Fitting Complex PK/PD Models with Stan

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Fitting Complex PK/PD Models with Stan"

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 non-analytic 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 Non-analytic 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 well-posed 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 one-at-a-time to no greater success.

35 The Gibbs sampler updates each parameter one-at-a-time 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 high-performance 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 one-compartment Michaelis-Menten 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

Tutorial 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 information

Lecture 2: From Linear Regression to Kalman Filter and Beyond

Lecture 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 information

Hamiltonian Monte Carlo

Hamiltonian Monte Carlo Hamiltonian Monte Carlo within Stan Daniel Lee Columbia University, Statistics Department bearlee@alum.mit.edu BayesComp mc-stan.org Why MCMC? Have data. Have a rich statistical model. No analytic solution.

More information

Markov Chain Monte Carlo, Numerical Integration

Markov 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 information

Bayesian Inference: Probit and Linear Probability Models

Bayesian Inference: Probit and Linear Probability Models Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 5-1-2014 Bayesian Inference: Probit and Linear Probability Models Nate Rex Reasch Utah State University Follow

More information

DAG models and Markov Chain Monte Carlo methods a short overview

DAG 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 information

Example: Ground Motion Attenuation

Example: 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 information

Bagging During Markov Chain Monte Carlo for Smoother Predictions

Bagging 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 information

Lecture 2: From Linear Regression to Kalman Filter and Beyond

Lecture 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 information

Machine Learning Techniques for Computer Vision

Machine 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 information

Characterizing Travel Time Reliability and Passenger Path Choice in a Metro Network

Characterizing 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, Singapore-ETH Centre lijun.sun@ivt.baug.ethz.ch National University of Singapore

More information

Notes on pseudo-marginal methods, variational Bayes and ABC

Notes on pseudo-marginal methods, variational Bayes and ABC Notes on pseudo-marginal methods, variational Bayes and ABC Christian Andersson Naesseth October 3, 2016 The Pseudo-Marginal Framework Assume we are interested in sampling from the posterior distribution

More information

Bayesian networks: approximate inference

Bayesian networks: approximate inference Bayesian networks: approximate inference Machine Intelligence Thomas D. Nielsen September 2008 Approximative inference September 2008 1 / 25 Motivation Because of the (worst-case) intractability of exact

More information

Approximate Inference using MCMC

Approximate 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 information

Chapter 2. Data Analysis

Chapter 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 information

Petr Volf. Model for Difference of Two Series of Poisson-like Count Data

Petr Volf. Model for Difference of Two Series of Poisson-like 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 e-mail: volf@utia.cas.cz Model for Difference of Two Series of Poisson-like

More information

Multilevel Statistical Models: 3 rd edition, 2003 Contents

Multilevel 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 information

Notes on Noise Contrastive Estimation (NCE)

Notes on Noise Contrastive Estimation (NCE) Notes on Noise Contrastive Estimation NCE) David Meyer dmm@{-4-5.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 information

Inversion Base Height. Daggot Pressure Gradient Visibility (miles)

Inversion 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 information

MCMC notes by Mark Holder

MCMC 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 information

Bayesian GLMs and Metropolis-Hastings Algorithm

Bayesian 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 information

MCMC algorithms for fitting Bayesian models

MCMC 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 information

Markov Networks.

Markov 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 information

Nonparametric Drift Estimation for Stochastic Differential Equations

Nonparametric 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 information

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA

PATTERN 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 information

Bayesian Prediction of Code Output. ASA Albuquerque Chapter Short Course October 2014

Bayesian 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 information

Karl-Rudolf Koch Introduction to Bayesian Statistics Second Edition

Karl-Rudolf Koch Introduction to Bayesian Statistics Second Edition Karl-Rudolf Koch Introduction to Bayesian Statistics Second Edition Karl-Rudolf Koch Introduction to Bayesian Statistics Second, updated and enlarged Edition With 17 Figures Professor Dr.-Ing., Dr.-Ing.

More information

Risk Estimation and Uncertainty Quantification by Markov Chain Monte Carlo Methods

Risk 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/risk-and-uncertainty/staff/k-zuev/

More information

Bayes Networks 6.872/HST.950

Bayes Networks 6.872/HST.950 Bayes Networks 6.872/HST.950 What Probabilistic Models Should We Use? Full joint distribution Completely expressive Hugely data-hungry Exponential computational complexity Naive Bayes (full conditional

More information

When 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 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 information

MATH 12 CLASS 5 NOTES, SEP

MATH 12 CLASS 5 NOTES, SEP MATH 12 CLASS 5 NOTES, SEP 30 2011 Contents 1. Vector-valued functions 1 2. Differentiating and integrating vector-valued functions 3 3. Velocity and Acceleration 4 Over the past two weeks we have developed

More information

Statistical Inference and Methods

Statistical 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 information

Adaptive Monte Carlo methods

Adaptive Monte Carlo methods Adaptive Monte Carlo methods Jean-Michel Marin Projet Select, INRIA Futurs, Université Paris-Sud joint with Randal Douc (École Polytechnique), Arnaud Guillin (Université de Marseille) and Christian Robert

More information

Introduction to Markov Chain Monte Carlo

Introduction 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 information

Paul Karapanagiotidis ECO4060

Paul Karapanagiotidis ECO4060 Paul Karapanagiotidis ECO4060 The way forward 1) Motivate why Markov-Chain Monte Carlo (MCMC) is useful for econometric modeling 2) Introduce Markov-Chain Monte Carlo (MCMC) - Metropolis-Hastings (MH)

More information

General Construction of Irreversible Kernel in Markov Chain Monte Carlo

General 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 information

Bayesian Monte Carlo Filtering for Stochastic Volatility Models

Bayesian 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 information

Definition 3 (Continuity). A function f is continuous at c if lim x c f(x) = f(c).

Definition 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 information

An ABC interpretation of the multiple auxiliary variable method

An ABC interpretation of the multiple auxiliary variable method School of Mathematical and Physical Sciences Department of Mathematics and Statistics Preprint MPS-2016-07 27 April 2016 An ABC interpretation of the multiple auxiliary variable method by Dennis Prangle

More information

Bayesian Inference. Chapter 4: Regression and Hierarchical Models

Bayesian 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 information

Bayesian linear regression

Bayesian 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 information

Efficiency and Reliability of Bayesian Calibration of Energy Supply System Models

Efficiency 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 information

Package lmm. R topics documented: March 19, Version 0.4. Date Title Linear mixed models. Author Joseph L. Schafer

Package 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 2012-3-19 Title Linear mixed models Author Joseph L. Schafer Maintainer Jing hua Zhao Depends R (>= 2.0.0) Description Some

More information

IEEE 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 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 information

Comparison of Three Calculation Methods for a Bayesian Inference of Two Poisson Parameters

Comparison of Three Calculation Methods for a Bayesian Inference of Two Poisson Parameters Journal of Modern Applied Statistical Methods Volume 13 Issue 1 Article 26 5-1-2014 Comparison of Three Calculation Methods for a Bayesian Inference of Two Poisson Parameters Yohei Kawasaki Tokyo University

More information

Neutral Bayesian reference models for incidence rates of (rare) clinical events

Neutral 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 information

Approximate Bayesian Computation: a simulation based approach to inference

Approximate 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 information

CPSC 540: Machine Learning

CPSC 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 information

SAMPLING ALGORITHMS. In general. Inference in Bayesian models

SAMPLING 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 information

Der SPP 1167-PQP und die stochastische Wettervorhersage

Der SPP 1167-PQP und die stochastische Wettervorhersage Der SPP 1167-PQP und die stochastische Wettervorhersage Andreas Hense 9. November 2007 Overview The priority program SPP1167: mission and structure The stochastic weather forecasting Introduction, Probabilities

More information

Density Estimation. Seungjin Choi

Density Estimation. Seungjin Choi Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/

More information

Bayesian course - problem set 5 (lecture 6)

Bayesian 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 information

Bayesian Networks BY: MOHAMAD ALSABBAGH

Bayesian 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 information

Accelerated Gibbs Sampling for Infinite Sparse Factor Analysis

Accelerated Gibbs Sampling for Infinite Sparse Factor Analysis LLNL-TR-499647 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 information

Introduction to Bayesian methods in inverse problems

Introduction 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 information

PROBABILISTIC PROGRAMMING: BAYESIAN MODELLING MADE EASY

PROBABILISTIC 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 information

Markov Chain Monte Carlo Methods for Stochastic

Markov 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 information

NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET. Directional Metropolis Hastings updates for posteriors with nonlinear likelihoods

NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET. Directional Metropolis Hastings updates for posteriors with nonlinear likelihoods NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET Directional Metropolis Hastings updates for posteriors with nonlinear likelihoods by Håkon Tjelmeland and Jo Eidsvik PREPRINT STATISTICS NO. 5/2004 NORWEGIAN

More information

PROBABILISTIC PROGRAMMING: BAYESIAN MODELLING MADE EASY. Arto Klami

PROBABILISTIC 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 information

Lecture 6: Bayesian Inference in SDE Models

Lecture 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 information

Sampling Algorithms for Probabilistic Graphical models

Sampling 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 information

Factorization of Seperable and Patterned Covariance Matrices for Gibbs Sampling

Factorization 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, 228-77 California Institute of

More information

Sequential Monte Carlo and Particle Filtering. Frank Wood Gatsby, November 2007

Sequential 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 information

Learning Static Parameters in Stochastic Processes

Learning 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 information

MCMC Methods: Gibbs and Metropolis

MCMC Methods: Gibbs and Metropolis MCMC Methods: Gibbs and Metropolis Patrick Breheny February 28 Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/30 Introduction As we have seen, the ability to sample from the posterior distribution

More information

Partial factor modeling: predictor-dependent shrinkage for linear regression

Partial factor modeling: predictor-dependent shrinkage for linear regression modeling: predictor-dependent 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 information

Latent Dirichlet Allocation (LDA)

Latent 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 & Pre-Processing Natural Language

More information

Uncertainty Quantification in Performance Evaluation of Manufacturing Processes

Uncertainty 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 information

Bayesian Inference. Chapter 1. Introduction and basic concepts

Bayesian 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 information

Bayesian room-acoustic modal analysis

Bayesian room-acoustic modal analysis Bayesian room-acoustic 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 information

Computer Practical: Metropolis-Hastings-based MCMC

Computer Practical: Metropolis-Hastings-based MCMC Computer Practical: Metropolis-Hastings-based MCMC Andrea Arnold and Franz Hamilton North Carolina State University July 30, 2016 A. Arnold / F. Hamilton (NCSU) MH-based MCMC July 30, 2016 1 / 19 Markov

More information

Bayesian inference & process convolution models Dave Higdon, Statistical Sciences Group, LANL

Bayesian 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 information

Bayesian lithology/fluid inversion comparison of two algorithms

Bayesian lithology/fluid inversion comparison of two algorithms Comput Geosci () 14:357 367 DOI.07/s596-009-9155-9 REVIEW PAPER Bayesian lithology/fluid inversion comparison of two algorithms Marit Ulvmoen Hugo Hammer Received: 2 April 09 / Accepted: 17 August 09 /

More information

CS 188: Artificial Intelligence Spring Announcements

CS 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 information

Use of Eigen values and eigen vectors to calculate higher transition probabilities

Use of Eigen values and eigen vectors to calculate higher transition probabilities The Lecture Contains : Markov-Bernoulli 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 information

Introduction to Graphical Models

Introduction to Graphical Models Introduction to Graphical Models The 15 th Winter School of Statistical Physics POSCO International Center & POSTECH, Pohang 2018. 1. 9 (Tue.) Yung-Kyun Noh GENERALIZATION FOR PREDICTION 2 Probabilistic

More information

Bayesian Computations for DSGE Models

Bayesian 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 information

Sampling Rejection Sampling Importance Sampling Markov Chain Monte Carlo. Sampling Methods. Oliver Schulte - CMPT 419/726. Bishop PRML Ch.

Sampling 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 information

Reinforcement Learning

Reinforcement Learning Reinforcement Learning Model-Based Reinforcement Learning Model-based, PAC-MDP, sample complexity, exploration/exploitation, RMAX, E3, Bayes-optimal, Bayesian RL, model learning Vien Ngo MLR, University

More information

1. INTRODUCTION Propp and Wilson (1996,1998) described a protocol called \coupling from the past" (CFTP) for exact sampling from a distribution using

1. 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 steady-state

More information

Introduction to Stochastic Gradient Markov Chain Monte Carlo Methods

Introduction to Stochastic Gradient Markov Chain Monte Carlo Methods Introduction to Stochastic Gradient Markov Chain Monte Carlo Methods Changyou Chen Department of Electrical and Computer Engineering, Duke University cc448@duke.edu Duke-Tsinghua Machine Learning Summer

More information

Bayesian inference: what it means and why we care

Bayesian 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 Paris-Dauphine 6 November 2017 Mathematical Coffees Robin Ryder (Dauphine)

More information

Markov chain Monte Carlo

Markov 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 information

Variational Principal Components

Variational 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 information

Problems 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 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 information

Machine Learning. Lecture 4: Regularization and Bayesian Statistics. Feng Li. https://funglee.github.io

Machine 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 information

Introduction to Stan for Markov Chain Monte Carlo

Introduction 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/education-and-outreach.shtml

More information

Monte Carlo Methods. Geoff Gordon February 9, 2006

Monte 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 information

NATIONAL UNIVERSITY OF SINGAPORE PC5215 NUMERICAL RECIPES WITH APPLICATIONS. (Semester I: AY ) Time Allowed: 2 Hours

NATIONAL 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 information

RANDOM TOPICS. stochastic gradient descent & Monte Carlo

RANDOM 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 information

Measurement 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 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 information

Bayesian Inference using Markov Chain Monte Carlo in Phylogenetic Studies

Bayesian 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 information

The Polya-Gamma Gibbs Sampler for Bayesian. Logistic Regression is Uniformly Ergodic

The Polya-Gamma Gibbs Sampler for Bayesian. Logistic Regression is Uniformly Ergodic he Polya-Gamma 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 information

Monetary 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 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 information

Evaluation 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 Evaluation of the Fisher information matrix in nonlinear mixed eect models without linearization Sebastian Ueckert, Marie-Karelle Riviere, and France Mentré INSERM, IAME, UMR 1137, F-75018 Paris, France;

More information

A Brief Introduction to Graphical Models. Presenter: Yijuan Lu November 12,2004

A 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 information

A Bayesian Probit Model with Spatial Dependencies

A 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 information

Model Counting for Logical Theories

Model 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 (MPI-SWS) Kaiserslautern

More information

Dynamic Data Driven Simulations in Stochastic Environments

Dynamic Data Driven Simulations in Stochastic Environments Computing 77, 321 333 (26) Digital Object Identifier (DOI) 1.17/s67-6-165-3 Dynamic Data Driven Simulations in Stochastic Environments C. Douglas, Lexington, Y. Efendiev, R. Ewing, College Station, V.

More information

Hamiltonian Monte Carlo with Fewer Momentum Reversals

Hamiltonian Monte Carlo with Fewer Momentum Reversals Hamiltonian Monte Carlo with ewer Momentum Reversals Jascha Sohl-Dickstein December 6, 2 Hamiltonian dynamics with partial momentum refreshment, in the style of Horowitz, Phys. ett. B, 99, explore the

More information