Parallel Particle Filter in Julia
|
|
|
- Lester Rogers
- 5 years ago
- Views:
Transcription
1 Parallel Particle Filter in Julia Gustavo Goretkin December 12, / 27
2 First a disclaimer The project in a sentence. workings 2 / 27
3 First a disclaimer First a disclaimer The project in a sentence. workings This project is not finished. 3 / 27
4 The project in a sentence. First a disclaimer The project in a sentence. workings Implement a particle filter in Julia that takes advantage of distributed-memory parallelism. 4 / 27
5 Particle Filter First a disclaimer The project in a sentence. workings An approximation to the general Bayes filter 5 / 27
6 Particle Filter First a disclaimer The project in a sentence. workings An approximation to the general Bayes filter Track the state of a dynamical system 5 / 27
7 Particle Filter First a disclaimer The project in a sentence. workings An approximation to the general Bayes filter Track the state of a dynamical system but the state is not directly observable but the dynamical system is noisy 5 / 27
8 Particle Filter First a disclaimer The project in a sentence. workings An approximation to the general Bayes filter Track the state of a dynamical system but the state is not directly observable but the dynamical system is noisy Same concept as the Kalman filter, but fewer assumptions 5 / 27
9 Particle Filter First a disclaimer The project in a sentence. workings An approximation to the general Bayes filter Track the state of a dynamical system but the state is not directly observable but the dynamical system is noisy Same concept as the Kalman filter, but fewer assumptions but the system dynamics may be non-linear the observation function may be non-linear the process noise and and observation noise may be non-gaussian the hypothesis is not confined to be Gaussian can have multimodal hypotheses 5 / 27
10 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 6 / 27
11 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 2. Propagate each particle independently according to system dynamics. This is the a priori hypothesis. 6 / 27
12 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 2. Propagate each particle independently according to system dynamics. This is the a priori hypothesis. 3. Make an observation and weight each particle by the likelihood of the the particle p(observation particle) 6 / 27
13 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 2. Propagate each particle independently according to system dynamics. This is the a priori hypothesis. 3. Make an observation and weight each particle by the likelihood of the the particle p(observation particle) 4. Resample n particles according to their weights. This represents the a posteriori hypothesis at t n. 6 / 27
14 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 2. Propagate each particle independently according to system dynamics. This is the a priori hypothesis. 3. Make an observation and weight each particle by the likelihood of the the particle p(observation particle) 4. Resample n particles according to their weights. This represents the a posteriori hypothesis at t n. (a) many different resampling techniques with different computation complexities and variances. 6 / 27
15 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 2. Propagate each particle independently according to system dynamics. This is the a priori hypothesis. 3. Make an observation and weight each particle by the likelihood of the the particle p(observation particle) 4. Resample n particles according to their weights. This represents the a posteriori hypothesis at t n. (a) (b) many different resampling techniques with different computation complexities and variances. the part that is not embarrassingly parallel. 6 / 27
16 Particle Filter workings First a disclaimer The project in a sentence. workings 1. Start with a set of n particles at step t n 1. These particles represent the hypothesis at that time. 2. Propagate each particle independently according to system dynamics. This is the a priori hypothesis. 3. Make an observation and weight each particle by the likelihood of the the particle p(observation particle) 4. Resample n particles according to their weights. This represents the a posteriori hypothesis at t n. (a) (b) many different resampling techniques with different computation complexities and variances. the part that is not embarrassingly parallel. 6 / 27
17 7 / 27
18 1 1 Taken from Probabilistic Robotics / 27
19 9 / 27
20 10 / 27
21 11 / 27
22 12 / 27
23 13 / 27
24 14 / 27
25 15 / 27
26 16 / 27
27 17 / 27
28 18 / 27
29 19 / 27
30 Absolute Value observation n = 1000 Square Value observation n = 500 Square Value observation n = / 27
31 Absolute Value observation n = / 27
32 Square Value observation n = / 27
33 Square Value observation n = / 27
34 Goal of the project Current State of the Future Work 24 / 27
35 Goal of the project Implement a practical parallel particle filter for distributed memory systems Goal of the project Current State of the Future Work 25 / 27
36 Goal of the project Goal of the project Current State of the Future Work Implement a practical parallel particle filter for distributed memory systems Bayesian Filtering libraries exist 25 / 27
37 Goal of the project Goal of the project Current State of the Future Work Implement a practical parallel particle filter for distributed memory systems Bayesian Filtering libraries exist I didn t find any for distributed memory systems. 25 / 27
38 Goal of the project Goal of the project Current State of the Future Work Implement a practical parallel particle filter for distributed memory systems Bayesian Filtering libraries exist I didn t find any for distributed memory systems. Maybe for a good reason / 27
39 Goal of the project Goal of the project Current State of the Future Work Implement a practical parallel particle filter for distributed memory systems Bayesian Filtering libraries exist I didn t find any for distributed memory systems. Maybe for a good reason... Create a framework general enough so that different parameters of the particle filter can be tested plugging in different sampling techniques, noise parameters, etc. 25 / 27
40 Goal of the project Goal of the project Current State of the Future Work Implement a practical parallel particle filter for distributed memory systems Bayesian Filtering libraries exist I didn t find any for distributed memory systems. Maybe for a good reason... Create a framework general enough so that different parameters of the particle filter can be tested plugging in different sampling techniques, noise parameters, etc. benchmark computational performance and estimation performance 25 / 27
41 Goal of the project Goal of the project Current State of the Future Work Implement a practical parallel particle filter for distributed memory systems Bayesian Filtering libraries exist I didn t find any for distributed memory systems. Maybe for a good reason... Create a framework general enough so that different parameters of the particle filter can be tested plugging in different sampling techniques, noise parameters, etc. benchmark computational performance and estimation performance Provide a use case of Julia. Hopefully create an elegant implementation that might attract others to the platform especially think about Monte Carlo Bayesian inference 25 / 27
42 Current State of the Implementation still pending. I m still learning Julia. Goal of the project Current State of the Future Work 26 / 27
43 Current State of the Goal of the project Current State of the Future Work Implementation still pending. I m still learning Julia. SIMD approach. Each compute node starts with a balanced set of particles. The state propagation and observation functions happen independently on each machine. After an independent resampling step, particles will likely be unbalanced. 26 / 27
44 Current State of the Goal of the project Current State of the Future Work Implementation still pending. I m still learning Julia. SIMD approach. Each compute node starts with a balanced set of particles. The state propagation and observation functions happen independently on each machine. After an independent resampling step, particles will likely be unbalanced. Balancing between the two largest offenders 26 / 27
45 Current State of the Goal of the project Current State of the Future Work Implementation still pending. I m still learning Julia. SIMD approach. Each compute node starts with a balanced set of particles. The state propagation and observation functions happen independently on each machine. After an independent resampling step, particles will likely be unbalanced. Balancing between the two largest offenders DArray primitives implemented argmin, argmax, cumsum 26 / 27
46 Current State of the Goal of the project Current State of the Future Work Implementation still pending. I m still learning Julia. SIMD approach. Each compute node starts with a balanced set of particles. The state propagation and observation functions happen independently on each machine. After an independent resampling step, particles will likely be unbalanced. Balancing between the two largest offenders DArray primitives implemented argmin, argmax, cumsum Other primitives binary search, independent sampling, redistribution 26 / 27
47 Current State of the Goal of the project Current State of the Future Work Implementation still pending. I m still learning Julia. SIMD approach. Each compute node starts with a balanced set of particles. The state propagation and observation functions happen independently on each machine. After an independent resampling step, particles will likely be unbalanced. Balancing between the two largest offenders DArray primitives implemented argmin, argmax, cumsum Other primitives binary search, independent sampling, redistribution 26 / 27
48 Future Work Goal of the project Current State of the Future Work Consider other rebalancing approaches solving a minimum-cost graph flow problem to rebalance particles 27 / 27
49 Future Work Goal of the project Current State of the Future Work Consider other rebalancing approaches solving a minimum-cost graph flow problem to rebalance particles Think about memory allocation minimize memory allocation per iteration. 27 / 27
50 Future Work Goal of the project Current State of the Future Work Consider other rebalancing approaches solving a minimum-cost graph flow problem to rebalance particles Think about memory allocation minimize memory allocation per iteration. Think differently about parallelism. I m stuck in the MPI mentality probably not exploiting DArrays enough. 27 / 27
51 Future Work Goal of the project Current State of the Future Work Consider other rebalancing approaches solving a minimum-cost graph flow problem to rebalance particles Think about memory allocation minimize memory allocation per iteration. Think differently about parallelism. I m stuck in the MPI mentality probably not exploiting DArrays enough. 27 / 27
CIS 390 Fall 2016 Robotics: Planning and Perception Final Review Questions
CIS 390 Fall 2016 Robotics: Planning and Perception Final Review Questions December 14, 2016 Questions Throughout the following questions we will assume that x t is the state vector at time t, z t is the
Bayesian Learning (II)
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning (II) Niels Landwehr Overview Probabilities, expected values, variance Basic concepts of Bayesian learning MAP
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
Notes on Machine Learning for and
Notes on Machine Learning for 16.410 and 16.413 (Notes adapted from Tom Mitchell and Andrew Moore.) Choosing Hypotheses Generally want the most probable hypothesis given the training data Maximum a posteriori
Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization
Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization Wolfram Burgard, Cyrill Stachniss, Maren Bennewitz, Kai Arras 1 Motivation Recall: Discrete filter Discretize the
Introduction to Bayesian Learning. Machine Learning Fall 2018
Introduction to Bayesian Learning Machine Learning Fall 2018 1 What we have seen so far What does it mean to learn? Mistake-driven learning Learning by counting (and bounding) number of mistakes PAC learnability
Bayesian Networks: Construction, Inference, Learning and Causal Interpretation. Volker Tresp Summer 2016
Bayesian Networks: Construction, Inference, Learning and Causal Interpretation Volker Tresp Summer 2016 1 Introduction So far we were mostly concerned with supervised learning: we predicted one or several
Lecture : Probabilistic Machine Learning
Lecture : Probabilistic Machine Learning Riashat Islam Reasoning and Learning Lab McGill University September 11, 2018 ML : Many Methods with Many Links Modelling Views of Machine Learning Machine Learning
AUTOMOTIVE ENVIRONMENT SENSORS
AUTOMOTIVE ENVIRONMENT SENSORS Lecture 5. Localization BME KÖZLEKEDÉSMÉRNÖKI ÉS JÁRMŰMÉRNÖKI KAR 32708-2/2017/INTFIN SZÁMÚ EMMI ÁLTAL TÁMOGATOTT TANANYAG Related concepts Concepts related to vehicles moving
Bayesian Networks: Construction, Inference, Learning and Causal Interpretation. Volker Tresp Summer 2014
Bayesian Networks: Construction, Inference, Learning and Causal Interpretation Volker Tresp Summer 2014 1 Introduction So far we were mostly concerned with supervised learning: we predicted one or several
Particle 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
2D Image Processing (Extended) Kalman and particle filter
2D Image Processing (Extended) Kalman and particle filter Prof. Didier Stricker Dr. Gabriele Bleser Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz
Introduction 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
Kalman filtering and friends: Inference in time series models. Herke van Hoof slides mostly by Michael Rubinstein
Kalman filtering and friends: Inference in time series models Herke van Hoof slides mostly by Michael Rubinstein Problem overview Goal Estimate most probable state at time k using measurement up to time
Naïve Bayes classification
Naïve Bayes classification 1 Probability theory Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. Examples: A person s height, the outcome of a coin toss
PILCO: A Model-Based and Data-Efficient Approach to Policy Search
PILCO: A Model-Based and Data-Efficient Approach to Policy Search (M.P. Deisenroth and C.E. Rasmussen) CSC2541 November 4, 2016 PILCO Graphical Model PILCO Probabilistic Inference for Learning COntrol
Bayesian Regression Linear and Logistic Regression
When we want more than point estimates Bayesian Regression Linear and Logistic Regression Nicole Beckage Ordinary Least Squares Regression and Lasso Regression return only point estimates But what if we
Part 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
Probabilistic 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
Autonomous Mobile Robot Design
Autonomous Mobile Robot Design Topic: Particle Filter for Localization Dr. Kostas Alexis (CSE) These slides relied on the lectures from C. Stachniss, and the book Probabilistic Robotics from Thurn et al.
Bayesian Learning. CSL603 - Fall 2017 Narayanan C Krishnan
Bayesian Learning CSL603 - Fall 2017 Narayanan C Krishnan ckn@iitrpr.ac.in Outline Bayes Theorem MAP Learners Bayes optimal classifier Naïve Bayes classifier Example text classification Bayesian networks
Modeling and state estimation Examples State estimation Probabilities Bayes filter Particle filter. Modeling. CSC752 Autonomous Robotic Systems
Modeling CSC752 Autonomous Robotic Systems Ubbo Visser Department of Computer Science University of Miami February 21, 2017 Outline 1 Modeling and state estimation 2 Examples 3 State estimation 4 Probabilities
Bayesian Networks Inference with Probabilistic Graphical Models
4190.408 2016-Spring Bayesian Networks Inference with Probabilistic Graphical Models Byoung-Tak Zhang intelligence Lab Seoul National University 4190.408 Artificial (2016-Spring) 1 Machine Learning? Learning
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
Approximate Inference Part 1 of 2
Approximate Inference Part 1 of 2 Tom Minka Microsoft Research, Cambridge, UK Machine Learning Summer School 2009 http://mlg.eng.cam.ac.uk/mlss09/ Bayesian paradigm Consistent use of probability theory
Recent Advances in Bayesian Inference Techniques
Recent Advances in Bayesian Inference Techniques Christopher M. Bishop Microsoft Research, Cambridge, U.K. research.microsoft.com/~cmbishop SIAM Conference on Data Mining, April 2004 Abstract Bayesian
Approximate Inference Part 1 of 2
Approximate Inference Part 1 of 2 Tom Minka Microsoft Research, Cambridge, UK Machine Learning Summer School 2009 http://mlg.eng.cam.ac.uk/mlss09/ 1 Bayesian paradigm Consistent use of probability theory
Naïve Bayes classification. p ij 11/15/16. Probability theory. Probability theory. Probability theory. X P (X = x i )=1 i. Marginal Probability
Probability theory Naïve Bayes classification Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s height, the outcome of a coin toss Distinguish
Bayesian Machine Learning
Bayesian Machine Learning Andrew Gordon Wilson ORIE 6741 Lecture 4 Occam s Razor, Model Construction, and Directed Graphical Models https://people.orie.cornell.edu/andrew/orie6741 Cornell University September
Probabilistic numerics for deep learning
Presenter: Shijia Wang Department of Engineering Science, University of Oxford rning (RLSS) Summer School, Montreal 2017 Outline 1 Introduction Probabilistic Numerics 2 Components Probabilistic modeling
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen. Bayesian Learning. Tobias Scheffer, Niels Landwehr
Universität Potsdam Institut für Informatik Lehrstuhl Maschinelles Lernen Bayesian Learning Tobias Scheffer, Niels Landwehr Remember: Normal Distribution Distribution over x. Density function with parameters
Pattern 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
Approximate Inference
Approximate Inference Simulation has a name: sampling Sampling is a hot topic in machine learning, and it s really simple Basic idea: Draw N samples from a sampling distribution S Compute an approximate
Probabilistic Graphical Models
Probabilistic Graphical Models Introduction. Basic Probability and Bayes Volkan Cevher, Matthias Seeger Ecole Polytechnique Fédérale de Lausanne 26/9/2011 (EPFL) Graphical Models 26/9/2011 1 / 28 Outline
Lecture: Gaussian Process Regression. STAT 6474 Instructor: Hongxiao Zhu
Lecture: Gaussian Process Regression STAT 6474 Instructor: Hongxiao Zhu Motivation Reference: Marc Deisenroth s tutorial on Robot Learning. 2 Fast Learning for Autonomous Robots with Gaussian Processes
Results: MCMC Dancers, q=10, n=500
Motivation Sampling Methods for Bayesian Inference How to track many INTERACTING targets? A Tutorial Frank Dellaert Results: MCMC Dancers, q=10, n=500 1 Probabilistic Topological Maps Results Real-Time
Mathematical Formulation of Our Example
Mathematical Formulation of Our Example We define two binary random variables: open and, where is light on or light off. Our question is: What is? Computer Vision 1 Combining Evidence Suppose our robot
2D Image Processing. Bayes filter implementation: Kalman filter
2D Image Processing Bayes filter implementation: Kalman filter Prof. Didier Stricker Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz http://av.dfki.de
Why 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
Inference and estimation in probabilistic time series models
1 Inference and estimation in probabilistic time series models David Barber, A Taylan Cemgil and Silvia Chiappa 11 Time series The term time series refers to data that can be represented as a sequence
Mobile Robot Localization
Mobile Robot Localization 1 The Problem of Robot Localization Given a map of the environment, how can a robot determine its pose (planar coordinates + orientation)? Two sources of uncertainty: - observations
9/12/17. Types of learning. Modeling data. Supervised learning: Classification. Supervised learning: Regression. Unsupervised learning: Clustering
Types of learning Modeling data Supervised: we know input and targets Goal is to learn a model that, given input data, accurately predicts target data Unsupervised: we know the input only and want to make
Bayesian Methods in Artificial Intelligence
WDS'10 Proceedings of Contributed Papers, Part I, 25 30, 2010. ISBN 978-80-7378-139-2 MATFYZPRESS Bayesian Methods in Artificial Intelligence M. Kukačka Charles University, Faculty of Mathematics and Physics,
Bayesian Deep Learning
Bayesian Deep Learning Mohammad Emtiyaz Khan AIP (RIKEN), Tokyo http://emtiyaz.github.io emtiyaz.khan@riken.jp June 06, 2018 Mohammad Emtiyaz Khan 2018 1 What will you learn? Why is Bayesian inference
Markov chain Monte Carlo methods for visual tracking
Markov chain Monte Carlo methods for visual tracking Ray Luo rluo@cory.eecs.berkeley.edu Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720
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
CONDENSATION Conditional Density Propagation for Visual Tracking
CONDENSATION Conditional Density Propagation for Visual Tracking Michael Isard and Andrew Blake Presented by Neil Alldrin Department of Computer Science & Engineering University of California, San Diego
Statistical learning. Chapter 20, Sections 1 4 1
Statistical learning Chapter 20, Sections 1 4 Chapter 20, Sections 1 4 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete
Bayesian Learning Extension
Bayesian Learning Extension This document will go over one of the most useful forms of statistical inference known as Baye s Rule several of the concepts that extend from it. Named after Thomas Bayes this
Probabilistic Graphical Models. Guest Lecture by Narges Razavian Machine Learning Class April
Probabilistic Graphical Models Guest Lecture by Narges Razavian Machine Learning Class April 14 2017 Today What is probabilistic graphical model and why it is useful? Bayesian Networks Basic Inference
Mobile Robot Localization
Mobile Robot Localization 1 The Problem of Robot Localization Given a map of the environment, how can a robot determine its pose (planar coordinates + orientation)? Two sources of uncertainty: - observations
Lecture 9: PGM Learning
13 Oct 2014 Intro. to Stats. Machine Learning COMP SCI 4401/7401 Table of Contents I Learning parameters in MRFs 1 Learning parameters in MRFs Inference and Learning Given parameters (of potentials) and
Probabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 13: Learning in Gaussian Graphical Models, Non-Gaussian Inference, Monte Carlo Methods Some figures
Graphical Models for Collaborative Filtering
Graphical Models for Collaborative Filtering Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Sequence modeling HMM, Kalman Filter, etc.: Similarity: the same graphical model topology,
MODULE -4 BAYEIAN LEARNING
MODULE -4 BAYEIAN LEARNING CONTENT Introduction Bayes theorem Bayes theorem and concept learning Maximum likelihood and Least Squared Error Hypothesis Maximum likelihood Hypotheses for predicting probabilities
Artificial Intelligence
Artificial Intelligence Roman Barták Department of Theoretical Computer Science and Mathematical Logic Summary of last lecture We know how to do probabilistic reasoning over time transition model P(X t
Managing Uncertainty
Managing Uncertainty Bayesian Linear Regression and Kalman Filter December 4, 2017 Objectives The goal of this lab is multiple: 1. First it is a reminder of some central elementary notions of Bayesian
Linear regression example Simple linear regression: f(x) = ϕ(x)t w w ~ N(0, ) The mean and covariance are given by E[f(x)] = ϕ(x)e[w] = 0.
![Linear regression example Simple linear regression: f(x) = ϕ(x)t w w ~ N(0, ) The mean and covariance are given by E[f(x)] = ϕ(x)e[w] = 0.](/thumbs/95/123587665.jpg "Linear regression example Simple linear regression: f(x) = ϕ(x)t w w ~ N(0, ) The mean and covariance are given by E[f(x)] = ϕ(x)e[w] = 0.")
Gaussian Processes Gaussian Process Stochastic process: basically, a set of random variables. may be infinite. usually related in some way. Gaussian process: each variable has a Gaussian distribution every
The Bayes classifier
The Bayes classifier Consider where is a random vector in is a random variable (depending on ) Let be a classifier with probability of error/risk given by The Bayes classifier (denoted ) is the optimal
CSCE 478/878 Lecture 6: Bayesian Learning
Bayesian Methods Not all hypotheses are created equal (even if they are all consistent with the training data) Outline CSCE 478/878 Lecture 6: Bayesian Learning Stephen D. Scott (Adapted from Tom Mitchell
Statistical learning. Chapter 20, Sections 1 3 1
Statistical learning Chapter 20, Sections 1 3 Chapter 20, Sections 1 3 1 Outline Bayesian learning Maximum a posteriori and maximum likelihood learning Bayes net learning ML parameter learning with complete
An 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
Introduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf
1 Introduction to Machine Learning Maximum Likelihood and Bayesian Inference Lecturers: Eran Halperin, Yishay Mansour, Lior Wolf 2013-14 We know that X ~ B(n,p), but we do not know p. We get a random sample
Introduction to Machine Learning. Maximum Likelihood and Bayesian Inference. Lecturers: Eran Halperin, Lior Wolf
1 Introduction to Machine Learning Maximum Likelihood and Bayesian Inference Lecturers: Eran Halperin, Lior Wolf 2014-15 We know that X ~ B(n,p), but we do not know p. We get a random sample from X, a
CSci 8980: Advanced Topics in Graphical Models Gaussian Processes
CSci 8980: Advanced Topics in Graphical Models Gaussian Processes Instructor: Arindam Banerjee November 15, 2007 Gaussian Processes Outline Gaussian Processes Outline Parametric Bayesian Regression Gaussian
Machine Learning for OR & FE
Machine Learning for OR & FE Hidden Markov Models Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com Additional References: David
Robotics. 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
Markov Models. CS 188: Artificial Intelligence Fall Example. Mini-Forward Algorithm. Stationary Distributions.
CS 88: Artificial Intelligence Fall 27 Lecture 2: HMMs /6/27 Markov Models A Markov model is a chain-structured BN Each node is identically distributed (stationarity) Value of X at a given time is called
2D Image Processing. Bayes filter implementation: Kalman filter
2D Image Processing Bayes filter implementation: Kalman filter Prof. Didier Stricker Dr. Gabriele Bleser Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche
Kernel methods, kernel SVM and ridge regression
Kernel methods, kernel SVM and ridge regression Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012 Collaborative Filtering 2 Collaborative Filtering R: rating matrix; U: user factor;
Kernel adaptive Sequential Monte Carlo
Kernel adaptive Sequential Monte Carlo Ingmar Schuster (Paris Dauphine) Heiko Strathmann (University College London) Brooks Paige (Oxford) Dino Sejdinovic (Oxford) December 7, 2015 1 / 36 Section 1 Outline
COMP90051 Statistical Machine Learning
COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Trevor Cohn 24. Hidden Markov Models & message passing Looking back Representation of joint distributions Conditional/marginal independence
Tracking using CONDENSATION: Conditional Density Propagation
Tracking using CONDENSATION: Conditional Density Propagation Goal Model-based visual tracking in dense clutter at near video frae rates M. Isard and A. Blake, CONDENSATION Conditional density propagation
Probabilistic Classification
Bayesian Networks Probabilistic Classification Goal: Gather Labeled Training Data Build/Learn a Probability Model Use the model to infer class labels for unlabeled data points Example: Spam Filtering...
Efficient Monitoring for Planetary Rovers
International Symposium on Artificial Intelligence and Robotics in Space (isairas), May, 2003 Efficient Monitoring for Planetary Rovers Vandi Verma vandi@ri.cmu.edu Geoff Gordon ggordon@cs.cmu.edu Carnegie
The 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
Human 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
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
Symbolic Variable Elimination in Discrete and Continuous Graphical Models. Scott Sanner Ehsan Abbasnejad
Symbolic Variable Elimination in Discrete and Continuous Graphical Models Scott Sanner Ehsan Abbasnejad Inference for Dynamic Tracking No one previously did this inference exactly in closed-form! Exact
Machine Learning, Midterm Exam: Spring 2008 SOLUTIONS. Q Topic Max. Score Score. 1 Short answer questions 20.
10-601 Machine Learning, Midterm Exam: Spring 2008 Please put your name on this cover sheet If you need more room to work out your answer to a question, use the back of the page and clearly mark on the
Why 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
DEPARTMENT OF COMPUTER SCIENCE Autumn Semester MACHINE LEARNING AND ADAPTIVE INTELLIGENCE
Data Provided: None DEPARTMENT OF COMPUTER SCIENCE Autumn Semester 203 204 MACHINE LEARNING AND ADAPTIVE INTELLIGENCE 2 hours Answer THREE of the four questions. All questions carry equal weight. Figures
Bayesian Learning. Two Roles for Bayesian Methods. Bayes Theorem. Choosing Hypotheses
Bayesian Learning Two Roles for Bayesian Methods Probabilistic approach to inference. Quantities of interest are governed by prob. dist. and optimal decisions can be made by reasoning about these prob.
Linear 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
Machine Learning for Data Science (CS4786) Lecture 24
Machine Learning for Data Science (CS4786) Lecture 24 HMM Particle Filter Course Webpage : http://www.cs.cornell.edu/courses/cs4786/2017fa/ Rejection Sampling Rejection Sampling Sample variables from joint
From Bayes to Extended Kalman Filter
From Bayes to Extended Kalman Filter Michal Reinštein Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception http://cmp.felk.cvut.cz/
Machine Learning 4771
Machine Learning 4771 Instructor: Tony Jebara Topic 11 Maximum Likelihood as Bayesian Inference Maximum A Posteriori Bayesian Gaussian Estimation Why Maximum Likelihood? So far, assumed max (log) likelihood
Inference Control and Driving of Natural Systems
Inference Control and Driving of Natural Systems MSci/MSc/MRes nick.jones@imperial.ac.uk The fields at play We will be drawing on ideas from Bayesian Cognitive Science (psychology and neuroscience), Biological
Particle Filters; Simultaneous Localization and Mapping (Intelligent Autonomous Robotics) Subramanian Ramamoorthy School of Informatics
Particle Filters; Simultaneous Localization and Mapping (Intelligent Autonomous Robotics) Subramanian Ramamoorthy School of Informatics Recap: State Estimation using Kalman Filter Project state and error
STA 4273H: Sta-s-cal Machine Learning
STA 4273H: Sta-s-cal Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 2 In our
Sequential Bayesian Updating
BS2 Statistical Inference, Lectures 14 and 15, Hilary Term 2009 May 28, 2009 We consider data arriving sequentially X 1,..., X n,... and wish to update inference on an unknown parameter θ online. In a
Gaussian Process Optimization with Mutual Information
Gaussian Process Optimization with Mutual Information Emile Contal 1 Vianney Perchet 2 Nicolas Vayatis 1 1 CMLA Ecole Normale Suprieure de Cachan & CNRS, France 2 LPMA Université Paris Diderot & CNRS,
Particle lter for mobile robot tracking and localisation
Particle lter for mobile robot tracking and localisation Tinne De Laet K.U.Leuven, Dept. Werktuigkunde 19 oktober 2005 Particle lter 1 Overview Goal Introduction Particle lter Simulations Particle lter
y Xw 2 2 y Xw λ w 2 2
CS 189 Introduction to Machine Learning Spring 2018 Note 4 1 MLE and MAP for Regression (Part I) So far, we ve explored two approaches of the regression framework, Ordinary Least Squares and Ridge Regression:
Sequential 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
Nonparametric Bayesian Methods (Gaussian Processes)
[70240413 Statistical Machine Learning, Spring, 2015] Nonparametric Bayesian Methods (Gaussian Processes) Jun Zhu dcszj@mail.tsinghua.edu.cn http://bigml.cs.tsinghua.edu.cn/~jun State Key Lab of Intelligent
CPSC 340: Machine Learning and Data Mining. MLE and MAP Fall 2017
CPSC 340: Machine Learning and Data Mining MLE and MAP Fall 2017 Assignment 3: Admin 1 late day to hand in tonight, 2 late days for Wednesday. Assignment 4: Due Friday of next week. Last Time: Multi-Class
Generative Clustering, Topic Modeling, & Bayesian Inference
Generative Clustering, Topic Modeling, & Bayesian Inference INFO-4604, Applied Machine Learning University of Colorado Boulder December 12-14, 2017 Prof. Michael Paul Unsupervised Naïve Bayes Last week
Naïve Bayes Lecture 17
Naïve Bayes Lecture 17 David Sontag New York University Slides adapted from Luke Zettlemoyer, Carlos Guestrin, Dan Klein, and Mehryar Mohri Bayesian Learning Use Bayes rule! Data Likelihood Prior Posterior
Point Estimation. Vibhav Gogate The University of Texas at Dallas
Point Estimation Vibhav Gogate The University of Texas at Dallas Some slides courtesy of Carlos Guestrin, Chris Bishop, Dan Weld and Luke Zettlemoyer. Basics: Expectation and Variance Binary Variables