Bayesian Hierarchical Modeling for Longitudinal Frequency Data Joseph Jordan Advisor: John C. Kern II Department of Mathematics and Computer Science Duquesne.

Slides:

Advertisements

Similar presentations

Model checking in mixture models via mixed predictive p-values Alex Lewin and Sylvia Richardson, Centre for Biostatistics, Imperial College, London Mixed.

Advertisements

Probabilistic models Jouni Tuomisto THL. Outline Deterministic models with probabilistic parameters Hierarchical Bayesian models Bayesian belief nets.

1 Bayesian methods for parameter estimation and data assimilation with crop models David Makowski and Daniel Wallach INRA, France September 2006.

Bayesian Statistics Simon French

Comments on Hierarchical models, and the need for Bayes Peter Green, University of Bristol, UK IWSM, Chania, July 2002.

Bayesian Estimation in MARK

Part 24: Bayesian Estimation 24-1/35 Econometrics I Professor William Greene Stern School of Business Department of Economics.

Gibbs Sampling Qianji Zheng Oct. 5th, 2010.

1 Graphical Diagnostic Tools for Evaluating Latent Class Models: An Application to Depression in the ECA Study Elizabeth S. Garrett Department of Biostatistics.

Introduction to Sampling based inference and MCMC Ata Kaban School of Computer Science The University of Birmingham.

CHAPTER 16 MARKOV CHAIN MONTE CARLO

Bayesian statistics – MCMC techniques

Gaussian Processes to Speed up Hamiltonian Monte Carlo Matthieu Lê Journal Club 11/04/141 Neal, Radford M (2011). " MCMC Using Hamiltonian Dynamics. "

Stochastic approximate inference Kay H. Brodersen Computational Neuroeconomics Group Department of Economics University of Zurich Machine Learning and.

BAYESIAN INFERENCE Sampling techniques

Industrial Engineering College of Engineering Bayesian Kernel Methods for Binary Classification and Online Learning Problems Theodore Trafalis Workshop.

Use of Bayesian Methods for Markov Modelling in Cost Effectiveness Analysis: An application to taxane use in advanced breast cancer Nicola Cooper, Keith.

CS774. Markov Random Field : Theory and Application Lecture 16 Kyomin Jung KAIST Nov

Computational statistics 2009 Random walk. Computational statistics 2009 Random walk with absorbing barrier.

Computational statistics, course introduction Course contents  Monte Carlo Methods  Random number generation  Simulation methodology  Bootstrap  Markov.

Exploring subjective probability distributions using Bayesian statistics Tom Griffiths Department of Psychology Cognitive Science Program University of.

Using ranking and DCE data to value health states on the QALY scale using conventional and Bayesian methods Theresa Cain.

Part 23: Simulation Based Estimation 23-1/26 Econometrics I Professor William Greene Stern School of Business Department of Economics.

Generalised Evidence Synthesis Keith Abrams, Cosetta Minelli, Nicola Cooper & Alex Sutton Medical Statistics Group Department of Health Sciences, University.

Everything you ever wanted to know about BUGS, R2winBUGS, and Adaptive Rejection Sampling A Presentation by Keith Betts.

Research Topics for Spatial temporal processes Time Series:Time Series: nonlinear state space M. West (1996) In Bayesian Statistics 5, Oxford University.

CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Statistical inference.

Image Analysis and Markov Random Fields (MRFs) Quanren Xiong.

1 Spatial and Spatio-temporal modeling of the abundance of spawning coho salmon on the Oregon coast R Ruben Smith Don L. Stevens Jr. September.

Bayes Factor Based on Han and Carlin (2001, JASA).

Bayesian Statistics in Clinical Trials Case Studies: Agenda

Modeling Menstrual Cycle Length in Pre- and Peri-Menopausal Women Michael Elliott Xiaobi Huang Sioban Harlow University of Michigan School of Public Health.

Material Model Parameter Identification via Markov Chain Monte Carlo Christian Knipprath 1 Alexandros A. Skordos – ACCIS,

Queensland University of Technology CRICOS No J Towards Likelihood Free Inference Tony Pettitt QUT, Brisbane Joint work with.

Introduction to MCMC and BUGS. Computational problems More parameters -> even more parameter combinations Exact computation and grid approximation become.

St5219: Bayesian hierarchical modelling lecture 2.1.

1 Gil McVean Tuesday 24 th February 2009 Markov Chain Monte Carlo.

Monte Carlo Methods1 T Special Course In Information Science II Tomas Ukkonen

Simulation of the matrix Bingham-von Mises- Fisher distribution, with applications to multivariate and relational data Discussion led by Chunping Wang.

Mixture Models, Monte Carlo, Bayesian Updating and Dynamic Models Mike West Computing Science and Statistics, Vol. 24, pp , 1993.

Fast Simulators for Assessment and Propagation of Model Uncertainty* Jim Berger, M.J. Bayarri, German Molina June 20, 2001 SAMO 2001, Madrid *Project of.

Integrating Topics and Syntax -Thomas L

“Users of scooters must transfer to a fixed seat” (Notice in Brisbane Taxi)

October 28, 2008 Model qualification and assumption checking 1 To Validate or not to Validate? If all models are wrong but some.

1 Tree Crown Extraction Using Marked Point Processes Guillaume Perrin Xavier Descombes – Josiane Zerubia ARIANA, joint research group CNRS/INRIA/UNSA INRIA.

The generalization of Bayes for continuous densities is that we have some density f(y|  ) where y and  are vectors of data and parameters with  being.

And now…. …for something completely different (Monty Python)

Probabilistic models Jouni Tuomisto THL. Outline Deterministic models with probabilistic parameters Hierarchical Bayesian models Bayesian belief nets.

Markov Chain Monte Carlo for LDA C. Andrieu, N. D. Freitas, and A. Doucet, An Introduction to MCMC for Machine Learning, R. M. Neal, Probabilistic.

Lecture #9: Introduction to Markov Chain Monte Carlo, part 3

A shared random effects transition model for longitudinal count data with informative missingness Jinhui Li Joint work with Yingnian Wu, Xiaowei Yang.

1 Chapter 8: Model Inference and Averaging Presented by Hui Fang.

Stochastic Frontier Models

CS Statistical Machine learning Lecture 25 Yuan (Alan) Qi Purdue CS Nov

An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis B. P. Carlin and A. E. Gelfand Statistics and Computing 1991 A Generic Approach to Posterior.

Introduction: Metropolis-Hasting Sampler Purpose--To draw samples from a probability distribution There are three steps 1Propose a move from x to y 2Accept.

Efficiency Measurement William Greene Stern School of Business New York University.

Hierarchical Models. Conceptual: What are we talking about? – What makes a statistical model hierarchical? – How does that fit into population analysis?

Generalization Performance of Exchange Monte Carlo Method for Normal Mixture Models Kenji Nagata, Sumio Watanabe Tokyo Institute of Technology.

Introduction to Sampling based inference and MCMC

Lecture 1 Statistical Modelling and Inference

MCMC Output & Metropolis-Hastings Algorithm Part I

“Bayesian Inference Using Gibbs Sampling”

Introducing Bayesian Approaches to Twin Data Analysis

Ch3: Model Building through Regression

Introduction to the bayes Prefix in Stata 15

Linear and generalized linear mixed effects models

Remember that our objective is for some density f(y|) for observations where y and  are vectors of data and parameters,  being sampled from a prior.

Predictive distributions

CS639: Data Management for Data Science

Presentation transcript:

Bayesian Hierarchical Modeling for Longitudinal Frequency Data Joseph Jordan Advisor: John C. Kern II Department of Mathematics and Computer Science Duquesne University May 6, 2005

Outline Motivation The Model Model Simulation Model Implementation Metropolis-Hastings Sampling Algorithm Results Conclusion References

Motivation Yale University Study: The Patrick & Catherine Weldon Donaghue Medical Foundation Menopausal women in breast cancer remission Acupuncture relief of menopausal symptoms Unlike previous models, this model explicitly recognizes time dependence through prior distributions

Model Simulations: Study Information Individuals randomly assigned to 1 of 3 groups Length of Study: 13 weeks (1 week baseline followed by 12 weeks of “treatment” Measurement: Hot flush frequency (91 observations)

Motivation: Study Samples Education Group: 6 individuals given weekly educational sessions Treatment Group: 16 individuals given weekly acupuncture on effective bodily areas Placebo Group: 17 individuals given weekly acupuncture on non-effective bodily areas

Motivation: Actual Subject Profile

Mean Hot Flush Frequencies

The Model:

The Model: Prior Distributions

The Model: Prior Distributions (Non-Informative)

Model Simulation:  j =.5,  j =.9,  2 j =.5

Model Simulation:  j =.5,  j =.5,  2 j =.5

Model Implementation: Markov Chain Monte Carlo Metropolis-Hastings Sampling: Gibbs Sampling:

Metropolis-Hastings Sampling: Requirements MUST know posterior distribution for parameter (product of likelihood and prior distributions) Computational precision issues – utilize natural logs For example:

Metropolis-Hastings Sampling: Algorithm

Gibbs Sampling: Requirements Requirement: MUST know full conditional distribution for parameter Sample from full conditional distribution; ALWAYS accept  * I For Example:

Gibbs Sampling: Full Conditional Distributions

Metropolis-Hastings Likelihood for  ij  ij : mean hot flush freq on days i and 2i-1 for i=1,…,44, with  45j representing the mean hot flush freq for days 89, 90, 91

Metropolis-Hastings Prior for  ij

Metropolis-Hastings Difference in log posterior densities evaluated at  * ij and  c ij

Metropolis-Hastings Likelihood for  j

Metropolis-Hastings Prior for  j

Metropolis-Hastings Difference in log posterior densities evaluated at  * j and  c j

Metropolis-Hastings Updating  j Same likelihood as  j

Metropolis-Hastings Updating  2 j Same likelihood as  j

Metropolis-Hastings Updating  0j Same posterior as  ij ’s

Metropolis-Hastings Likelihood Distribution for 

Metropolis-Hastings Prior Distribution for 

Metropolis-Hastings Updating  Same likelihood as  Uniform prior

Metropolis-Hastings Updating a and b Uniform Prior Same likelihood and prior for b

Hastings Ratios

Results Treatment Group

Results Placebo Group

Results Education Group

Results Boxplot for  0 ’s

Results Boxplot for Exponentiated  0

References Borgesi, J A Piecewise Linear Generalized Poisson Regression Approach to Modeling Longitudinal Frequency Data. Unpublished masters thesis, Duquesne University, Pittsburgh, PA, USA. Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B Bayesian Data Analysis. London: Chapman and Hall. Gilks, W.R., Richardson, S., and Spiegelhalter, D.J Markov Chain Monte Carlo in Practice. London: Chapman and Hall. Kern, J. and S.M. Cohen Menopausal symptom relief with acupuncture: modeling longitudinal frequency data. Vol 34, 3: Communications in Statistics: Simulation and Computation.