Fitting Complex PK/PD Models with Stan

 Philippa Harrison
 10 months ago
 Views:
Transcription
1 Fitting Complex PK/PD Models with Stan University of Warwick Bayes Pharma Novartis Basel Campus May 20, 2015
2 Because of the clinical applications, precise PK/PD modeling is incredibly important. PK/PD Model
3 Because of the clinical applications, precise PK/PD modeling is incredibly important. PK/PD Model Treatment R(T, )!
4 Robust treatment decisions requires modeling both the latent PK/PD process and the measurement. PK/PD Model Treatment R(T, )!
5 Robust treatment decisions requires modeling both the latent PK/PD process and the measurement. PK/PD Model Data D Treatment R(T, )!
6 Robust treatment decisions requires modeling both the latent PK/PD process and the measurement. PK/PD Model Measurement Model (D ) Data D Treatment R(T, )!
7 Analytic models are common due to their computational convenience but ultimately too restrictive. PK/PD Model Measurement Model (D ) Data D Treatment R(T, )!
8 Analytic PK/PD models, for example, are exactly solvable. PK/PD Model Measurement Model (D ) Data D Treatment R(T, )!
9 Analytic PK/PD models, for example, are exactly solvable. dc dt = V m V C K m + C
10 These analytic models, however, can t capture many nonlinear effects in more realistic PK/PD systems. dc dt = V m V C K m + C + k V e kt
11 Similarly, analytic measurement models are straightforward to fit. PK/PD Model Measurement Model (D ) Data D Treatment R(T, )!
12 Similarly, analytic measurement models are straightforward to fit. bc t,n (V,V m )=N (C(t, V, V m ), )
13 Similarly, analytic measurement models are straightforward to fit. bc t,n (V,V m )=N (C(t, V, V m ), ) NX C(t, V, V m ) 1 N Ĉ t,n (V,V m ) n=1
14 Similarly, analytic measurement models are straightforward to fit. bc t,n (V,V m )=N (C(t, V, V m ), ) NX C(t, V, V m ) 1 N Ĉ t,n (V,V m ) V,V m C 1 1 N n=1 NX n=1 Ĉ t,n (V,V m )!
15 But for population data analytic measurement models either introduce bias or suffer from large uncertainties. dc dt = V m V C K m + C
16 But for population data analytic measurement models either introduce bias or suffer from large uncertainties. dc dt = V n m V n C K m + C
17 But for population data analytic measurement models either introduce bias or suffer from large uncertainties. dc dt = V m n V n C K m + C
18 But for population data analytic measurement models either introduce bias or suffer from large uncertainties. V 1,V 1 m V n,v n m V N,V N m......
19 If we want to maximize the utility of the data then we need nonanalytic measurement models. V 1,V 1 m V n,v n m V N,V N m µ V,! V µ Vm,! Vm V n log N (µ V,! V ) V n m log N (µ Vm,! Vm )
20 Nonanalytic models are practically difficult because they are computationally demanding to fit. PK/PD Model Measurement Model (D ) Data D Treatment R(T, )!
21 Recall that in Bayesian inference our complete model is specified with a posterior distribution.
22 Recall that in Bayesian inference our complete model is specified with a posterior distribution. ( )
23 Recall that in Bayesian inference our complete model is specified with a posterior distribution. (D ) ( )
24 Recall that in Bayesian inference our complete model is specified with a posterior distribution. ( D) / (D ) ( )
25 And all wellposed statistical manipulations are expectations with respect to the posterior. Z E[f] = d ( D)f( )
26 Expectations, however, are computationally demanding when the model is complex. Z E[f] = d ( D)f( )
27 The problem is that probability concentrates on a nonlinear surface called the typical set.
28 In order to estimate expectations we can use Markov chain Monte Carlo to find and explore the typical set.
29 In order to estimate expectations we can use Markov chain Monte Carlo to find and explore the typical set.
30 Given the complexity of the PK/PD model, this exploration has to be extremely efficient to be practical.
31 Random Walk Metropolis explores with an ineffective guided diffusion.
32 Random Walk Metropolis explores with an ineffective guided diffusion.
33 Random Walk Metropolis explores with an ineffective guided diffusion.
34 The Gibbs sampler updates each parameter oneatatime to no greater success.
35 The Gibbs sampler updates each parameter oneatatime to no greater success.
36 In order to explore efficiently we need coherent exploration.
37 In order to explore efficiently we need coherent exploration.
38
39 Modeling Language
40 Modeling Language Hamiltonian Monte Carlo
41 Modeling Language Hamiltonian Monte Carlo
42 A strongly typed modeling language allows users to specify complex models with minimal effort.
43 Stan not only admits the specification of a system of ordinary differential equations functions { real[] onecomp_mm_infusion(real t, real[] y, real[] theta, real[] x_r, int[] x_i) {! real dydt[1];! if (t < x_r[2]) dydt[1] <  theta[2] * y[1] / ( theta[1] * (theta[3] + y[1]) ) + x_r[1] / (theta[1] * x_r[2]); else dydt[1] <  theta[2] * y[1] / ( theta[1] * (theta[3] + y[1]) );! return dydt;! } }
44 But also highperformance integrators. transformed parameters { for (p in 1:N_patients) { V[p] < exp(mu_v + eta_v[p] * omega_v); V_m[p] < exp(mu_v_m + eta_v_m[p] * omega_v_m);! } theta[1] < V[p]; theta[2] < V_m[p]; theta[3] < K_m; C < integrate_ode(onecomp_mm_infusion, C0, t0, t, theta, x_r, x_i);
45 Consequently, even complex statistical modeling of PK/PD systems is straightforward in Stan. model { eta_v ~ normal(0, 1); mu_v ~ normal(log(5), 1); sigma_v ~ cauchy(0, 1); eta_v_m ~ normal(0, 1); mu_v_m ~ cauchy(0, 1); sigma_v_m ~ cauchy(0, 1);! } K_m ~ cauchy(0, 1); for (p in 1:N_patients) for (n in 1:N_t) C_hat[p, n] ~ lognormal(log(c[p, n]), 0.15);
46 Modeling Language Hamiltonian Monte Carlo
47 Once a posterior has been specified, Stan implements Hamiltonian Monte Carlo to estimate expectations.
48 Let s consider a onecompartment MichaelisMenten clearance model with a constant infusion. dc dt = V m V C K m + C + I(t) I(t) = 8 < : D VT, 0, t apple 0 0 <t<t 0, T apple t
49 Data is simulated for ten patients, with each patient given their own V and V m. dc n dt = V n m V n C n K m + C n + I(t) Ĉ n (t) log N (C n (t), )
50 Data is simulated for ten patients, with each patient given their own V and V m. dc n dt = V n m V n C n K m + C n + I(t) Ĉ n (t) log N (C n (t), ) V n log N (µ V,! V ) V n m log N (µ Vm,! Vm )
51 Within a few minutes Stan generates all of the samples we need to infer the individual patient concentrations Concentration(mg/L) Time(days)
52 As well as simulated data for constructing posterior predictive checks Concentration(mg/L) Time(days)
53 We can also analyze the system parameters for each individual patient V(L)
54 As well as the population parameters µ V (L) V (L)
55 As well as the population parameters µ Vm (mg/day) Vm (mg/day)
56
Tutorial on Probabilistic Programming with PyMC3
185.A83 Machine Learning for Health Informatics 2017S, VU, 2.0 h, 3.0 ECTS Tutorial 0204.04.2017 Tutorial on Probabilistic Programming with PyMC3 florian.endel@tuwien.ac.at http://hcikdd.org/machinelearningforhealthinformaticscourse
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 informationHamiltonian Monte Carlo
Hamiltonian Monte Carlo within Stan Daniel Lee Columbia University, Statistics Department bearlee@alum.mit.edu BayesComp mcstan.org Why MCMC? Have data. Have a rich statistical model. No analytic solution.
More informationMarkov Chain Monte Carlo, Numerical Integration
Markov Chain Monte Carlo, Numerical Integration (See Statistics) Trevor Gallen Fall 2015 1 / 1 Agenda Numerical Integration: MCMC methods Estimating Markov Chains Estimating latent variables 2 / 1 Numerical
More informationBayesian Inference: Probit and Linear Probability Models
Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 512014 Bayesian Inference: Probit and Linear Probability Models Nate Rex Reasch Utah State University Follow
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: DAGMCLecture.tex
More informationExample: Ground Motion Attenuation
Example: Ground Motion Attenuation Problem: Predict the probability distribution for Peak Ground Acceleration (PGA), the level of ground shaking caused by an earthquake Earthquake records are used to update
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 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 informationMachine Learning Techniques for Computer Vision
Machine Learning Techniques for Computer Vision Part 2: Unsupervised Learning Microsoft Research Cambridge x 3 1 0.5 0.2 0 0.5 0.3 0 0.5 1 ECCV 2004, Prague x 2 x 1 Overview of Part 2 Mixture models EM
More informationCharacterizing Travel Time Reliability and Passenger Path Choice in a Metro Network
Characterizing Travel Time Reliability and Passenger Path Choice in a Metro Network Lijun SUN Future Cities Laboratory, SingaporeETH Centre lijun.sun@ivt.baug.ethz.ch National University of Singapore
More informationNotes on pseudomarginal methods, variational Bayes and ABC
Notes on pseudomarginal methods, variational Bayes and ABC Christian Andersson Naesseth October 3, 2016 The PseudoMarginal Framework Assume we are interested in sampling from the posterior distribution
More informationBayesian networks: approximate inference
Bayesian networks: approximate inference Machine Intelligence Thomas D. Nielsen September 2008 Approximative inference September 2008 1 / 25 Motivation Because of the (worstcase) intractability of exact
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 informationChapter 2. Data Analysis
Chapter 2 Data Analysis 2.1. Density Estimation and Survival Analysis The most straightforward application of BNP priors for statistical inference is in density estimation problems. Consider the generic
More informationPetr Volf. Model for Difference of Two Series of Poissonlike Count Data
Petr Volf Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Pod vodárenskou věží 4, 182 8 Praha 8 email: volf@utia.cas.cz Model for Difference of Two Series of Poissonlike
More informationMultilevel Statistical Models: 3 rd edition, 2003 Contents
Multilevel Statistical Models: 3 rd edition, 2003 Contents Preface Acknowledgements Notation Two and three level models. A general classification notation and diagram Glossary Chapter 1 An introduction
More informationNotes on Noise Contrastive Estimation (NCE)
Notes on Noise Contrastive Estimation NCE) David Meyer dmm@{45.net,uoregon.edu,...} March 0, 207 Introduction In this note we follow the notation used in [2]. Suppose X x, x 2,, x Td ) is a sample of
More informationInversion Base Height. Daggot Pressure Gradient Visibility (miles)
Stanford University June 2, 1998 Bayesian Backtting: 1 Bayesian Backtting Trevor Hastie Stanford University Rob Tibshirani University of Toronto Email: trevor@stat.stanford.edu Ftp: stat.stanford.edu:
More informationMCMC notes by Mark Holder
MCMC notes by Mark Holder Bayesian inference Ultimately, we want to make probability statements about true values of parameters, given our data. For example P(α 0 < α 1 X). According to Bayes theorem:
More informationBayesian GLMs and MetropolisHastings Algorithm
Bayesian GLMs and MetropolisHastings Algorithm We have seen that with conjugate or semiconjugate prior distributions the Gibbs sampler can be used to sample from the posterior distribution. In situations,
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 informationMarkov Networks.
Markov Networks www.biostat.wisc.edu/~dpage/cs760/ Goals for the lecture you should understand the following concepts Markov network syntax Markov network semantics Potential functions Partition function
More informationNonparametric Drift Estimation for Stochastic Differential Equations
Nonparametric Drift Estimation for Stochastic Differential Equations Gareth Roberts 1 Department of Statistics University of Warwick Brazilian Bayesian meeting, March 2010 Joint work with O. Papaspiliopoulos,
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Contents in latter part Linear Dynamical Systems What is different from HMM? Kalman filter Its strength and limitation Particle Filter
More informationBayesian Prediction of Code Output. ASA Albuquerque Chapter Short Course October 2014
Bayesian Prediction of Code Output ASA Albuquerque Chapter Short Course October 2014 Abstract This presentation summarizes Bayesian prediction methodology for the Gaussian process (GP) surrogate representation
More informationKarlRudolf Koch Introduction to Bayesian Statistics Second Edition
KarlRudolf Koch Introduction to Bayesian Statistics Second Edition KarlRudolf Koch Introduction to Bayesian Statistics Second, updated and enlarged Edition With 17 Figures Professor Dr.Ing., Dr.Ing.
More informationRisk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods
Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods Konstantin Zuev Institute for Risk and Uncertainty University of Liverpool http://www.liv.ac.uk/riskanduncertainty/staff/kzuev/
More informationBayes Networks 6.872/HST.950
Bayes Networks 6.872/HST.950 What Probabilistic Models Should We Use? Full joint distribution Completely expressive Hugely datahungry Exponential computational complexity Naive Bayes (full conditional
More informationWhen one of things that you don t know is the number of things that you don t know
When one of things that you don t know is the number of things that you don t know Malcolm Sambridge Research School of Earth Sciences, Australian National University Canberra, Australia. Sambridge. M.,
More informationMATH 12 CLASS 5 NOTES, SEP
MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vectorvalued functions 1 2. Differentiating and integrating vectorvalued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed
More informationStatistical Inference and Methods
Department of Mathematics Imperial College London d.stephens@imperial.ac.uk http://stats.ma.ic.ac.uk/ das01/ 31st January 2006 Part VI Session 6: Filtering and Time to Event Data Session 6: Filtering and
More informationAdaptive Monte Carlo methods
Adaptive Monte Carlo methods JeanMichel Marin Projet Select, INRIA Futurs, Université ParisSud joint with Randal Douc (École Polytechnique), Arnaud Guillin (Université de Marseille) and Christian Robert
More informationIntroduction to Markov Chain Monte Carlo
Introduction to Markov Chain Monte Carlo Jim Albert March 18, 2018 A Selected Data Problem Here is an interesting problem with selected data. Suppose you are measuring the speeds of cars driving on an
More informationPaul Karapanagiotidis ECO4060
Paul Karapanagiotidis ECO4060 The way forward 1) Motivate why MarkovChain Monte Carlo (MCMC) is useful for econometric modeling 2) Introduce MarkovChain Monte Carlo (MCMC)  MetropolisHastings (MH)
More informationGeneral Construction of Irreversible Kernel in Markov Chain Monte Carlo
General Construction of Irreversible Kernel in Markov Chain Monte Carlo Metropolis heat bath Suwa Todo Department of Applied Physics, The University of Tokyo Department of Physics, Boston University (from
More informationBayesian Monte Carlo Filtering for Stochastic Volatility Models
Bayesian Monte Carlo Filtering for Stochastic Volatility Models Roberto Casarin CEREMADE University Paris IX (Dauphine) and Dept. of Economics University Ca Foscari, Venice Abstract Modelling of the financial
More informationDefinition 3 (Continuity). A function f is continuous at c if lim x c f(x) = f(c).
Functions of Several Variables A function of several variables is just what it sounds like. It may be viewed in at least three different ways. We will use a function of two variables as an example. z =
More informationAn ABC interpretation of the multiple auxiliary variable method
School of Mathematical and Physical Sciences Department of Mathematics and Statistics Preprint MPS201607 27 April 2016 An ABC interpretation of the multiple auxiliary variable method by Dennis Prangle
More informationBayesian Inference. Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Ausín and Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative
More informationBayesian linear regression
Bayesian linear regression Linear regression is the basis of most statistical modeling. The model is Y i = X T i β + ε i, where Y i is the continuous response X i = (X i1,..., X ip ) T is the corresponding
More informationEfficiency and Reliability of Bayesian Calibration of Energy Supply System Models
Efficiency and Reliability of Bayesian Calibration of Energy Supply System Models Kathrin Menberg 1,2, Yeonsook Heo 2, Ruchi Choudhary 1 1 University of Cambridge, Department of Engineering, Cambridge,
More informationPackage lmm. R topics documented: March 19, Version 0.4. Date Title Linear mixed models. Author Joseph L. Schafer
Package lmm March 19, 2012 Version 0.4 Date 2012319 Title Linear mixed models Author Joseph L. Schafer Maintainer Jing hua Zhao Depends R (>= 2.0.0) Description Some
More informationIEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 3, MARCH
IEEE TRANSACTIONS ON AUDIO, SPEECH, AND LANGUAGE PROCESSING, VOL. 18, NO. 3, MARCH 2010 589 Gamma Markov Random Fields for Audio Source Modeling Onur Dikmen, Member, IEEE, and A. Taylan Cemgil, Member,
More informationComparison of Three Calculation Methods for a Bayesian Inference of Two Poisson Parameters
Journal of Modern Applied Statistical Methods Volume 13 Issue 1 Article 26 512014 Comparison of Three Calculation Methods for a Bayesian Inference of Two Poisson Parameters Yohei Kawasaki Tokyo University
More informationNeutral Bayesian reference models for incidence rates of (rare) clinical events
Neutral Bayesian reference models for incidence rates of (rare) clinical events Jouni Kerman Statistical Methodology, Novartis Pharma AG, Basel BAYES2012, May 10, Aachen Outline Motivation why reference
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 informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and NonParametric 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 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 informationDer SPP 1167PQP und die stochastische Wettervorhersage
Der SPP 1167PQP und die stochastische Wettervorhersage Andreas Hense 9. November 2007 Overview The priority program SPP1167: mission and structure The stochastic weather forecasting Introduction, Probabilities
More informationDensity Estimation. Seungjin Choi
Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongamro, Namgu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/
More informationBayesian course  problem set 5 (lecture 6)
Bayesian course  problem set 5 (lecture 6) Ben Lambert November 30, 2016 1 Stan entry level: discoveries data The file prob5 discoveries.csv contains data on the numbers of great inventions and scientific
More informationBayesian Networks BY: MOHAMAD ALSABBAGH
Bayesian Networks BY: MOHAMAD ALSABBAGH Outlines Introduction Bayes Rule Bayesian Networks (BN) Representation Size of a Bayesian Network Inference via BN BN Learning Dynamic BN Introduction Conditional
More informationAccelerated Gibbs Sampling for Infinite Sparse Factor Analysis
LLNLTR499647 Accelerated Gibbs Sampling for Infinite Sparse Factor Analysis D. M. Andrzejewski September 19, 2011 Disclaimer This document was prepared as an account of work sponsored by an agency of
More informationIntroduction to Bayesian methods in inverse problems
Introduction to Bayesian methods in inverse problems Ville Kolehmainen 1 1 Department of Applied Physics, University of Eastern Finland, Kuopio, Finland March 4 2013 Manchester, UK. Contents Introduction
More informationPROBABILISTIC PROGRAMMING: BAYESIAN MODELLING MADE EASY
PROBABILISTIC PROGRAMMING: BAYESIAN MODELLING MADE EASY Arto Klami Adapted from my talk in AIHelsinki seminar Dec 15, 2016 1 MOTIVATING INTRODUCTION Most of the artificial intelligence success stories
More informationMarkov Chain Monte Carlo Methods for Stochastic
Markov Chain Monte Carlo Methods for Stochastic Optimization i John R. Birge The University of Chicago Booth School of Business Joint work with Nicholas Polson, Chicago Booth. JRBirge U Florida, Nov 2013
More informationNORGES TEKNISKNATURVITENSKAPELIGE UNIVERSITET. Directional Metropolis Hastings updates for posteriors with nonlinear likelihoods
NORGES TEKNISKNATURVITENSKAPELIGE UNIVERSITET Directional Metropolis Hastings updates for posteriors with nonlinear likelihoods by Håkon Tjelmeland and Jo Eidsvik PREPRINT STATISTICS NO. 5/2004 NORWEGIAN
More informationPROBABILISTIC PROGRAMMING: BAYESIAN MODELLING MADE EASY. Arto Klami
PROBABILISTIC PROGRAMMING: BAYESIAN MODELLING MADE EASY Arto Klami 1 PROBABILISTIC PROGRAMMING Probabilistic programming is to probabilistic modelling as deep learning is to neural networks (Antti Honkela,
More informationLecture 6: Bayesian Inference in SDE Models
Lecture 6: Bayesian Inference in SDE Models Bayesian Filtering and Smoothing Point of View Simo Särkkä Aalto University Simo Särkkä (Aalto) Lecture 6: Bayesian Inference in SDEs 1 / 45 Contents 1 SDEs
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 informationFactorization of Seperable and Patterned Covariance Matrices for Gibbs Sampling
Monte Carlo Methods Appl, Vol 6, No 3 (2000), pp 205 210 c VSP 2000 Factorization of Seperable and Patterned Covariance Matrices for Gibbs Sampling Daniel B Rowe H & SS, 22877 California Institute of
More informationSequential Monte Carlo and Particle Filtering. Frank Wood Gatsby, November 2007
Sequential Monte Carlo and Particle Filtering Frank Wood Gatsby, November 2007 Importance Sampling Recall: Let s say that we want to compute some expectation (integral) E p [f] = p(x)f(x)dx and we remember
More informationLearning Static Parameters in Stochastic Processes
Learning Static Parameters in Stochastic Processes Bharath Ramsundar December 14, 2012 1 Introduction Consider a Markovian stochastic process X T evolving (perhaps nonlinearly) over time variable T. We
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 informationPartial factor modeling: predictordependent shrinkage for linear regression
modeling: predictordependent shrinkage for linear Richard Hahn, Carlos Carvalho and Sayan Mukherjee JASA 2013 Review by Esther Salazar Duke University December, 2013 Factor framework The factor framework
More informationLatent Dirichlet Allocation (LDA)
Latent Dirichlet Allocation (LDA) A review of topic modeling and customer interactions application 3/11/2015 1 Agenda Agenda Items 1 What is topic modeling? Intro Text Mining & PreProcessing Natural Language
More informationUncertainty Quantification in Performance Evaluation of Manufacturing Processes
Uncertainty Quantification in Performance Evaluation of Manufacturing Processes Manufacturing Systems October 27, 2014 Saideep Nannapaneni, Sankaran Mahadevan Vanderbilt University, Nashville, TN Acknowledgement
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 informationBayesian roomacoustic modal analysis
Bayesian roomacoustic modal analysis Wesley Henderson a) Jonathan Botts b) Ning Xiang c) Graduate Program in Architectural Acoustics, School of Architecture, Rensselaer Polytechnic Institute, Troy, New
More informationComputer Practical: MetropolisHastingsbased MCMC
Computer Practical: MetropolisHastingsbased MCMC Andrea Arnold and Franz Hamilton North Carolina State University July 30, 2016 A. Arnold / F. Hamilton (NCSU) MHbased MCMC July 30, 2016 1 / 19 Markov
More informationBayesian inference & process convolution models Dave Higdon, Statistical Sciences Group, LANL
1 Bayesian inference & process convolution models Dave Higdon, Statistical Sciences Group, LANL 2 MOVING AVERAGE SPATIAL MODELS Kernel basis representation for spatial processes z(s) Define m basis functions
More informationBayesian lithology/fluid inversion comparison of two algorithms
Comput Geosci () 14:357 367 DOI.07/s59600991559 REVIEW PAPER Bayesian lithology/fluid inversion comparison of two algorithms Marit Ulvmoen Hugo Hammer Received: 2 April 09 / Accepted: 17 August 09 /
More informationCS 188: Artificial Intelligence Spring Announcements
CS 188: Artificial Intelligence Spring 2011 Lecture 18: HMMs and Particle Filtering 4/4/2011 Pieter Abbeel  UC Berkeley Many slides over this course adapted from Dan Klein, Stuart Russell, Andrew Moore
More informationUse of Eigen values and eigen vectors to calculate higher transition probabilities
The Lecture Contains : MarkovBernoulli Chain Note Assignments Random Walks which are correlated Actual examples of Markov Chains Examples Use of Eigen values and eigen vectors to calculate higher transition
More informationIntroduction to Graphical Models
Introduction to Graphical Models The 15 th Winter School of Statistical Physics POSCO International Center & POSTECH, Pohang 2018. 1. 9 (Tue.) YungKyun Noh GENERALIZATION FOR PREDICTION 2 Probabilistic
More informationBayesian Computations for DSGE Models
Bayesian Computations for DSGE Models Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 This Lecture is Based on Bayesian Estimation of DSGE Models Edward P. Herbst &
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 informationReinforcement Learning
Reinforcement Learning ModelBased Reinforcement Learning Modelbased, PACMDP, sample complexity, exploration/exploitation, RMAX, E3, Bayesoptimal, Bayesian RL, model learning Vien Ngo MLR, University
More information1. INTRODUCTION Propp and Wilson (1996,1998) described a protocol called \coupling from the past" (CFTP) for exact sampling from a distribution using
Ecient Use of Exact Samples by Duncan J. Murdoch* and Jerey S. Rosenthal** Abstract Propp and Wilson (1996,1998) described a protocol called coupling from the past (CFTP) for exact sampling from the steadystate
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 DukeTsinghua Machine Learning Summer
More informationBayesian inference: what it means and why we care
Bayesian inference: what it means and why we care Robin J. Ryder Centre de Recherche en Mathématiques de la Décision Université ParisDauphine 6 November 2017 Mathematical Coffees Robin Ryder (Dauphine)
More informationMarkov chain Monte Carlo
Markov chain Monte Carlo Peter Beerli October 10, 2005 [this chapter is highly influenced by chapter 1 in Markov chain Monte Carlo in Practice, eds Gilks W. R. et al. Chapman and Hall/CRC, 1996] 1 Short
More informationVariational Principal Components
Variational Principal Components Christopher M. Bishop Microsoft Research 7 J. J. Thomson Avenue, Cambridge, CB3 0FB, U.K. cmbishop@microsoft.com http://research.microsoft.com/ cmbishop In Proceedings
More informationProblems with Penalised Maximum Likelihood and Jeffrey s Priors to Account For Separation in Large Datasets with Rare Events
Problems with Penalised Maximum Likelihood and Jeffrey s Priors to Account For Separation in Large Datasets with Rare Events Liam F. McGrath September 15, 215 Abstract When separation is a problem in binary
More informationMachine Learning. Lecture 4: Regularization and Bayesian Statistics. Feng Li. https://funglee.github.io
Machine Learning Lecture 4: Regularization and Bayesian Statistics Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 207 Overfitting Problem
More informationIntroduction to Stan for Markov Chain Monte Carlo
Introduction to Stan for Markov Chain Monte Carlo Matthew Simpson Department of Statistics, University of Missouri April 25, 2017 These slides and accompanying R and Stan files are available at http://stsn.missouri.edu/educationandoutreach.shtml
More informationMonte Carlo Methods. Geoff Gordon February 9, 2006
Monte Carlo Methods Geoff Gordon ggordon@cs.cmu.edu February 9, 2006 Numerical integration problem 5 4 3 f(x,y) 2 1 1 0 0.5 0 X 0.5 1 1 0.8 0.6 0.4 Y 0.2 0 0.2 0.4 0.6 0.8 1 x X f(x)dx Used for: function
More informationNATIONAL UNIVERSITY OF SINGAPORE PC5215 NUMERICAL RECIPES WITH APPLICATIONS. (Semester I: AY ) Time Allowed: 2 Hours
NATIONAL UNIVERSITY OF SINGAPORE PC55 NUMERICAL RECIPES WITH APPLICATIONS (Semester I: AY 0) Time Allowed: Hours INSTRUCTIONS TO CANDIDATES. This examination paper contains FIVE questions and comprises
More informationRANDOM TOPICS. stochastic gradient descent & Monte Carlo
RANDOM TOPICS stochastic gradient descent & Monte Carlo MASSIVE MODEL FITTING nx minimize f(x) = 1 n i=1 f i (x) Big! (over 100K) minimize 1 least squares 2 kax bk2 = X i 1 2 (a ix b i ) 2 minimize 1 SVM
More informationMeasurement Error and Linear Regression of Astronomical Data. Brandon Kelly Penn State Summer School in Astrostatistics, June 2007
Measurement Error and Linear Regression of Astronomical Data Brandon Kelly Penn State Summer School in Astrostatistics, June 2007 Classical Regression Model Collect n data points, denote i th pair as (η
More informationBayesian Inference using Markov Chain Monte Carlo in Phylogenetic Studies
Bayesian Inference using Markov Chain Monte Carlo in Phylogenetic Studies 1 What is phylogeny? Essay written for the course in Markov Chains 2004 Torbjörn Karfunkel Phylogeny is the evolutionary development
More informationThe PolyaGamma Gibbs Sampler for Bayesian. Logistic Regression is Uniformly Ergodic
he PolyaGamma Gibbs Sampler for Bayesian Logistic Regression is Uniformly Ergodic Hee Min Choi and James P. Hobert Department of Statistics University of Florida August 013 Abstract One of the most widely
More informationMonetary and Exchange Rate Policy Under Remittance Fluctuations. Technical Appendix and Additional Results
Monetary and Exchange Rate Policy Under Remittance Fluctuations Technical Appendix and Additional Results Federico Mandelman February In this appendix, I provide technical details on the Bayesian estimation.
More informationEvaluation of the Fisher information matrix in nonlinear mixed eect models without linearization
Evaluation of the Fisher information matrix in nonlinear mixed eect models without linearization Sebastian Ueckert, MarieKarelle Riviere, and France Mentré INSERM, IAME, UMR 1137, F75018 Paris, France;
More informationA Brief Introduction to Graphical Models. Presenter: Yijuan Lu November 12,2004
A Brief Introduction to Graphical Models Presenter: Yijuan Lu November 12,2004 References Introduction to Graphical Models, Kevin Murphy, Technical Report, May 2001 Learning in Graphical Models, Michael
More informationA Bayesian Probit Model with Spatial Dependencies
A Bayesian Probit Model with Spatial Dependencies Tony E. Smith Department of Systems Engineering University of Pennsylvania Philadephia, PA 19104 email: tesmith@ssc.upenn.edu James P. LeSage Department
More informationModel Counting for Logical Theories
Model Counting for Logical Theories Wednesday Dmitry Chistikov Rayna Dimitrova Department of Computer Science University of Oxford, UK Max Planck Institute for Software Systems (MPISWS) Kaiserslautern
More informationDynamic Data Driven Simulations in Stochastic Environments
Computing 77, 321 333 (26) Digital Object Identifier (DOI) 1.17/s6761653 Dynamic Data Driven Simulations in Stochastic Environments C. Douglas, Lexington, Y. Efendiev, R. Ewing, College Station, V.
More informationHamiltonian Monte Carlo with Fewer Momentum Reversals
Hamiltonian Monte Carlo with ewer Momentum Reversals Jascha SohlDickstein December 6, 2 Hamiltonian dynamics with partial momentum refreshment, in the style of Horowitz, Phys. ett. B, 99, explore the
More information