Bayesian modelling. Hans-Peter Helfrich. University of Bonn. Theodor-Brinkmann-Graduate School
|
|
- Hector Neal
- 6 years ago
- Views:
Transcription
1 Bayesian modelling Hans-Peter Helfrich University of Bonn Theodor-Brinkmann-Graduate School H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 1 / 22
2 Overview 1 Bayesian modelling 2 Examples 3 WinBUGS - A Bayesian modelling framework 4 Markov Chain Monte Carlo Algorithm 5 References H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 2 / 22
3 Bayesian modelling Objectives [Van Oijen et al., 2005] Bridging the gaps between models and data Bayesian calibration Inferring the parameter from the data (outcome) Bayesian model Parameters θ 1,..., θ n ModelL(y θ) Data y = (y 1,..., y m ) Main steps (cf. sses/ps and pdf/stb.pdf) 1 Process model 2 Data model 3 Prior density distribution H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 3 / 22
4 Bayes theorem Statistical inference can be done via Bayes theorem. The posterior distribution is given by p(θ y) L(y θ)p(θ) up to a normalizing factor. The posterior density function contains all statistical information for providing mean values, medians, and credible intervals. Sampling methods In general, the normalizing factor cannot be explicitly calculated. That can be done by sampling methods that give samples with the posterior density distribution. Mainly, two methods can be used Gibbs sampling Markov Chain Monte Carlo Method H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 4 / 22
5 Linear regression Process model We have the simple model x i = at i + b, i = 1,..., n. Data model The theoretical outcomes x 1,..., x n are disturbed by random errors ε 1,..., ε n : y i = x i + ε i, i = 1,..., n. We assume that random errors are independent and normally distributed 1 L(y θ) = ( ( 2π) n σ exp (y 1 x 1 ) 2 ) n 2σ 2 exp ( (y n x n ) 2 ) 2σ 2 with the parameter vector θ = (a, b, σ). H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 5 / 22
6 Prior For the parameters to be estimated, prior densities should be specified. If no information is available, we may choose the non-informative prior, i.e., p(θ) = const. Several experiments Bayes methods allow updating the prior information. For the first data set y 1, we get by Bayes theorem p 1 (θ y 1 ) L 1 (y 1 θ)p(θ) For the second data set y 2, we may take p 1 (θ y 1 ) as prior density to obtain and so on. p 2 (θ y 2 ) L 2 (y 2 θ)p 1 (θ y 1 ) H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 6 / 22
7 Nonlinear models Distance between two walls An electronic measuring device measures the distance between two points by sending an electromagnetic beam. Assume we want to measure the perpendicular distance θ between two walls. In practice, the beam is not perpendicular to the walls. Assume we have a displacement e at our measurement. z = θ 2 + e 2 θ e For a displacement e, we get by the theorem of Pythagoras as length z z = θ 2 + e 2 H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 7 / 22
8 Model specification Model specification Our model looks like that e N (0, σ 2 p) z = θ 2 + e 2 We may introduce for z an additional error caused by the measuring device y N (z, σ 2 m) e N (0, σ 2 p) z = θ 2 + e 2. We can simulate the density distribution of the measurements with a random number generator. For example, we can use in R the function rnorm() for getting normally distributed samples. H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 8 / 22
9 Distribution of the error Error caused by displacement Density Distance [m] Distribution of the error σ p = 0.3, σ m = 0, σ = H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 9 / 22
10 Distribution of the error Error caused by displacement and by device Density Distance [m] Distribution of the error σ p = 0.3, σ m = 0.005, σ = H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 9 / 22
11 WinBUGS - A Bayesian modelling framework WinBUGS [Lunn et al., 2000] WinBUGS is a fully extensible modular framework for constructing and analysing Bayesian full probability models. Models may be specified either textually via the BUGS language or pictorially using a graphical interface called DoodleBUGS. BUGS is an acronym for Bayesian inference Using Gibbs Sampling. Linear model In WinBUGS, a linear model is specified by model { } y dnorm(x, tau) x <- a*t + b H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 10 / 22
12 Linear regression We extend the model by incorporating N observations and by assigning priors. model { for (j in 1:N) { y[j] dnorm(x[j], tau) x[j] <- a*t[j] + b } a dunif(0,a_max) b dunif(0,b_max) tau dunif(0,tau_max) } For a, b and τ we assign non-informative priors. By dnorm(x[j], tau) the Gaussian density is specified, where τ denotes the precision 1/σ 2 H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 11 / 22
13 Nonlinear model As before, we get by the theorem of Pythagoras the length z z = θ 2 + e 2, where e denotes the displacement, and θ denotes the distance between the two walls. Distance meter model: BUGS code model { y dnorm(z, tau_m) e dnorm(0, tau_p) z <- sqrt(theta * theta + e * e) theta dunif(a, b) } H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 12 / 22
14 Distance meter example We extend the model by incorporating N observations. model { for (j in 1:N) { y[j] dnorm(z[j], tau_m) e[j] dnorm(0, tau_p) z[j] <- sqrt(theta * theta + e[j] * e[j]) } theta dunif(a, b) } The displacement e which is different for each measurement changes the observed distance to a value z. The distributions of the errors are specified by normal distributions with precisions τ p and τ m. In the last line, a uniform prior distribution of the unknown distance is given. H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 13 / 22
15 Visualization Directed graph by Doodle H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 14 / 22
16 Markov Chain Monte Carlo Algorithms [Hastings, 1970] Markov Chains [Gelman et al., 2004] We consider a system which may be at time t in a certain state X (t). First we consider a finite set of states, say i = 1,..., S. The states may be thought as different locations. Transition matrix P The probability that we come from location i to the location j is denoted by p ij. The matrix P = (p ij ) is called the transition matrix. A Markov chain is characterized by assuming that the state X (t + 1) is determined by X (t), and does not explicitly depend on former states. Starting distribution By π 1,..., π S we denote the probabilities for being at time t at locations 1,..., S. H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 15 / 22
17 Coming to location 1 The probability for coming from location i to location 1 is p i1. Since we are at i with probability π i, we get π 1 p 11 + π 2 p π S p S1 as probability for coming to location 1 at time t + 1. In a similiar way, we get the probability for coming to location j. Distribution at time t + 1 The probability that at time t + 1 the location j is reached is given by π i p ij. In matrix notation, we have πp. i H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 16 / 22
18 Irreducibility For two steps, the transition matrix is given by P 2, for three steps by P 3 and so on. A Markov chain is said to be irreducible if there exists for every i and every j some n such that p n ij is positive. Stationarity We call the distribution π stationary if π j = i π i p ij. It means that the distribution does not change in further steps. Construction of the transition matrix For a distribution π i, we are looking for transition matrix which has the given distribution as stationary distribution. H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 17 / 22
19 Markov Chain Monte Carlo Algorithm Jumping rule We choose a transition matrix Q = (q ij ) called jumping or proposal distribution, which must be symmetric. If we are at location i we go to location j with probability q ij. That means each row of Q gives a probability distribution Two stage algorithm We choose a jumping matrix Q = {q ij } and consider the algorithm 1 Assume that X (t) = i and select a state j using a distribution given by the ith row of Q 2 Take X (t + 1) = j with probability α ij and X (t + 1) = i with probability 1 α ij, where α ij = { 1 (πj /π i 1) π j /π i, (π j /π i < 1) H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 18 / 22
20 Markov Chain Monte Carlo Algorithm Samples from a given distribution Assume we have a probability distribution (π 1,..., π S ). The general form of our algorithm is the following: 1 Set k = 1, X (k) = i where i {1,..., S} is chosen at random. 2 Draw a sample j at random with probability q ij, set k k Calculate the odds ratio r = π j /π i If r 1 set If r < 1 set 4 Set i = X (k) 5 Go to Step 2 X (k) = X (k) = j { j with probability r i with probability 1 r H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 19 / 22
21 Explanation of the acceptance and rejection rules Reversibility We consider matrices P which satisfy the reversibility condition π i p ij = π j p ji which implies stationarity. We put p ij = q ij α ij, j = i. Since j p ji = 1, this property ensures stationarity π j = i π i p ij. Theorem The transition matrix q ij α ij satisfies the reversibility condition. Generalization to a continuous density distribution The algorithm can be extented for a continuous distribution. This gives th famous Metropolis algorithm. H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 20 / 22
22 Metropolis algoritm Assume we have a probability distribution π(x) = π(x 1, x 2,..., x n ) for all vectors x = (x 1, x 2,..., x n ) in a given set U R n. Choose a function q(x, y), which is symmetric such that q(., y) is for each y a probability distribution. The general form of our algorithm is the following: 1 Set k = 1, X (k) = x where x U is chosen at random 2 Draw a sample y q(., x) at random, set k k Calculate the odds ratio r = π(y)/π(x) If r 1 set If r < 1 set 4 Set x = X (k) 5 Go to Step 2 X (k) = X (k) = y { y with probability r x with probability 1 r H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 21 / 22
23 References Gelman, A. B., Carlin, J. S., Stern, H., and Rubin, D. (2004). Bayesian Data Analysis. Chapman & Hall. Hastings, W. K. (1970). Monte carlo sampling methods using Markov chains and their applications. Biometrika, 57: Lunn, D. J., Thomas, A., Best, N., and Spiegelhalter, D. (2000). Winbugs - a Bayesian modelling framework: Concepts, structure, and extensibility. Statistics and Computing (2000), 10: Van Oijen, M., Rougier, J., and Smith, R. (2005). Bayesian calibration of process-based forest models: bridging the gap between models and data. Tree Physiology, 25(7): H.-P. Helfrich (University of Bonn) Bayesian modelling Brinkmann School 22 / 22
eqr094: Hierarchical MCMC for Bayesian System Reliability
eqr094: Hierarchical MCMC for Bayesian System Reliability Alyson G. Wilson Statistical Sciences Group, Los Alamos National Laboratory P.O. Box 1663, MS F600 Los Alamos, NM 87545 USA Phone: 505-667-9167
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 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 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 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 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 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 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 Inference. Chapter 1. Introduction and basic concepts
Bayesian Inference Chapter 1. Introduction and basic concepts M. Concepción Ausín Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master
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 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 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 informationBayesian Graphical Models
Graphical Models and Inference, Lecture 16, Michaelmas Term 2009 December 4, 2009 Parameter θ, data X = x, likelihood L(θ x) p(x θ). Express knowledge about θ through prior distribution π on θ. Inference
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
MARKOV CHAIN MONTE CARLO RYAN WANG Abstract. This paper gives a brief introduction to Markov Chain Monte Carlo methods, which offer a general framework for calculating difficult integrals. We start with
More informationAdvanced Statistical Modelling
Markov chain Monte Carlo (MCMC) Methods and Their Applications in Bayesian Statistics School of Technology and Business Studies/Statistics Dalarna University Borlänge, Sweden. Feb. 05, 2014. Outlines 1
More informationMarkov Chain Monte Carlo
Chapter 5 Markov Chain Monte Carlo MCMC is a kind of improvement of the Monte Carlo method By sampling from a Markov chain whose stationary distribution is the desired sampling distributuion, it is possible
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 informationTheory of Stochastic Processes 8. Markov chain Monte Carlo
Theory of Stochastic Processes 8. Markov chain Monte Carlo Tomonari Sei sei@mist.i.u-tokyo.ac.jp Department of Mathematical Informatics, University of Tokyo June 8, 2017 http://www.stat.t.u-tokyo.ac.jp/~sei/lec.html
More informationParameter Estimation. William H. Jefferys University of Texas at Austin Parameter Estimation 7/26/05 1
Parameter Estimation William H. Jefferys University of Texas at Austin bill@bayesrules.net Parameter Estimation 7/26/05 1 Elements of Inference Inference problems contain two indispensable elements: Data
More informationMarkov Chain Monte Carlo methods
Markov Chain Monte Carlo methods Tomas McKelvey and Lennart Svensson Signal Processing Group Department of Signals and Systems Chalmers University of Technology, Sweden November 26, 2012 Today s learning
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 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 informationBayesian Phylogenetics:
Bayesian Phylogenetics: an introduction Marc A. Suchard msuchard@ucla.edu UCLA Who is this man? How sure are you? The one true tree? Methods we ve learned so far try to find a single tree that best describes
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 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 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 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 informationA note on Reversible Jump Markov Chain Monte Carlo
A note on Reversible Jump Markov Chain Monte Carlo Hedibert Freitas Lopes Graduate School of Business The University of Chicago 5807 South Woodlawn Avenue Chicago, Illinois 60637 February, 1st 2006 1 Introduction
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 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 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 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 informationMarkov Chains and MCMC
Markov Chains and MCMC Markov chains Let S = {1, 2,..., N} be a finite set consisting of N states. A Markov chain Y 0, Y 1, Y 2,... is a sequence of random variables, with Y t S for all points in time
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 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 informationStatistics & Data Sciences: First Year Prelim Exam May 2018
Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book
More informationMarkov chain Monte Carlo methods in atmospheric remote sensing
1 / 45 Markov chain Monte Carlo methods in atmospheric remote sensing Johanna Tamminen johanna.tamminen@fmi.fi ESA Summer School on Earth System Monitoring and Modeling July 3 Aug 11, 212, Frascati July,
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 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 informationStatistical Machine Learning Lecture 8: Markov Chain Monte Carlo Sampling
1 / 27 Statistical Machine Learning Lecture 8: Markov Chain Monte Carlo Sampling Melih Kandemir Özyeğin University, İstanbul, Turkey 2 / 27 Monte Carlo Integration The big question : Evaluate E p(z) [f(z)]
More informationSupplement to A Hierarchical Approach for Fitting Curves to Response Time Measurements
Supplement to A Hierarchical Approach for Fitting Curves to Response Time Measurements Jeffrey N. Rouder Francis Tuerlinckx Paul L. Speckman Jun Lu & Pablo Gomez May 4 008 1 The Weibull regression model
More informationBayesian Estimation for the Generalized Logistic Distribution Type-II Censored Accelerated Life Testing
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 20, 969-986 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.39111 Bayesian Estimation for the Generalized Logistic Distribution Type-II
More informationMCMC: Markov Chain Monte Carlo
I529: Machine Learning in Bioinformatics (Spring 2013) MCMC: Markov Chain Monte Carlo Yuzhen Ye School of Informatics and Computing Indiana University, Bloomington Spring 2013 Contents Review of Markov
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 informationLecture 1 Bayesian inference
Lecture 1 Bayesian inference olivier.francois@imag.fr April 2011 Outline of Lecture 1 Principles of Bayesian inference Classical inference problems (frequency, mean, variance) Basic simulation algorithms
More informationBayesian model selection: methodology, computation and applications
Bayesian model selection: methodology, computation and applications David Nott Department of Statistics and Applied Probability National University of Singapore Statistical Genomics Summer School Program
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 informationWho was Bayes? Bayesian Phylogenetics. What is Bayes Theorem?
Who was Bayes? Bayesian Phylogenetics Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison October 6, 2011 The Reverand Thomas Bayes was born in London in 1702. He was the
More informationBayesian Estimation with Sparse Grids
Bayesian Estimation with Sparse Grids Kenneth L. Judd and Thomas M. Mertens Institute on Computational Economics August 7, 27 / 48 Outline Introduction 2 Sparse grids Construction Integration with sparse
More informationBayesian Phylogenetics
Bayesian Phylogenetics Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison October 6, 2011 Bayesian Phylogenetics 1 / 27 Who was Bayes? The Reverand Thomas Bayes was born
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 informationBUGS Bayesian inference Using Gibbs Sampling
BUGS Bayesian inference Using Gibbs Sampling Glen DePalma Department of Statistics May 30, 2013 www.stat.purdue.edu/~gdepalma 1 / 20 Bayesian Philosophy I [Pearl] turned Bayesian in 1971, as soon as I
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 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 informationMarkov Chain Monte Carlo in Practice
Markov Chain Monte Carlo in Practice Edited by W.R. Gilks Medical Research Council Biostatistics Unit Cambridge UK S. Richardson French National Institute for Health and Medical Research Vilejuif France
More informationLecture 8: The Metropolis-Hastings Algorithm
30.10.2008 What we have seen last time: Gibbs sampler Key idea: Generate a Markov chain by updating the component of (X 1,..., X p ) in turn by drawing from the full conditionals: X (t) j Two drawbacks:
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 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 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 informationMonte Carlo Methods. Leon Gu CSD, CMU
Monte Carlo Methods Leon Gu CSD, CMU Approximate Inference EM: y-observed variables; x-hidden variables; θ-parameters; E-step: q(x) = p(x y, θ t 1 ) M-step: θ t = arg max E q(x) [log p(y, x θ)] θ Monte
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 informationMetropolis Hastings. Rebecca C. Steorts Bayesian Methods and Modern Statistics: STA 360/601. Module 9
Metropolis Hastings Rebecca C. Steorts Bayesian Methods and Modern Statistics: STA 360/601 Module 9 1 The Metropolis-Hastings algorithm is a general term for a family of Markov chain simulation 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 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 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 informationA Search and Jump Algorithm for Markov Chain Monte Carlo Sampling. Christopher Jennison. Adriana Ibrahim. Seminar at University of Kuwait
A Search and Jump Algorithm for Markov Chain Monte Carlo Sampling Christopher Jennison Department of Mathematical Sciences, University of Bath, UK http://people.bath.ac.uk/mascj Adriana Ibrahim Institute
More informationMcGill University. Department of Epidemiology and Biostatistics. Bayesian Analysis for the Health Sciences. Course EPIB-682.
McGill University Department of Epidemiology and Biostatistics Bayesian Analysis for the Health Sciences Course EPIB-682 Lawrence Joseph Intro to Bayesian Analysis for the Health Sciences EPIB-682 2 credits
More informationBayesian GLMs and Metropolis-Hastings Algorithm
Bayesian GLMs and Metropolis-Hastings Algorithm We have seen that with conjugate or semi-conjugate prior distributions the Gibbs sampler can be used to sample from the posterior distribution. In situations,
More informationProbabilistic Graphical Networks: Definitions and Basic Results
This document gives a cursory overview of Probabilistic Graphical Networks. The material has been gleaned from different sources. I make no claim to original authorship of this material. Bayesian Graphical
More informationLecture 5: Spatial probit models. James P. LeSage University of Toledo Department of Economics Toledo, OH
Lecture 5: Spatial probit models James P. LeSage University of Toledo Department of Economics Toledo, OH 43606 jlesage@spatial-econometrics.com March 2004 1 A Bayesian spatial probit model with individual
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 information7. Estimation and hypothesis testing. Objective. Recommended reading
7. Estimation and hypothesis testing Objective In this chapter, we show how the election of estimators can be represented as a decision problem. Secondly, we consider the problem of hypothesis testing
More informationBayesian inference for multivariate extreme value distributions
Bayesian inference for multivariate extreme value distributions Sebastian Engelke Clément Dombry, Marco Oesting Toronto, Fields Institute, May 4th, 2016 Main motivation For a parametric model Z F θ of
More informationApril 20th, Advanced Topics in Machine Learning California Institute of Technology. Markov Chain Monte Carlo for Machine Learning
for for Advanced Topics in California Institute of Technology April 20th, 2017 1 / 50 Table of Contents for 1 2 3 4 2 / 50 History of methods for Enrico Fermi used to calculate incredibly accurate predictions
More informationSC7/SM6 Bayes Methods HT18 Lecturer: Geoff Nicholls Lecture 2: Monte Carlo Methods Notes and Problem sheets are available at http://www.stats.ox.ac.uk/~nicholls/bayesmethods/ and via the MSc weblearn pages.
More informationTutorial on Probabilistic Programming with PyMC3
185.A83 Machine Learning for Health Informatics 2017S, VU, 2.0 h, 3.0 ECTS Tutorial 02-04.04.2017 Tutorial on Probabilistic Programming with PyMC3 florian.endel@tuwien.ac.at http://hci-kdd.org/machine-learning-for-health-informatics-course
More informationConvex Optimization CMU-10725
Convex Optimization CMU-10725 Simulated Annealing Barnabás Póczos & Ryan Tibshirani Andrey Markov Markov Chains 2 Markov Chains Markov chain: Homogen Markov chain: 3 Markov Chains Assume that the state
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference Sudipto Banerjee and Andrew O. Finley 2 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department
More informationReconstruction of individual patient data for meta analysis via Bayesian approach
Reconstruction of individual patient data for meta analysis via Bayesian approach Yusuke Yamaguchi, Wataru Sakamoto and Shingo Shirahata Graduate School of Engineering Science, Osaka University Masashi
More informationA Geometric Interpretation of the Metropolis Hastings Algorithm
Statistical Science 2, Vol. 6, No., 5 9 A Geometric Interpretation of the Metropolis Hastings Algorithm Louis J. Billera and Persi Diaconis Abstract. The Metropolis Hastings algorithm transforms a given
More informationApproximate Bayesian Computation: a simulation based approach to inference
Approximate Bayesian Computation: a simulation based approach to inference Richard Wilkinson Simon Tavaré 2 Department of Probability and Statistics University of Sheffield 2 Department of Applied Mathematics
More informationMSc MT15. Further Statistical Methods: MCMC. Lecture 5-6: Markov chains; Metropolis Hastings MCMC. Notes and Practicals available at
MSc MT15. Further Statistical Methods: MCMC Lecture 5-6: Markov chains; Metropolis Hastings MCMC Notes and Practicals available at www.stats.ox.ac.uk\ nicholls\mscmcmc15 Markov chain Monte Carlo Methods
More informationConvergence Rate of Markov Chains
Convergence Rate of Markov Chains Will Perkins April 16, 2013 Convergence Last class we saw that if X n is an irreducible, aperiodic, positive recurrent Markov chain, then there exists a stationary distribution
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 informationLabor-Supply Shifts and Economic Fluctuations. Technical Appendix
Labor-Supply Shifts and Economic Fluctuations Technical Appendix Yongsung Chang Department of Economics University of Pennsylvania Frank Schorfheide Department of Economics University of Pennsylvania January
More informationINTRODUCTION TO BAYESIAN STATISTICS
INTRODUCTION TO BAYESIAN STATISTICS Sarat C. Dass Department of Statistics & Probability Department of Computer Science & Engineering Michigan State University TOPICS The Bayesian Framework Different Types
More informationMarkov Chain Monte Carlo and Applied Bayesian Statistics
Markov Chain Monte Carlo and Applied Bayesian Statistics Trinity Term 2005 Prof. Gesine Reinert Markov chain Monte Carlo is a stochastic simulation technique that is very useful for computing inferential
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 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 informationMarkov chain Monte Carlo
Markov chain Monte Carlo Karl Oskar Ekvall Galin L. Jones University of Minnesota March 12, 2019 Abstract Practically relevant statistical models often give rise to probability distributions that are analytically
More informationAnswers and expectations
Answers and expectations For a function f(x) and distribution P(x), the expectation of f with respect to P is The expectation is the average of f, when x is drawn from the probability distribution P E
More informationDavid Giles Bayesian Econometrics
9. Model Selection - Theory David Giles Bayesian Econometrics One nice feature of the Bayesian analysis is that we can apply it to drawing inferences about entire models, not just parameters. Can't do
More informationBayesian model selection in graphs by using BDgraph package
Bayesian model selection in graphs by using BDgraph package A. Mohammadi and E. Wit March 26, 2013 MOTIVATION Flow cytometry data with 11 proteins from Sachs et al. (2005) RESULT FOR CELL SIGNALING DATA
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 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 informationMolecular Epidemiology Workshop: Bayesian Data Analysis
Molecular Epidemiology Workshop: Bayesian Data Analysis Jay Taylor and Ananias Escalante School of Mathematical and Statistical Sciences Center for Evolutionary Medicine and Informatics Arizona State University
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 informationBayesian inference & Markov chain Monte Carlo. Note 1: Many slides for this lecture were kindly provided by Paul Lewis and Mark Holder
Bayesian inference & Markov chain Monte Carlo Note 1: Many slides for this lecture were kindly provided by Paul Lewis and Mark Holder Note 2: Paul Lewis has written nice software for demonstrating Markov
More informationThe Effects of Monetary Policy on Stock Market Bubbles: Some Evidence
The Effects of Monetary Policy on Stock Market Bubbles: Some Evidence Jordi Gali Luca Gambetti ONLINE APPENDIX The appendix describes the estimation of the time-varying coefficients VAR model. The model
More information