Markov chain Monte Carlo methods
|
|
|
- Brianna McGee
- 5 years ago
- Views:
Transcription
1 Markov chain Monte Carlo methods (supplementary material) see also the applet February 9 6
2 Independent Hastings Metropolis Sampler Outline Independent Hastings Metropolis Sampler Symmetric random walk HM algorithm Gibbs sampler Burn in
3 Independent Hastings Metropolis Sampler Independent Hastings Metropolis Sampler Target density: π = N (, ) on R Proposal distribution: q(x) = N (, σ )[x]. Different values of σ are considered in turn: σ < σ >
4 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
5 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
6 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
7 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
8 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
9 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
10 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
11 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
12 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
13 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
14 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
15 Independent Hastings Metropolis Sampler Here are few iterations of the algorithm: proposed points (blue square) accepted or not as the new value of the chain (red star) proposal distribution (dots) and target distribution (dash-dots)
16 Independent Hastings Metropolis Sampler Variance of the proposal distribution : σ =. The histogram of the first samples of the chain compared to the target distribution (red, solid line) and to the proposal distribution (red, dashed line)
17 Independent Hastings Metropolis Sampler Variance of the proposal distribution : σ = Histogram of the first samples of the chain compared to the target distribution (red, solid line) and to the proposal distribution (red, dashed line)
18 Symmetric random walk HM algorithm Outline Independent Hastings Metropolis Sampler Symmetric random walk HM algorithm Algorithm Size of the jump Heavy-tailed target distributions Gibbs sampler Burn in
19 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
20 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
21 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
22 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
23 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
24 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
25 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
26 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
27 Symmetric random walk HM algorithm Algorithm Algorithm Proposed points (blue squares) accepted or not as the new values of the chain (red star) proposed distribution (dashed line) and target distribution (solid line)
28 Symmetric random walk HM algorithm Size of the jump How to choose the size of the jump (/3) Target density: π = N (, ) on R Symmetric proposal distribution: q(y) = N (, σ )[y]. Different values of σ are considered.
29 Symmetric random walk HM algorithm Size of the jump How to choose the size of the jump (/3) Target distribution: N (, Σ) Proposal distribution: N (, I). 4 Level curves of the target and proposal densities
30 Markov chain Monte Carlo methods Symmetric random walk HM algorithm Size of the jump How to choose the size of the jump (3/3) Proposal distribution: Nd (, σ I). Target distribution: Nd (, I) d= (α =.4) d= 8 (α =.) d= 3 (α = ) d= (α =.36) d= 8 (α =.7) d= 3 (α =.4) 4 4 d= 8 (α =.3) Different values of d : projection of the chain on R [top] σ does not depend on d; d =, 8, 3 (left to right). The mean acceptance rate is resp. α =.4;.; [bottom] σ = c/ d with d =, 8, 3, 8 (left to right). The mean acceptance rate is resp..36;.7;.4;.3 3 4
31 Symmetric random walk HM algorithm Size of the jump Towards adaptive methods (/4) A classical receipe in the case of Symmetric Random Walk HM with Gaussian proposal N d (, Γ) with Γ =.38 d Σ π where Σ π is the covariance matrix of π. Usually, this matrix is unknown
32 Symmetric random walk HM algorithm Size of the jump Towards adaptive methods (/4) Dimension d =. (projection of the two main directions) Target distribution π = N d (, Γ) Proposal distribution q(x, y) = N d (x,.38 d I).
33 Symmetric random walk HM algorithm Size of the jump Towards adaptive methods (3/4) Dimension d =. (projection of the two main directions) Target distribution π = N d (, Γ) Proposal distribution q(x, y) = N d (x,.38 d Γ).
34 Symmetric random walk HM algorithm Size of the jump Towards adaptive methods (4/4) Dimension d =. (projection of the two main directions) Target distribution π = N d (, Γ) Proposal distribution q(x, y) = N d (x, c t Γ t ).
35 Markov chain Monte Carlo methods Symmetric random walk HM algorithm Size of the jump Multimodal target densities Target distribution (left) Samples from a SRWHM with proposal N (, σ I) (right) Target density : mixture of dim Gaussian Hastings Metropolis draws means of the components draws means of the components
36 Symmetric random walk HM algorithm Heavy-tailed target distributions Heavy-tailed target distributions target distribution: π(x) /( + x ) (on R + ) P(X > 3) =. proposal: q(x) N (, σ ) Displayed: paths of length, started at X = (close to the mode) the auto-correlation function qq-plots to compare the chain to an i.i.d. sample
37 Gibbs sampler Outline Independent Hastings Metropolis Sampler Symmetric random walk HM algorithm Gibbs sampler Burn in
38 Gibbs sampler The Gibbs sampler, for a toy example Target distribution on R : π(x, y) exp ) ( x a y b approximated by a Gibbs sampler: Conditional distribution of X given Y : N (, a ) Conditional distribution of Y given X : N (, b ) we choose a = b =
39 Gibbs sampler
40 Gibbs sampler
41 Gibbs sampler
42 Gibbs sampler
43 Gibbs sampler
44 Gibbs sampler After draws (red); comparison to the i.i.d. case (blue)
45 Gibbs sampler Correlated case Target distribution on R : [ ]) a ρab (X, Y ) N (, ρab b ρ [, ]. Sampled by a Gibbs algorithm Conditional distribution of X given Y : N (ρay/b, a ( ρ )) Conditional distribution of Y given X: N (ρbx/a, b ( ρ )) we choose a = b = ρ =.95
46 Markov chain Monte Carlo methods Gibbs sampler After draws (red); comparison to the i.i.d. case (blue) x 4 x 4 Estimation of E[X] (left) and E[Y ] (right), by the Gibbs sampler (blue) and by i.i.d. draws (red)
47 Gibbs sampler Ising model (/) Target distribution on {, } d say: d black or white vertices π(x) exp β i j I xi x j so that the conditional distribution of x i given the other vertices P(x i = x i ) exp ( β Card{j i, x j = })
48 Gibbs sampler Ising model (/) Run: 4 chains of length, for 4 different values of β when β increases, the correlation increases. Displayed: the number of black points vs the number of iterations.
49 Burn in Outline Independent Hastings Metropolis Sampler Symmetric random walk HM algorithm Gibbs sampler Burn in
50 Markov chain Monte Carlo methods Burn in Burn-in period (/) Illustration in the case of a symmetric random walk HM: Target distribution π(x) = exp( x) (on R + ). Proposal distribution: N (, ) Tow paths started resp. at : X = 5 and X =. Displayed: the path (top/bottom, left) and the estimation of the expectation of π (top/bottom, right)
51 Burn in Burn-in period (/) Paths and auto-correlation function
Image analysis. Computer Vision and Classification Image Segmentation. 7 Image analysis
7 Computer Vision and Classification 413 / 458 Computer Vision and Classification The k-nearest-neighbor method The k-nearest-neighbor (knn) procedure has been used in data analysis and machine learning
MCMC Methods for data modeling
MCMC Methods for data modeling Kenneth Scerri Department of Automatic Control and Systems Engineering Introduction 1. Symposium on Data Modelling 2. Outline: a. Definition and uses of MCMC b. MCMC algorithms
1 Methods for Posterior Simulation
1 Methods for Posterior Simulation Let p(θ y) be the posterior. simulation. Koop presents four methods for (posterior) 1. Monte Carlo integration: draw from p(θ y). 2. Gibbs sampler: sequentially drawing
Overview. Monte Carlo Methods. Statistics & Bayesian Inference Lecture 3. Situation At End Of Last Week
Statistics & Bayesian Inference Lecture 3 Joe Zuntz Overview Overview & Motivation Metropolis Hastings Monte Carlo Methods Importance sampling Direct sampling Gibbs sampling Monte-Carlo Markov Chains Emcee
Convergence and Efficiency of Adaptive MCMC. Jinyoung Yang
Convergence and Efficiency of Adaptive MCMC by Jinyoung Yang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Statistical Sciences University
MSA101/MVE Lecture 5
MSA101/MVE187 2017 Lecture 5 Petter Mostad Chalmers University September 12, 2017 1 / 15 Importance sampling MC integration computes h(x)f (x) dx where f (x) is a probability density function, by simulating
arxiv: v3 [stat.co] 27 Apr 2012
![arxiv: v3 [stat.co] 27 Apr 2012](/thumbs/86/93793267.jpg "arxiv: v3 [stat.co] 27 Apr 2012")
A multi-point Metropolis scheme with generic weight functions arxiv:1112.4048v3 stat.co 27 Apr 2012 Abstract Luca Martino, Victor Pascual Del Olmo, Jesse Read Department of Signal Theory and Communications,
Chapter 1. Introduction
Chapter 1 Introduction A Monte Carlo method is a compuational method that uses random numbers to compute (estimate) some quantity of interest. Very often the quantity we want to compute is the mean of
Monte Carlo sampling
1 y u theta 0 x 1 Monte Carlo sampling Problem 1 Suppose we want to sample uniformly at random from the triangle defined by the points (0,0), (0,1), (1,0). First Sampling Algorithm: We decide to do this
Package atmcmc. February 19, 2015
Type Package Package atmcmc February 19, 2015 Title Automatically Tuned Markov Chain Monte Carlo Version 1.0 Date 2014-09-16 Author Jinyoung Yang Maintainer Jinyoung Yang
Quantitative Biology II!
Quantitative Biology II! Lecture 3: Markov Chain Monte Carlo! March 9, 2015! 2! Plan for Today!! Introduction to Sampling!! Introduction to MCMC!! Metropolis Algorithm!! Metropolis-Hastings Algorithm!!
An Introduction to Markov Chain Monte Carlo
An Introduction to Markov Chain Monte Carlo Markov Chain Monte Carlo (MCMC) refers to a suite of processes for simulating a posterior distribution based on a random (ie. monte carlo) process. In other
Probabilistic Graphical Models
Probabilistic Graphical Models Lecture 17 EM CS/CNS/EE 155 Andreas Krause Announcements Project poster session on Thursday Dec 3, 4-6pm in Annenberg 2 nd floor atrium! Easels, poster boards and cookies
Package gibbs.met documentation
Version 1.1-2 Package gibbs.met documentation Title Naive Gibbs Sampling with Metropolis Steps Author Longhai Li of March 26, 2008 Maintainer Longhai Li
Convexization in Markov Chain Monte Carlo
in Markov Chain Monte Carlo 1 IBM T. J. Watson Yorktown Heights, NY 2 Department of Aerospace Engineering Technion, Israel August 23, 2011 Problem Statement MCMC processes in general are governed by non
Statistical Matching using Fractional Imputation
Statistical Matching using Fractional Imputation Jae-Kwang Kim 1 Iowa State University 1 Joint work with Emily Berg and Taesung Park 1 Introduction 2 Classical Approaches 3 Proposed method 4 Application:
Package gibbs.met. February 19, 2015
Version 1.1-3 Title Naive Gibbs Sampling with Metropolis Steps Author Longhai Li Package gibbs.met February 19, 2015 Maintainer Longhai Li Depends R (>=
Monte Carlo for Spatial Models
Monte Carlo for Spatial Models Murali Haran Department of Statistics Penn State University Penn State Computational Science Lectures April 2007 Spatial Models Lots of scientific questions involve analyzing
Bayesian Statistics Group 8th March Slice samplers. (A very brief introduction) The basic idea
Bayesian Statistics Group 8th March 2000 Slice samplers (A very brief introduction) The basic idea lacements To sample from a distribution, simply sample uniformly from the region under the density function
Approximate Bayesian Computation. Alireza Shafaei - April 2016
Approximate Bayesian Computation Alireza Shafaei - April 2016 The Problem Given a dataset, we are interested in. The Problem Given a dataset, we are interested in. The Problem Given a dataset, we are interested
Package dirmcmc. R topics documented: February 27, 2017
Type Package Title Directional Metropolis Hastings Algorithm Version 1.3.3 Date 2017-02-17 Author Abhirup Mallik Package dirmcmc February 27, 2017 Maintainer Abhirup Mallik
arxiv: v2 [stat.co] 19 Feb 2016
![arxiv: v2 [stat.co] 19 Feb 2016](/thumbs/73/68585270.jpg "arxiv: v2 [stat.co] 19 Feb 2016")
Noname manuscript No. (will be inserted by the editor) Issues in the Multiple Try Metropolis mixing L. Martino F. Louzada Received: date / Accepted: date arxiv:158.4253v2 [stat.co] 19 Feb 216 Abstract
Optimization Methods III. The MCMC. Exercises.
Aula 8. Optimization Methods III. Exercises. 0 Optimization Methods III. The MCMC. Exercises. Anatoli Iambartsev IME-USP Aula 8. Optimization Methods III. Exercises. 1 [RC] A generic Markov chain Monte
Markov Chain Monte Carlo (part 1)
Markov Chain Monte Carlo (part 1) Edps 590BAY Carolyn J. Anderson Department of Educational Psychology c Board of Trustees, University of Illinois Spring 2018 Depending on the book that you select for
Stochastic Simulation: Algorithms and Analysis
Soren Asmussen Peter W. Glynn Stochastic Simulation: Algorithms and Analysis et Springer Contents Preface Notation v xii I What This Book Is About 1 1 An Illustrative Example: The Single-Server Queue 1
Level-set MCMC Curve Sampling and Geometric Conditional Simulation
Level-set MCMC Curve Sampling and Geometric Conditional Simulation Ayres Fan John W. Fisher III Alan S. Willsky February 16, 2007 Outline 1. Overview 2. Curve evolution 3. Markov chain Monte Carlo 4. Curve
Clustering Relational Data using the Infinite Relational Model
Clustering Relational Data using the Infinite Relational Model Ana Daglis Supervised by: Matthew Ludkin September 4, 2015 Ana Daglis Clustering Data using the Infinite Relational Model September 4, 2015
Markov Random Fields and Gibbs Sampling for Image Denoising
Markov Random Fields and Gibbs Sampling for Image Denoising Chang Yue Electrical Engineering Stanford University changyue@stanfoed.edu Abstract This project applies Gibbs Sampling based on different Markov
Computer vision: models, learning and inference. Chapter 10 Graphical Models
Computer vision: models, learning and inference Chapter 10 Graphical Models Independence Two variables x 1 and x 2 are independent if their joint probability distribution factorizes as Pr(x 1, x 2 )=Pr(x
MCMC Diagnostics. Yingbo Li MATH Clemson University. Yingbo Li (Clemson) MCMC Diagnostics MATH / 24
MCMC Diagnostics Yingbo Li Clemson University MATH 9810 Yingbo Li (Clemson) MCMC Diagnostics MATH 9810 1 / 24 Convergence to Posterior Distribution Theory proves that if a Gibbs sampler iterates enough,
Network-based auto-probit modeling for protein function prediction
Network-based auto-probit modeling for protein function prediction Supplementary material Xiaoyu Jiang, David Gold, Eric D. Kolaczyk Derivation of Markov Chain Monte Carlo algorithm with the GO annotation
Curve Sampling and Geometric Conditional Simulation
Curve Sampling and Geometric Conditional Simulation Ayres Fan Joint work with John Fisher, William Wells, Jonathan Kane, and Alan Willsky S S G Massachusetts Institute of Technology September 19, 2007
GAMES Webinar: Rendering Tutorial 2. Monte Carlo Methods. Shuang Zhao
GAMES Webinar: Rendering Tutorial 2 Monte Carlo Methods Shuang Zhao Assistant Professor Computer Science Department University of California, Irvine GAMES Webinar Shuang Zhao 1 Outline 1. Monte Carlo integration
Statistical techniques for data analysis in Cosmology
Statistical techniques for data analysis in Cosmology arxiv:0712.3028; arxiv:0911.3105 Numerical recipes (the bible ) Licia Verde ICREA & ICC UB-IEEC http://icc.ub.edu/~liciaverde outline Lecture 1: Introduction
Detecting Piecewise Linear Networks Using Reversible Jump Markov Chain Monte Carlo
Clemson University TigerPrints All Theses Theses 8-2010 Detecting Piecewise Linear Networks Using Reversible Jump Markov Chain Monte Carlo Akshay Apte Clemson University, apte.aa@gmail.com Follow this
Graphical Models, Bayesian Method, Sampling, and Variational Inference
Graphical Models, Bayesian Method, Sampling, and Variational Inference With Application in Function MRI Analysis and Other Imaging Problems Wei Liu Scientific Computing and Imaging Institute University
Linear Modeling with Bayesian Statistics
Linear Modeling with Bayesian Statistics Bayesian Approach I I I I I Estimate probability of a parameter State degree of believe in specific parameter values Evaluate probability of hypothesis given the
INTRO TO THE METROPOLIS ALGORITHM
INTRO TO THE METROPOLIS ALGORITHM A famous reliability experiment was performed where n = 23 ball bearings were tested and the number of revolutions were recorded. The observations in ballbearing2.dat
Short-Cut MCMC: An Alternative to Adaptation
Short-Cut MCMC: An Alternative to Adaptation Radford M. Neal Dept. of Statistics and Dept. of Computer Science University of Toronto http://www.cs.utoronto.ca/ radford/ Third Workshop on Monte Carlo Methods,
Clustering web search results
Clustering K-means Machine Learning CSE546 Emily Fox University of Washington November 4, 2013 1 Clustering images Set of Images [Goldberger et al.] 2 1 Clustering web search results 3 Some Data 4 2 K-means
Missing Data Analysis for the Employee Dataset
Missing Data Analysis for the Employee Dataset 67% of the observations have missing values! Modeling Setup Random Variables: Y i =(Y i1,...,y ip ) 0 =(Y i,obs, Y i,miss ) 0 R i =(R i1,...,r ip ) 0 ( 1
GT "Calcul Ensembliste"
GT "Calcul Ensembliste" Beyond the bounded error framework for non linear state estimation Fahed Abdallah Université de Technologie de Compiègne 9 Décembre 2010 Fahed Abdallah GT "Calcul Ensembliste" 9
Semiparametric Mixed Effecs with Hierarchical DP Mixture
Semiparametric Mixed Effecs with Hierarchical DP Mixture R topics documented: April 21, 2007 hdpm-package........................................ 1 hdpm............................................ 2 hdpmfitsetup........................................
Bayesian Spatiotemporal Modeling with Hierarchical Spatial Priors for fmri
Bayesian Spatiotemporal Modeling with Hierarchical Spatial Priors for fmri Galin L. Jones 1 School of Statistics University of Minnesota March 2015 1 Joint with Martin Bezener and John Hughes Experiment
An Efficient Model Selection for Gaussian Mixture Model in a Bayesian Framework
IEEE SIGNAL PROCESSING LETTERS, VOL. XX, NO. XX, XXX 23 An Efficient Model Selection for Gaussian Mixture Model in a Bayesian Framework Ji Won Yoon arxiv:37.99v [cs.lg] 3 Jul 23 Abstract In order to cluster
Clustering Lecture 5: Mixture Model
Clustering Lecture 5: Mixture Model Jing Gao SUNY Buffalo 1 Outline Basics Motivation, definition, evaluation Methods Partitional Hierarchical Density-based Mixture model Spectral methods Advanced topics
AN ADAPTIVE POPULATION IMPORTANCE SAMPLER. Luca Martino*, Victor Elvira\ David Luengcfi, Jukka Corander*
AN ADAPTIVE POPULATION IMPORTANCE SAMPLER Luca Martino*, Victor Elvira\ David Luengcfi, Jukka Corander* * Dep. of Mathematics and Statistics, University of Helsinki, 00014 Helsinki (Finland). t Dep. of
Expectation Maximization (EM) and Gaussian Mixture Models
Expectation Maximization (EM) and Gaussian Mixture Models Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer 1 2 3 4 5 6 7 8 Unsupervised Learning Motivation
CSCI 599 Class Presenta/on. Zach Levine. Markov Chain Monte Carlo (MCMC) HMM Parameter Es/mates
CSCI 599 Class Presenta/on Zach Levine Markov Chain Monte Carlo (MCMC) HMM Parameter Es/mates April 26 th, 2012 Topics Covered in this Presenta2on A (Brief) Review of HMMs HMM Parameter Learning Expecta2on-
Modified Metropolis-Hastings algorithm with delayed rejection
Modified Metropolis-Hastings algorithm with delayed reection K.M. Zuev & L.S. Katafygiotis Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong, China ABSTRACT: The
Digital Image Processing Laboratory: Markov Random Fields and MAP Image Segmentation
Purdue University: Digital Image Processing Laboratories Digital Image Processing Laboratory: Markov Random Fields and MAP Image Segmentation December, 205 Introduction This laboratory explores the use
Monte Carlo Methods and Statistical Computing: My Personal E
Monte Carlo Methods and Statistical Computing: My Personal Experience Department of Mathematics & Statistics Indian Institute of Technology Kanpur November 29, 2014 Outline Preface 1 Preface 2 3 4 5 6
Monte Carlo Techniques for Bayesian Statistical Inference A comparative review
1 Monte Carlo Techniques for Bayesian Statistical Inference A comparative review BY Y. FAN, D. WANG School of Mathematics and Statistics, University of New South Wales, Sydney 2052, AUSTRALIA 18th January,
Scalable Bayes Clustering for Outlier Detection Under Informative Sampling
Scalable Bayes Clustering for Outlier Detection Under Informative Sampling Based on JMLR paper of T. D. Savitsky Terrance D. Savitsky Office of Survey Methods Research FCSM - 2018 March 7-9, 2018 1 / 21
Use of Extreme Value Statistics in Modeling Biometric Systems
Use of Extreme Value Statistics in Modeling Biometric Systems Similarity Scores Two types of matching: Genuine sample Imposter sample Matching scores Enrolled sample 0.95 0.32 Probability Density Decision
Probability Model for 2 RV s
Probability Model for 2 RV s The joint probability mass function of X and Y is P X,Y (x, y) = P [X = x, Y= y] Joint PMF is a rule that for any x and y, gives the probability that X = x and Y= y. 3 Example:
Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin
Clustering K-means Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin 2005-2014 1 Clustering images Set of Images [Goldberger et al.] Carlos Guestrin 2005-2014
INLA: Integrated Nested Laplace Approximations
INLA: Integrated Nested Laplace Approximations John Paige Statistics Department University of Washington October 10, 2017 1 The problem Markov Chain Monte Carlo (MCMC) takes too long in many settings.
GiRaF: a toolbox for Gibbs Random Fields analysis
GiRaF: a toolbox for Gibbs Random Fields analysis Julien Stoehr *1, Pierre Pudlo 2, and Nial Friel 1 1 University College Dublin 2 Aix-Marseille Université February 24, 2016 Abstract GiRaF package offers
Package bayesdccgarch
Version 2.0 Date 2016-01-29 Package bayesdccgarch February 7, 2016 Title The Bayesian Dynamic Conditional Correlation GARCH Model Bayesian estimation of dynamic conditional correlation GARCH model for
Hierarchical Bayesian Modeling with Ensemble MCMC. Eric B. Ford (Penn State) Bayesian Computing for Astronomical Data Analysis June 12, 2014
Hierarchical Bayesian Modeling with Ensemble MCMC Eric B. Ford (Penn State) Bayesian Computing for Astronomical Data Analysis June 12, 2014 Simple Markov Chain Monte Carlo Initialise chain with θ 0 (initial
CS281 Section 9: Graph Models and Practical MCMC
CS281 Section 9: Graph Models and Practical MCMC Scott Linderman November 11, 213 Now that we have a few MCMC inference algorithms in our toolbox, let s try them out on some random graph models. Graphs
K-Means and Gaussian Mixture Models
K-Means and Gaussian Mixture Models David Rosenberg New York University June 15, 2015 David Rosenberg (New York University) DS-GA 1003 June 15, 2015 1 / 43 K-Means Clustering Example: Old Faithful Geyser
Parallel Multivariate Slice Sampling
Parallel Multivariate Slice Sampling Matthew M. Tibbits Department of Statistics Pennsylvania State University mmt143@stat.psu.edu Murali Haran Department of Statistics Pennsylvania State University mharan@stat.psu.edu
Statistical modelling of the patterns of planar sections of prostatic capillaries on the basis of Strauss hard-core processes
TITLE Statistical modelling of the patterns of planar sections of prostatic capillaries on the basis of Strauss hard-core processes AUTHORS Torsten Mattfeldt 1 Stefanie Eckel 2 Frank Fleischer 3 Volker
Debugging MCMC Code. Charles J. Geyer. April 15, 2017
Debugging MCMC Code Charles J. Geyer April 15, 2017 1 Introduction This document discusses debugging Markov chain Monte Carlo code using the R contributed package mcmc (Version 0.9-5) for examples. It
Fitting Social Network Models Using the Varying Truncation S. Truncation Stochastic Approximation MCMC Algorithm
Fitting Social Network Models Using the Varying Truncation Stochastic Approximation MCMC Algorithm May. 17, 2012 1 This talk is based on a joint work with Dr. Ick Hoon Jin Abstract The exponential random
Clustering K-means. Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, Carlos Guestrin
Clustering K-means Machine Learning CSEP546 Carlos Guestrin University of Washington February 18, 2014 Carlos Guestrin 2005-2014 1 Clustering images Set of Images [Goldberger et al.] Carlos Guestrin 2005-2014
Theoretical Concepts of Machine Learning
Theoretical Concepts of Machine Learning Part 2 Institute of Bioinformatics Johannes Kepler University, Linz, Austria Outline 1 Introduction 2 Generalization Error 3 Maximum Likelihood 4 Noise Models 5
A Nonparametric Bayesian Approach to Detecting Spatial Activation Patterns in fmri Data
A Nonparametric Bayesian Approach to Detecting Spatial Activation Patterns in fmri Data Seyoung Kim, Padhraic Smyth, and Hal Stern Bren School of Information and Computer Sciences University of California,
Visual Learning with Explicit and Implicit Manifolds
Visual Learning with Explicit and Implicit Manifolds --- A Mathematical Model of, Texton, and Primal Sketch Song-Chun Zhu Departments of Statistics and Computer Science University of California, Los Angeles
A Short History of Markov Chain Monte Carlo
A Short History of Markov Chain Monte Carlo Christian Robert and George Casella 2010 Introduction Lack of computing machinery, or background on Markov chains, or hesitation to trust in the practicality
No more questions will be added
CSC 2545, Spring 2017 Kernel Methods and Support Vector Machines Assignment 2 Due at the start of class, at 2:10pm, Thurs March 23. No late assignments will be accepted. The material you hand in should
Note Set 4: Finite Mixture Models and the EM Algorithm
Note Set 4: Finite Mixture Models and the EM Algorithm Padhraic Smyth, Department of Computer Science University of California, Irvine Finite Mixture Models A finite mixture model with K components, for
Parallel Adaptive Importance Sampling. Cotter, Colin and Cotter, Simon and Russell, Paul. MIMS EPrint:
Parallel Adaptive Importance Sampling Cotter, Colin and Cotter, Simon and Russell, Paul 2015 MIMS EPrint: 2015.82 Manchester Institute for Mathematical Sciences School of Mathematics The University of
Laboratorio di Problemi Inversi Esercitazione 4: metodi Bayesiani e importance sampling
Laboratorio di Problemi Inversi Esercitazione 4: metodi Bayesiani e importance sampling Luca Calatroni Dipartimento di Matematica, Universitá degli studi di Genova May 19, 2016. Luca Calatroni (DIMA, Unige)
R Programming Basics - Useful Builtin Functions for Statistics
R Programming Basics - Useful Builtin Functions for Statistics Vectorized Arithmetic - most arthimetic operations in R work on vectors. Here are a few commonly used summary statistics. testvect = c(1,3,5,2,9,10,7,8,6)
Slice sampler algorithm for generalized Pareto distribution
Slice sampler algorithm for generalized Pareto distribution Mohammad Rostami, Mohd Bakri Adam Yahya, Mohamed Hisham Yahya, Noor Akma Ibrahim Abstract In this paper, we developed the slice sampler algorithm
Expectation Maximization: Inferring model parameters and class labels
Expectation Maximization: Inferring model parameters and class labels Emily Fox University of Washington February 27, 2017 Mixture of Gaussian recap 1 2/26/17 Jumble of unlabeled images HISTOGRAM blue
Variability in Annual Temperature Profiles
Variability in Annual Temperature Profiles A Multivariate Spatial Analysis of Regional Climate Model Output Tamara Greasby, Stephan Sain Institute for Mathematics Applied to Geosciences, National Center
The Multiset Sampler
The Multiset Sampler Scotland C. Leman Department of Statistics Virginia Polytechnic Institute Blacksburg, VA 24061 Yuguo Chen Department of Statistics University of Illinois at Urbana-Champaign Champaign,
Bayesian Modelling with JAGS and R
Bayesian Modelling with JAGS and R Martyn Plummer International Agency for Research on Cancer Rencontres R, 3 July 2012 CRAN Task View Bayesian Inference The CRAN Task View Bayesian Inference is maintained
Statistical image models
Chapter 4 Statistical image models 4. Introduction 4.. Visual worlds Figure 4. shows images that belong to different visual worlds. The first world (fig. 4..a) is the world of white noise. It is the world
Institute for Statics und Dynamics of Structures Fuzzy Time Series
Institute for Statics und Dynamics of Structures Fuzzy Time Series Bernd Möller 1 Description of fuzzy time series 2 3 4 5 Conclusions Folie 2 von slide422 1 Description of fuzzy time series 2 3 4 5 Conclusions
Extracting Coactivated Features from Multiple Data Sets
Extracting Coactivated Features from Multiple Data Sets Michael U. Gutmann University of Helsinki michael.gutmann@helsinki.fi Aapo Hyvärinen University of Helsinki aapo.hyvarinen@helsinki.fi Michael U.
ADAPTIVE METROPOLIS-HASTINGS SAMPLING, OR MONTE CARLO KERNEL ESTIMATION
ADAPTIVE METROPOLIS-HASTINGS SAMPLING, OR MONTE CARLO KERNEL ESTIMATION CHRISTOPHER A. SIMS Abstract. A new algorithm for sampling from an arbitrary pdf. 1. Introduction Consider the standard problem of
Machine Learning and Data Mining. Clustering (1): Basics. Kalev Kask
Machine Learning and Data Mining Clustering (1): Basics Kalev Kask Unsupervised learning Supervised learning Predict target value ( y ) given features ( x ) Unsupervised learning Understand patterns of
AMCMC: An R interface for adaptive MCMC
AMCMC: An R interface for adaptive MCMC by Jeffrey S. Rosenthal * (February 2007) Abstract. We describe AMCMC, a software package for running adaptive MCMC algorithms on user-supplied density functions.
Discussion on Bayesian Model Selection and Parameter Estimation in Extragalactic Astronomy by Martin Weinberg
Discussion on Bayesian Model Selection and Parameter Estimation in Extragalactic Astronomy by Martin Weinberg Phil Gregory Physics and Astronomy Univ. of British Columbia Introduction Martin Weinberg reported
Mixture Models and the EM Algorithm
Mixture Models and the EM Algorithm Padhraic Smyth, Department of Computer Science University of California, Irvine c 2017 1 Finite Mixture Models Say we have a data set D = {x 1,..., x N } where x i is
A GENERAL GIBBS SAMPLING ALGORITHM FOR ANALYZING LINEAR MODELS USING THE SAS SYSTEM
A GENERAL GIBBS SAMPLING ALGORITHM FOR ANALYZING LINEAR MODELS USING THE SAS SYSTEM Jayawant Mandrekar, Daniel J. Sargent, Paul J. Novotny, Jeff A. Sloan Mayo Clinic, Rochester, MN 55905 ABSTRACT A general
Machine Learning A W 1sst KU. b) [1 P] Give an example for a probability distributions P (A, B, C) that disproves
![Machine Learning A W 1sst KU. b) [1 P] Give an example for a probability distributions P (A, B, C) that disproves](/thumbs/93/111796408.jpg "Machine Learning A W 1sst KU. b) [1 P] Give an example for a probability distributions P (A, B, C) that disproves")
Machine Learning A 708.064 11W 1sst KU Exercises Problems marked with * are optional. 1 Conditional Independence I [2 P] a) [1 P] Give an example for a probability distribution P (A, B, C) that disproves
The Rendering Equation and Path Tracing
The Rendering Equation and Path Tracing Louis Feng April 22, 2004 April 21, 2004 Realistic Image Synthesis (Spring 2004) 1 Topics The rendering equation Original form Meaning of the terms Integration Path
Artificial Neural Networks Unsupervised learning: SOM
Artificial Neural Networks Unsupervised learning: SOM 01001110 01100101 01110101 01110010 01101111 01101110 01101111 01110110 01100001 00100000 01110011 01101011 01110101 01110000 01101001 01101110 01100001
Estimating the Information Rate of Noisy Two-Dimensional Constrained Channels
Estimating the Information Rate of Noisy Two-Dimensional Constrained Channels Mehdi Molkaraie and Hans-Andrea Loeliger Dept. of Information Technology and Electrical Engineering ETH Zurich, Switzerland
Particle Filters for Visual Tracking
Particle Filters for Visual Tracking T. Chateau, Pascal Institute, Clermont-Ferrand 1 Content Particle filtering: a probabilistic framework SIR particle filter MCMC particle filter RJMCMC particle filter
10.4 Linear interpolation method Newton s method
10.4 Linear interpolation method The next best thing one can do is the linear interpolation method, also known as the double false position method. This method works similarly to the bisection method by
From Bayesian Analysis of Item Response Theory Models Using SAS. Full book available for purchase here.
From Bayesian Analysis of Item Response Theory Models Using SAS. Full book available for purchase here. Contents About this Book...ix About the Authors... xiii Acknowledgments... xv Chapter 1: Item Response
Analysis of Incomplete Multivariate Data
Analysis of Incomplete Multivariate Data J. L. Schafer Department of Statistics The Pennsylvania State University USA CHAPMAN & HALL/CRC A CR.C Press Company Boca Raton London New York Washington, D.C.
Image Segmentation using Gaussian Mixture Models
Image Segmentation using Gaussian Mixture Models Rahman Farnoosh, Gholamhossein Yari and Behnam Zarpak Department of Applied Mathematics, University of Science and Technology, 16844, Narmak,Tehran, Iran