Particle Filtering Approaches for Dynamic Stochastic Optimization
|
|
- Martha Burns
- 5 years ago
- Views:
Transcription
1 Particle Filtering Approaches for Dynamic Stochastic Optimization John R. Birge The University of Chicago Booth School of Business Joint work with Nicholas Polson, Chicago Booth. JRBirge I-Sim Workshop, Purdue U., July
2 Themes Dynamic stochastic optimization models face the curse of dimensionality Randomization can overcome some dimensionality effects With sufficient symmetry, the results can be quite effective Keys are conditioning properly on outcomes, appropriate re-sampling and split sampling, and uses of Gaussian mixtures JRBirge I-Sim Workshop, Purdue U., July
3 Basic problem Outline Problems of estimation Particle filters and sequential methods for simulation Optimization by slice sampling Dynamic Gaussian problems Extensions to Gaussian mixtures JRBirge I-Sim Workshop, Purdue U., July
4 General Problem: Find a set of controls, u t ; t = 0; : : : ; T, to T min E[ X 1 u t 2U t \A t ;x t+1 =g t (x t ;u t ) t=0 f t (x t+1 ; u t )] where A t is t -measurable (nonanticipative), U t represents the possibility of additional constraints on the controls, g t represents state dynamics, f t represents the cost of the current action and next state transition. Challenges: Rapid expansion of states in dimension of x t in each period, Exponential growth of trees over time. JRBirge I-Sim Workshop, Purdue U., July
5 First Problem: What is x t? Example: hidden Markov model Example: We begin with inventory x 0 (which may be uncertain) We make observations y 1,,y T based on reported sales and new inputs What is the distribution of actual inventory x t at any time t? JRBirge I-Sim Workshop, Purdue U., July
6 Bayesian Interpretation Suppose a distribution on x 0 : f(x 0 ) Assume some transition: p(x t x t-1 ) Observations: q(y t x t ) Goal: find P(x t y 1,,y t ( y T )) Idea: Use Bayes Rule to find the conditional probability: P(x t y t )=P(x t,y t )/P(y t ) / s q(y s x s )p(x s x s-1 )f(x 0 ) JRBirge I-Sim Workshop, Purdue U., July
7 Particle Methods Idea: Keep a limited number (N) of particles in each period Propagate forward to obtain new points Re-sample (re-weight) backwards to get consistent weights Re-balance if weights are low References: Gordon, Salmond, Smith (1993), Doucet (2008) JRBirge I-Sim Workshop, Purdue U., July
8 General Processes If the observation and transition functions, q and p are quite general, then P(x t y t ) may not be available analytically (E.g., no sales if stocked out, maximum limit, bulky demand) Alternatives: Markov chain Monte Carlo (MCMC) Particle methods JRBirge I-Sim Workshop, Purdue U., July
9 Sequential Importance Resampling (SIR) Method 1. Propagate. Draw x t+1;j ; j = 1; : : : ; N from f(x t+1;j jx t;j ). 2. Re-weight. Update weights, ^w t;j ; j = 1; : : : ; N, to ^w t+1;j = for j = 1; : : : ; N. w t;j g(y t+1 jx t+1 ) P N j=1 ^w t;jg(y t+1 jx t+1 ) 3. Re-sample. If a re-sampling condition is satis ed, draw N new samples x 0 t+1;j ; j = 1; : : : ; N from x t+1;j; j = 1; : : : ; N with probability distribution given by ^w t+1;j ; j = 1; : : : ; N; let x t+1;j = x 0 t+1;j and ^w t+1;j = 1=N for j = 1; : : : ; N. JRBirge I-Sim Workshop, Purdue U., July
10 Tractable Case: LQG Linear, quadratic Gaussian (LQG) Model Implications: All of the distributions are normal Can derive analytical formulas (Kalman filter): y t = H x t + v t ; x t =F x t-1 + u t ; v~n(0,q); u~n(o,r) Iterations: w t = y t -Hx t ; S=HP H T +R; K=P HTS -1 ; x t =x t +Kw t P =(I-KH)P ; x t =Fx t-1 ; P =FP F T +Q JRBirge RMS Phoenix, October
11 Inventory over 10 periods: - Actual (*) - Observed (+) Kalman estimate: x t ~ N(x t,p t ) Here: x 10 ~ N(13.8,0.62) Kalman Prediction JRBirge I-Sim Workshop, Purdue U., July
12 Comparison to MCMC/Particle Filter Idea: construct a sample distribution that fits the data using only information about one-step transitions Goal: maintain a constant number of samples across time instead of explosively growing tree Can it work? Why? JRBirge I-Sim Workshop, Purdue U., July
13 Propagate w t1, x 1 t w t2, x 2 t... w tn, x N t Method Structure Re-sample: x 1 w 1 t t+1 x 2 w 2 t t+1 y t x N w N t t+1 JRBirge I-Sim Workshop, Purdue U., July
14 Cum. Results N=50 Compare to Kalman Cumulative JRBirge I-Sim Workshop, Purdue U., July
15 Particle Frequency N=50 Particle (+) Kalman (*) JRBirge I-Sim Workshop, Purdue U., July
16 Other Uses of Particles Particle filtering: finding x t from y t =(y 1,,y t ) Particle smoothing: finding x t from y T =(y 1,,y T ) Particle learning: (Polson et al.) Find parameters µ t at the same time to explain the model. Simulation JRBirge I-Sim Workshop, Purdue U., July
17 Extensions to Optimization Problem: nd x to Pincus (1968): enough to sample from: min x f(x)subject to g(x) 0 ¼ (x) / expf (f(x))g with constraints (e.g., Geman/Geman (1985)) : ¼ ; (x) / exp f (f(x) + g(x))g Extension: Augment with latent variables ( ;!) then we can write ¼ ; (x) / expf (f(x) + g(x))g = e ax E! fe x2 2! ge bx E fe x2 2 g where the parameters (a; b) depend on ( ; ) and the functional forms of (f(x); g(x)). JRBirge I-Sim Workshop, Purdue U., July
18 Basic Optimization via Simulation Pincus results: Model: min F(x) Create a distribution P (x)/ exp(- F(x)) Then: lim 1 E P (x) = x * JRBirge I-Sim Workshop, Purdue U., July
19 Markov Chain Simulation From x t to x t+1 Sample over exp(- F(¹ x t +(1-¹) x t+1 ) ) Choose directions to obtain mixing in the region JRBirge I-Sim Workshop, Purdue U., July
20 Example Rosenbrock function: Á(x = (x 1 ; x 2 )) = (1 x 1 ) (x 2 x 2 1) 2 ; which has a global minimum at x = (1; 1). Speci cation for f: x t = fu if v e (Á(u)=Á(x t 1)) ; x t 1 otherwise; g where u» U([ 4; 4] 2 ), a uniform distribution over [ 4; 4] [ 4; 4], and v» U([0; 1]), a standard uniform draw. Note: corresponds to the Metropolis-Hastings method of Markov Chain Monte Carlo using the distribution proportional to e Á(x) for the acceptance criterion. JRBirge I-Sim Workshop, Purdue U., July
21 Test for Rosenbrock T=5000 iterations N=500 replications Re-sample when: Effective Sample Size<250 =2, 10, 100 JRBirge I-Sim Workshop, Purdue U., July
22 Rosenbrock Graphs = JRBirge I-Sim Workshop, Purdue U., July
23 Other Examples: Rastigrin JRBirge I-Sim Workshop, Purdue U., July
24 Himmelblau JRBirge I-Sim Workshop, Purdue U., July
25 Shubert JRBirge I-Sim Workshop, Purdue U., July
26 Combine Particles and Simulation Suppose N particles are used Consider a 2-stage problem f(x,») = c(x) + Q(x,y,») Distribution on x, y i,»: P(x,y 1, y N,»)/ i=1 N (c(x)+q(x,y i,» i )) Result: E P (x) x * JRBirge I-Sim Workshop, Purdue U., July
27 Implementation Propagation forward on y s: q(» j x 1 ) / { c(x) + Q(» j, y j, x)} p(» j ) where y j =argmin(c(x) + Q(» j, y j, x)) p( x» 1,y 1,,» N y N ) / j=1 N {c(x) + Q(» j, y j, x) }p(» j ) The particles concentrate the distribution on x* as in the deterministic optimization. JRBirge I-Sim Workshop, Purdue U., July
28 Example From B/Louveaux, Chapter 1 (Farmer): JRBirge I-Sim Workshop, Purdue U., July
29 Dynamic Models General Idea: Create a distribution over decision variables and random variables Use MCMC to sample with sufficient number of particles or annealing parameter marginal mean on optimum Extension for dynamics: Keep constant sample particle numbers Convergence in fixed number of particles per stage? JRBirge I-Sim Workshop, Purdue U., July
30 Example: Linear-Quadratic Gaussian All distributions are available in closed form x t =Ax t-1 +B t u t-1 +v t All normal distributions with risk-sensitive objective: -log E(exp(-(µ/2)(x 2T Qx 2 +u 1T Ru 1 ))) where u 1 is control in the first period Note: Equivalent to robust formulation Now, all can be found in closed form JRBirge I-Sim Workshop, Purdue U., July
31 Results for LQG Mean of u: m u 1 m u 1 = L 1(µ) x 1, where L 1 is the result from the Kalman filter Variance of u: V u (N) where (N) 0 as N 1 V u 1 m u 1 (µ) u 1 * as µ 0 where u 1 * is the two-stage risk-neutral solution. JRBirge I-Sim Workshop, Purdue U., July
32 Extensions to T Stages Can derive S t µ recursively to show that: p(x t x t-1,u t-1 )=K exp(-(1/2) ( s=1 t-1 (x tt Q s x s +u st R s u s )+x tt S t µ x t ) + (x t -Ax t-1 -Bu t-1 ) T V t-1 (x t -Ax t-1 -Bu t-1 )) For particle methods, j=1 N: Each x t j drawn consistently from this distribution JRBirge I-Sim Workshop, Purdue U., July
33 w t1, x t 1 u t 1 w t2, x t 2. u t 2 w tn, x t N u t N General Method Structure Propogate Re-sample: x 1 t+1 w t1, x 1 t x 2 t+1 u 1 t... x t+1 N w t2, x t 2 u t 2 JRBirge I-Sim Workshop, Purdue U., July w tn, x t N u t N
34 General Result If the propagation step is unbiased, then E(u 1 ) u 1 * The unbiased result is possible in LQG from the recursive structure. Harder to obtain in more general systems (without additional future sampling) JRBirge I-Sim Workshop, Purdue U., July
35 Two-Stage: Compare Direct MC Low Risk Aversion High Risk Aversion JRBirge I-Sim Workshop, Purdue U., July
36 Three Stage Example JRBirge I-Sim Workshop, Purdue U., July
37 Generalizing Objectives Direct: risk-sensitive exponential utility: exp(-µ Q(x,y,»)) - Q quadratic Objective: e -µ y - d E [exp((-µ 2 /2 j)(y j d) 2 )] where ~ Exp(2) Also, use: E[ e -µ y - d ]¼ 1 + µ y-d + O(1) JRBirge I-Sim Workshop, Purdue U., July
38 Further Generalizations Use objective approximations that preserve normal forms Extend LQG-like results to this general framework Find other frameworks where unbiased future samples are available Obtain sensitivity results on bias from imperfect future sampling Use optimization to pick ML instead JRBirge I-Sim Workshop, Purdue U., July
39 Conclusions Techniques from MCMC can be used effectively for optimizing to reduce dimension effects Results work well in situations with sufficient symmetry Keys are to effectively use re-sampling, split (and slice) sampling, and latent variables JRBirge I-Sim Workshop, Purdue U., July
40 Other Questions to Consider How can learning and parameter estimation be combined with optimization? What is the range of problems that can be approached in this manner? What are the best uses of optimization technology as part of this process? JRBirge I-Sim Workshop, Purdue U., July
41 Thank you! Questions? JRBirge I-Sim Workshop, Purdue U., July
Markov Chain Monte Carlo Methods for Stochastic Optimization
Markov Chain Monte Carlo Methods for Stochastic Optimization John R. Birge The University of Chicago Booth School of Business Joint work with Nicholas Polson, Chicago Booth. JRBirge U of Toronto, MIE,
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 informationParticle Filtering for Data-Driven Simulation and Optimization
Particle Filtering for Data-Driven Simulation and Optimization John R. Birge The University of Chicago Booth School of Business Includes joint work with Nicholas Polson. JRBirge INFORMS Phoenix, October
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 informationSequential Monte Carlo Methods for Bayesian Computation
Sequential Monte Carlo Methods for Bayesian Computation A. Doucet Kyoto Sept. 2012 A. Doucet (MLSS Sept. 2012) Sept. 2012 1 / 136 Motivating Example 1: Generic Bayesian Model Let X be a vector parameter
More informationAn introduction to Sequential Monte Carlo
An introduction to Sequential Monte Carlo Thang Bui Jes Frellsen Department of Engineering University of Cambridge Research and Communication Club 6 February 2014 1 Sequential Monte Carlo (SMC) methods
More informationThe Kalman Filter ImPr Talk
The Kalman Filter ImPr Talk Ged Ridgway Centre for Medical Image Computing November, 2006 Outline What is the Kalman Filter? State Space Models Kalman Filter Overview Bayesian Updating of Estimates Kalman
More informationLecture 8: Bayesian Estimation of Parameters in State Space Models
in State Space Models March 30, 2016 Contents 1 Bayesian estimation of parameters in state space models 2 Computational methods for parameter estimation 3 Practical parameter estimation in state space
More informationPattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
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 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 informationAuxiliary Particle Methods
Auxiliary Particle Methods Perspectives & Applications Adam M. Johansen 1 adam.johansen@bristol.ac.uk Oxford University Man Institute 29th May 2008 1 Collaborators include: Arnaud Doucet, Nick Whiteley
More information16 : Markov Chain Monte Carlo (MCMC)
10-708: Probabilistic Graphical Models 10-708, Spring 2014 16 : Markov Chain Monte Carlo MCMC Lecturer: Matthew Gormley Scribes: Yining Wang, Renato Negrinho 1 Sampling from low-dimensional distributions
More informationMonte Carlo Approximation of Monte Carlo Filters
Monte Carlo Approximation of Monte Carlo Filters Adam M. Johansen et al. Collaborators Include: Arnaud Doucet, Axel Finke, Anthony Lee, Nick Whiteley 7th January 2014 Context & Outline Filtering in State-Space
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 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 informationGraphical Models and Kernel Methods
Graphical Models and Kernel Methods Jerry Zhu Department of Computer Sciences University of Wisconsin Madison, USA MLSS June 17, 2014 1 / 123 Outline Graphical Models Probabilistic Inference Directed vs.
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 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 informationAn introduction to particle filters
An introduction to particle filters Andreas Svensson Department of Information Technology Uppsala University June 10, 2014 June 10, 2014, 1 / 16 Andreas Svensson - An introduction to particle filters Outline
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 informationRobotics. Mobile Robotics. Marc Toussaint U Stuttgart
Robotics Mobile Robotics State estimation, Bayes filter, odometry, particle filter, Kalman filter, SLAM, joint Bayes filter, EKF SLAM, particle SLAM, graph-based SLAM Marc Toussaint U Stuttgart DARPA Grand
More informationSensor Fusion: Particle Filter
Sensor Fusion: Particle Filter By: Gordana Stojceska stojcesk@in.tum.de Outline Motivation Applications Fundamentals Tracking People Advantages and disadvantages Summary June 05 JASS '05, St.Petersburg,
More informationTowards a Bayesian model for Cyber Security
Towards a Bayesian model for Cyber Security Mark Briers (mbriers@turing.ac.uk) Joint work with Henry Clausen and Prof. Niall Adams (Imperial College London) 27 September 2017 The Alan Turing Institute
More informationLecture 4: Dynamic models
linear s Lecture 4: s Hedibert Freitas Lopes The University of Chicago Booth School of Business 5807 South Woodlawn Avenue, Chicago, IL 60637 http://faculty.chicagobooth.edu/hedibert.lopes hlopes@chicagobooth.edu
More informationPattern Recognition and Machine Learning. Bishop Chapter 11: Sampling Methods
Pattern Recognition and Machine Learning Chapter 11: Sampling Methods Elise Arnaud Jakob Verbeek May 22, 2008 Outline of the chapter 11.1 Basic Sampling Algorithms 11.2 Markov Chain Monte Carlo 11.3 Gibbs
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 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 informationL09. PARTICLE FILTERING. NA568 Mobile Robotics: Methods & Algorithms
L09. PARTICLE FILTERING NA568 Mobile Robotics: Methods & Algorithms Particle Filters Different approach to state estimation Instead of parametric description of state (and uncertainty), use a set of state
More informationControlled sequential Monte Carlo
Controlled sequential Monte Carlo Jeremy Heng, Department of Statistics, Harvard University Joint work with Adrian Bishop (UTS, CSIRO), George Deligiannidis & Arnaud Doucet (Oxford) Bayesian Computation
More informationParticle Learning and Smoothing
Particle Learning and Smoothing Carlos Carvalho, Michael Johannes, Hedibert Lopes and Nicholas Polson This version: September 2009 First draft: December 2007 Abstract In this paper we develop particle
More informationIntroduction to Particle Filters for Data Assimilation
Introduction to Particle Filters for Data Assimilation Mike Dowd Dept of Mathematics & Statistics (and Dept of Oceanography Dalhousie University, Halifax, Canada STATMOS Summer School in Data Assimila5on,
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 11 Project
More informationAdvanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering
Advanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering Axel Gandy Department of Mathematics Imperial College London http://www2.imperial.ac.uk/~agandy London
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 informationSmoothers: Types and Benchmarks
Smoothers: Types and Benchmarks Patrick N. Raanes Oxford University, NERSC 8th International EnKF Workshop May 27, 2013 Chris Farmer, Irene Moroz Laurent Bertino NERSC Geir Evensen Abstract Talk builds
More informationECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering
ECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering Lecturer: Nikolay Atanasov: natanasov@ucsd.edu Teaching Assistants: Siwei Guo: s9guo@eng.ucsd.edu Anwesan Pal:
More informationLecture 6: Multiple Model Filtering, Particle Filtering and Other Approximations
Lecture 6: Multiple Model Filtering, Particle Filtering and Other Approximations Department of Biomedical Engineering and Computational Science Aalto University April 28, 2010 Contents 1 Multiple Model
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 informationA Review of Pseudo-Marginal Markov Chain Monte Carlo
A Review of Pseudo-Marginal Markov Chain Monte Carlo Discussed by: Yizhe Zhang October 21, 2016 Outline 1 Overview 2 Paper review 3 experiment 4 conclusion Motivation & overview Notation: θ denotes the
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 informationSlice Sampling Approaches for Stochastic Optimization
Slice Sampling Approaches for Stochastic Optimization John Birge, Nick Polson Chicago Booth INFORMS Philadelphia, October 2015 John Birge, Nick Polson (Chicago Booth) Split Sampling 05/01 1 / 28 Overview
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 informationCSC 446 Notes: Lecture 13
CSC 446 Notes: Lecture 3 The Problem We have already studied how to calculate the probability of a variable or variables using the message passing method. However, there are some times when the structure
More informationLinear Dynamical Systems
Linear Dynamical Systems Sargur N. srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/cse574/index.html Two Models Described by Same Graph Latent variables Observations
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 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 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 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 informationHuman Pose Tracking I: Basics. David Fleet University of Toronto
Human Pose Tracking I: Basics David Fleet University of Toronto CIFAR Summer School, 2009 Looking at People Challenges: Complex pose / motion People have many degrees of freedom, comprising an articulated
More informationMarkov Chain Monte Carlo
Markov Chain Monte Carlo (and Bayesian Mixture Models) David M. Blei Columbia University October 14, 2014 We have discussed probabilistic modeling, and have seen how the posterior distribution is the critical
More information6 Markov Chain Monte Carlo (MCMC)
6 Markov Chain Monte Carlo (MCMC) The underlying idea in MCMC is to replace the iid samples of basic MC methods, with dependent samples from an ergodic Markov chain, whose limiting (stationary) distribution
More informationNonlinear and/or Non-normal Filtering. Jesús Fernández-Villaverde University of Pennsylvania
Nonlinear and/or Non-normal Filtering Jesús Fernández-Villaverde University of Pennsylvania 1 Motivation Nonlinear and/or non-gaussian filtering, smoothing, and forecasting (NLGF) problems are pervasive
More informationBayesian Dynamic Linear Modelling for. Complex Computer Models
Bayesian Dynamic Linear Modelling for Complex Computer Models Fei Liu, Liang Zhang, Mike West Abstract Computer models may have functional outputs. With no loss of generality, we assume that a single computer
More informationEfficient Variational Inference in Large-Scale Bayesian Compressed Sensing
Efficient Variational Inference in Large-Scale Bayesian Compressed Sensing George Papandreou and Alan Yuille Department of Statistics University of California, Los Angeles ICCV Workshop on Information
More informationThe Hierarchical Particle Filter
and Arnaud Doucet http://go.warwick.ac.uk/amjohansen/talks MCMSki V Lenzerheide 7th January 2016 Context & Outline Filtering in State-Space Models: SIR Particle Filters [GSS93] Block-Sampling Particle
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 informationLecture 7: Optimal Smoothing
Department of Biomedical Engineering and Computational Science Aalto University March 17, 2011 Contents 1 What is Optimal Smoothing? 2 Bayesian Optimal Smoothing Equations 3 Rauch-Tung-Striebel Smoother
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 7 Approximate
More informationIntegrated Non-Factorized Variational Inference
Integrated Non-Factorized Variational Inference Shaobo Han, Xuejun Liao and Lawrence Carin Duke University February 27, 2014 S. Han et al. Integrated Non-Factorized Variational Inference February 27, 2014
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 16 Advanced topics in computational statistics 18 May 2017 Computer Intensive Methods (1) Plan of
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 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 informationMarkov Chain Monte Carlo Lecture 4
The local-trap problem refers to that in simulations of a complex system whose energy landscape is rugged, the sampler gets trapped in a local energy minimum indefinitely, rendering the simulation ineffective.
More informationWhy do we care? Measurements. Handling uncertainty over time: predicting, estimating, recognizing, learning. Dealing with time
Handling uncertainty over time: predicting, estimating, recognizing, learning Chris Atkeson 2004 Why do we care? Speech recognition makes use of dependence of words and phonemes across time. Knowing where
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 informationMachine Learning 4771
Machine Learning 4771 Instructor: ony Jebara Kalman Filtering Linear Dynamical Systems and Kalman Filtering Structure from Motion Linear Dynamical Systems Audio: x=pitch y=acoustic waveform Vision: x=object
More informationParticle Filters. Outline
Particle Filters M. Sami Fadali Professor of EE University of Nevada Outline Monte Carlo integration. Particle filter. Importance sampling. Degeneracy Resampling Example. 1 2 Monte Carlo Integration Numerical
More informationState Estimation using Moving Horizon Estimation and Particle Filtering
State Estimation using Moving Horizon Estimation and Particle Filtering James B. Rawlings Department of Chemical and Biological Engineering UW Math Probability Seminar Spring 2009 Rawlings MHE & PF 1 /
More informationMarkov Chains and MCMC
Markov Chains and MCMC CompSci 590.02 Instructor: AshwinMachanavajjhala Lecture 4 : 590.02 Spring 13 1 Recap: Monte Carlo Method If U is a universe of items, and G is a subset satisfying some property,
More informationExercises Tutorial at ICASSP 2016 Learning Nonlinear Dynamical Models Using Particle Filters
Exercises Tutorial at ICASSP 216 Learning Nonlinear Dynamical Models Using Particle Filters Andreas Svensson, Johan Dahlin and Thomas B. Schön March 18, 216 Good luck! 1 [Bootstrap particle filter for
More informationAn Brief Overview of Particle Filtering
1 An Brief Overview of Particle Filtering Adam M. Johansen a.m.johansen@warwick.ac.uk www2.warwick.ac.uk/fac/sci/statistics/staff/academic/johansen/talks/ May 11th, 2010 Warwick University Centre for Systems
More informationParticle Filtering a brief introductory tutorial. Frank Wood Gatsby, August 2007
Particle Filtering a brief introductory tutorial Frank Wood Gatsby, August 2007 Problem: Target Tracking A ballistic projectile has been launched in our direction and may or may not land near enough to
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 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 informationStochastic Gradient Descent
Stochastic Gradient Descent Machine Learning CSE546 Carlos Guestrin University of Washington October 9, 2013 1 Logistic Regression Logistic function (or Sigmoid): Learn P(Y X) directly Assume a particular
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 informationDensity 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 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 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 informationMonte Carlo Solution of Integral Equations
1 Monte Carlo Solution of Integral Equations of the second kind Adam M. Johansen a.m.johansen@warwick.ac.uk www2.warwick.ac.uk/fac/sci/statistics/staff/academic/johansen/talks/ March 7th, 2011 MIRaW Monte
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 informationWinter 2019 Math 106 Topics in Applied Mathematics. Lecture 9: Markov Chain Monte Carlo
Winter 2019 Math 106 Topics in Applied Mathematics Data-driven Uncertainty Quantification Yoonsang Lee (yoonsang.lee@dartmouth.edu) Lecture 9: Markov Chain Monte Carlo 9.1 Markov Chain A Markov Chain Monte
More information19 : Slice Sampling and HMC
10-708: Probabilistic Graphical Models 10-708, Spring 2018 19 : Slice Sampling and HMC Lecturer: Kayhan Batmanghelich Scribes: Boxiang Lyu 1 MCMC (Auxiliary Variables Methods) In inference, we are often
More informationComputer Intensive Methods in Mathematical Statistics
Computer Intensive Methods in Mathematical Statistics Department of mathematics johawes@kth.se Lecture 7 Sequential Monte Carlo methods III 7 April 2017 Computer Intensive Methods (1) Plan of today s lecture
More informationIntroduction to MCMC. DB Breakfast 09/30/2011 Guozhang Wang
Introduction to MCMC DB Breakfast 09/30/2011 Guozhang Wang Motivation: Statistical Inference Joint Distribution Sleeps Well Playground Sunny Bike Ride Pleasant dinner Productive day Posterior Estimation
More informationThe Unscented Particle Filter
The Unscented Particle Filter Rudolph van der Merwe (OGI) Nando de Freitas (UC Bereley) Arnaud Doucet (Cambridge University) Eric Wan (OGI) Outline Optimal Estimation & Filtering Optimal Recursive Bayesian
More information17 : Markov Chain Monte Carlo
10-708: Probabilistic Graphical Models, Spring 2015 17 : Markov Chain Monte Carlo Lecturer: Eric P. Xing Scribes: Heran Lin, Bin Deng, Yun Huang 1 Review of Monte Carlo Methods 1.1 Overview Monte Carlo
More informationApproximate Bayesian Computation and Particle Filters
Approximate Bayesian Computation and Particle Filters Dennis Prangle Reading University 5th February 2014 Introduction Talk is mostly a literature review A few comments on my own ongoing research See Jasra
More informationHidden Markov Models. Aarti Singh Slides courtesy: Eric Xing. Machine Learning / Nov 8, 2010
Hidden Markov Models Aarti Singh Slides courtesy: Eric Xing Machine Learning 10-701/15-781 Nov 8, 2010 i.i.d to sequential data So far we assumed independent, identically distributed data Sequential data
More informationIntroduction to Machine Learning
Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 25: Markov Chain Monte Carlo (MCMC) Course Review and Advanced Topics Many figures courtesy Kevin
More informationWhy do we care? Examples. Bayes Rule. What room am I in? Handling uncertainty over time: predicting, estimating, recognizing, learning
Handling uncertainty over time: predicting, estimating, recognizing, learning Chris Atkeson 004 Why do we care? Speech recognition makes use of dependence of words and phonemes across time. Knowing where
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 informationDensity Propagation for Continuous Temporal Chains Generative and Discriminative Models
$ Technical Report, University of Toronto, CSRG-501, October 2004 Density Propagation for Continuous Temporal Chains Generative and Discriminative Models Cristian Sminchisescu and Allan Jepson Department
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 informationExpectation propagation for signal detection in flat-fading channels
Expectation propagation for signal detection in flat-fading channels Yuan Qi MIT Media Lab Cambridge, MA, 02139 USA yuanqi@media.mit.edu Thomas Minka CMU Statistics Department Pittsburgh, PA 15213 USA
More informationParticle Filters: Convergence Results and High Dimensions
Particle Filters: Convergence Results and High Dimensions Mark Coates mark.coates@mcgill.ca McGill University Department of Electrical and Computer Engineering Montreal, Quebec, Canada Bellairs 2012 Outline
More informationSequential Monte Carlo Methods (for DSGE Models)
Sequential Monte Carlo Methods (for DSGE Models) Frank Schorfheide University of Pennsylvania, PIER, CEPR, and NBER October 23, 2017 Some References These lectures use material from our joint work: Tempered
More informationVariational Scoring of Graphical Model Structures
Variational Scoring of Graphical Model Structures Matthew J. Beal Work with Zoubin Ghahramani & Carl Rasmussen, Toronto. 15th September 2003 Overview Bayesian model selection Approximations using Variational
More information