte Carlo 2. Markov Chains 3. Gibbs Sampling 4. Basic Metropolis Algorithm 5. Metropolis-Hastings Algorithm 6. Slice
More informationStat 451 Lecture Notes Markov Chain Monte Carlo. Ryan Martin UIC
Stat 451 Lecture Notes 07 12 Markov Chain Monte Carlo Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapters 8 9 in Givens & Hoeting, Chapters 25 27 in Lange 2 Updated: April 4, 2016 1 / 42 Outline
More informationMarkov Chain Monte Carlo Inference. Siamak Ravanbakhsh Winter 2018
Graphical Models Markov Chain Monte Carlo Inference Siamak Ravanbakhsh Winter 2018 Learning objectives Markov chains the idea behind Markov Chain Monte Carlo (MCMC) two important examples: Gibbs sampling
More informationBagging During Markov Chain Monte Carlo for Smoother Predictions
Bagging During Markov Chain Monte Carlo for Smoother Predictions Herbert K. H. Lee University of California, Santa Cruz Abstract: Making good predictions from noisy data is a challenging problem. Methods
More informationMarkov chain Monte Carlo Lecture 9
Markov chain Monte Carlo Lecture 9 David Sontag New York University Slides adapted from Eric Xing and Qirong Ho (CMU) Limitations of Monte Carlo Direct (unconditional) sampling Hard to get rare events
More informationGraphical Models and Kernel Methods
Graphical Models and Kernel Methods Jerry Zhu Department of Computer Sciences University of Wisconsin Madison, USA MLSS June 17, 2014 1 / 123 Outline Graphical Models Probabilistic Inference Directed vs.
More informationMCMC Methods: Gibbs and Metropolis
MCMC Methods: Gibbs and Metropolis Patrick Breheny February 28 Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/30 Introduction As we have seen, the ability to sample from the posterior distribution
More information(5) Multi-parameter models - Gibbs sampling. ST440/540: Applied Bayesian Analysis
Summarizing a posterior Given the data and prior the posterior is determined Summarizing the posterior gives parameter estimates, intervals, and hypothesis tests Most of these computations are integrals
More informationLecture 4: Probabilistic Learning
DD2431 Autumn, 2015 1 Maximum Likelihood Methods Maximum A Posteriori Methods Bayesian methods 2 Classification vs Clustering Heuristic Example: K-means Expectation Maximization 3 Maximum Likelihood Methods
More informationBayesian data analysis in practice: Three simple examples
Bayesian data analysis in practice: Three simple examples Martin P. Tingley Introduction These notes cover three examples I presented at Climatea on 5 October 0. Matlab code is available by request to
More informationPreviously Monte Carlo Integration
Previously Simulation, sampling Monte Carlo Simulations Inverse cdf method Rejection sampling Today: sampling cont., Bayesian inference via sampling Eigenvalues and Eigenvectors Markov processes, PageRank
More informationMarkov chain Monte Carlo
1 / 26 Markov chain Monte Carlo Timothy Hanson 1 and Alejandro Jara 2 1 Division of Biostatistics, University of Minnesota, USA 2 Department of Statistics, Universidad de Concepción, Chile IAP-Workshop
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 13: Learning in Gaussian Graphical Models, Non-Gaussian Inference, Monte Carlo Methods Some figures
More informationComputational statistics
Computational statistics Markov Chain Monte Carlo methods Thierry Denœux March 2017 Thierry Denœux Computational statistics March 2017 1 / 71 Contents of this chapter When a target density f can be evaluated
More informationIntroduction to Probability and Statistics (Continued)
Introduction to Probability and Statistics (Continued) Prof. icholas Zabaras Center for Informatics and Computational Science https://cics.nd.edu/ University of otre Dame otre Dame, Indiana, USA Email:
More informationMonte Carlo in Bayesian Statistics
Monte Carlo in Bayesian Statistics Matthew Thomas SAMBa - University of Bath m.l.thomas@bath.ac.uk December 4, 2014 Matthew Thomas (SAMBa) Monte Carlo in Bayesian Statistics December 4, 2014 1 / 16 Overview
More informationMarkov Chain Monte Carlo (MCMC)
School of Computer Science 10-708 Probabilistic Graphical Models Markov Chain Monte Carlo (MCMC) Readings: MacKay Ch. 29 Jordan Ch. 21 Matt Gormley Lecture 16 March 14, 2016 1 Homework 2 Housekeeping Due
More informationNonparameteric Regression:
Nonparameteric Regression: Nadaraya-Watson Kernel Regression & Gaussian Process Regression Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro,
More informationSAMPLING ALGORITHMS. In general. Inference in Bayesian models
SAMPLING ALGORITHMS SAMPLING ALGORITHMS In general A sampling algorithm is an algorithm that outputs samples x 1, x 2,... from a given distribution P or density p. Sampling algorithms can for example be
More informationPart 1: Expectation Propagation
Chalmers Machine Learning Summer School Approximate message passing and biomedicine Part 1: Expectation Propagation Tom Heskes Machine Learning Group, Institute for Computing and Information Sciences Radboud
More informationMetropolis-Hastings Algorithm
Strength of the Gibbs sampler Metropolis-Hastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to
More informationSampling Methods (11/30/04)
CS281A/Stat241A: Statistical Learning Theory Sampling Methods (11/30/04) Lecturer: Michael I. Jordan Scribe: Jaspal S. Sandhu 1 Gibbs Sampling Figure 1: Undirected and directed graphs, respectively, with
More informationBayesian Inference in GLMs. Frequentists typically base inferences on MLEs, asymptotic confidence
Bayesian Inference in GLMs Frequentists typically base inferences on MLEs, asymptotic confidence limits, and log-likelihood ratio tests Bayesians base inferences on the posterior distribution of the unknowns
More informationMarkov Chain Monte Carlo
Markov Chain Monte Carlo (and Bayesian Mixture Models) David M. Blei Columbia University October 14, 2014 We have discussed probabilistic modeling, and have seen how the posterior distribution is the critical
More informationLearning the hyper-parameters. Luca Martino
Learning the hyper-parameters Luca Martino 2017 2017 1 / 28 Parameters and hyper-parameters 1. All the described methods depend on some choice of hyper-parameters... 2. For instance, do you recall λ (bandwidth
More informationINFINITE MIXTURES OF MULTIVARIATE GAUSSIAN PROCESSES
INFINITE MIXTURES OF MULTIVARIATE GAUSSIAN PROCESSES SHILIANG SUN Department of Computer Science and Technology, East China Normal University 500 Dongchuan Road, Shanghai 20024, China E-MAIL: slsun@cs.ecnu.edu.cn,
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 7 Approximate
More informationCalibration of Stochastic Volatility Models using Particle Markov Chain Monte Carlo Methods
Calibration of Stochastic Volatility Models using Particle Markov Chain Monte Carlo Methods Jonas Hallgren 1 1 Department of Mathematics KTH Royal Institute of Technology Stockholm, Sweden BFS 2012 June
More information27 : Distributed Monte Carlo Markov Chain. 1 Recap of MCMC and Naive Parallel Gibbs Sampling
10-708: Probabilistic Graphical Models 10-708, Spring 2014 27 : Distributed Monte Carlo Markov Chain Lecturer: Eric P. Xing Scribes: Pengtao Xie, Khoa Luu In this scribe, we are going to review the Parallel
More informationAUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET. Machine Learning T. Schön. (Chapter 11) AUTOMATIC CONTROL REGLERTEKNIK LINKÖPINGS UNIVERSITET
About the Eam I/II) ), Lecture 7 MCMC and Sampling Methods Thomas Schön Division of Automatic Control Linköping University Linköping, Sweden. Email: schon@isy.liu.se, Phone: 3-373, www.control.isy.liu.se/~schon/
More informationBayesian Machine Learning
Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 3 Stochastic Gradients, Bayesian Inference, and Occam s Razor https://people.orie.cornell.edu/andrew/orie6741 Cornell University August
More informationA quick introduction to Markov chains and Markov chain Monte Carlo (revised version)
A quick introduction to Markov chains and Markov chain Monte Carlo (revised version) Rasmus Waagepetersen Institute of Mathematical Sciences Aalborg University 1 Introduction These notes are intended to
More informationHastings-within-Gibbs Algorithm: Introduction and Application on Hierarchical Model
UNIVERSITY OF TEXAS AT SAN ANTONIO Hastings-within-Gibbs Algorithm: Introduction and Application on Hierarchical Model Liang Jing April 2010 1 1 ABSTRACT In this paper, common MCMC algorithms are introduced
More informationPattern Recognition and Machine Learning. Bishop Chapter 9: Mixture Models and EM
Pattern Recognition and Machine Learning Chapter 9: Mixture Models and EM Thomas Mensink Jakob Verbeek October 11, 27 Le Menu 9.1 K-means clustering Getting the idea with a simple example 9.2 Mixtures
More informationSampling Algorithms for Probabilistic Graphical models
Sampling Algorithms for Probabilistic Graphical models Vibhav Gogate University of Washington References: Chapter 12 of Probabilistic Graphical models: Principles and Techniques by Daphne Koller and Nir
More informationBayesian Methods in Multilevel Regression
Bayesian Methods in Multilevel Regression Joop Hox MuLOG, 15 september 2000 mcmc What is Statistics?! Statistics is about uncertainty To err is human, to forgive divine, but to include errors in your design
More informationA Review of Pseudo-Marginal Markov Chain Monte Carlo
A Review of Pseudo-Marginal Markov Chain Monte Carlo Discussed by: Yizhe Zhang October 21, 2016 Outline 1 Overview 2 Paper review 3 experiment 4 conclusion Motivation & overview Notation: θ denotes the
More informationMCMC and Gibbs Sampling. Kayhan Batmanghelich
MCMC and Gibbs Sampling Kayhan Batmanghelich 1 Approaches to inference l Exact inference algorithms l l l The elimination algorithm Message-passing algorithm (sum-product, belief propagation) The junction
More informationDAG models and Markov Chain Monte Carlo methods a short overview
DAG models and Markov Chain Monte Carlo methods a short overview Søren Højsgaard Institute of Genetics and Biotechnology University of Aarhus August 18, 2008 Printed: August 18, 2008 File: DAGMC-Lecture.tex
More informationMarkov Chain Monte Carlo Methods
Markov Chain Monte Carlo Methods John Geweke University of Iowa, USA 2005 Institute on Computational Economics University of Chicago - Argonne National Laboaratories July 22, 2005 The problem p (θ, ω I)
More informationThe Particle Filter. PD Dr. Rudolph Triebel Computer Vision Group. Machine Learning for Computer Vision
The Particle Filter Non-parametric implementation of Bayes filter Represents the belief (posterior) random state samples. by a set of This representation is approximate. Can represent distributions that
More informationAdaptive Monte Carlo methods
Adaptive Monte Carlo methods Jean-Michel Marin Projet Select, INRIA Futurs, Université Paris-Sud joint with Randal Douc (École Polytechnique), Arnaud Guillin (Université de Marseille) and Christian Robert
More informationIntroduction to Stochastic Gradient Markov Chain Monte Carlo Methods
Introduction to Stochastic Gradient Markov Chain Monte Carlo Methods Changyou Chen Department of Electrical and Computer Engineering, Duke University cc448@duke.edu Duke-Tsinghua Machine Learning Summer
More informationPattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
More informationBayesian Networks in Educational Assessment
Bayesian Networks in Educational Assessment Estimating Parameters with MCMC Bayesian Inference: Expanding Our Context Roy Levy Arizona State University Roy.Levy@asu.edu 2017 Roy Levy MCMC 1 MCMC 2 Posterior
More information