A latent variable model of configural conditioning
|
|
|
- Jennifer O’Neal’
- 5 years ago
- Views:
Transcription
1 A latent variable model of configural conditioning Aaron C. Courville obotics Institute, CMU Work with: Nathaniel D. Daw, UCL (Gatsby), David S. Touretzky, CMU and Geoff Gordon, CMU
2 Similarity & Discrimination in Animal Learning Similarity: How do animals respond to novel patterns of stimuli? Discrimination: How do animals learn to discriminate between overlapping patterns of stimuli? We recognize these issues as tradeoff between generalization and data-fitting 2
3 3
4 Perspectives on modeling conditioning Discriminative Bell Generative latent state Light Food Tone Bell Light Tone Food 4
5 Models of Conditioning: escorla-wagner (1972) Predicts reinforcement intensity as a linear function of stimuli, X = [A (light), B (bell),...]. V = i w i X i Learning rule is gradient descent on prediction error w i = α i β(r V )X i 5
6 Stimulus configurations Configural conditioning: discrimination and generalization between patterns of stimuli. Training: (XO) A+ B+ AB escorla-wagner: esponses (per min.) A / B AB esponse Strength A / B AB Trial Blocks Trial Blocks
7 Modeling Configurations Two dominant perspectives: 1. Added elements W, [W&, 1972] 2. Configural model [Pearce, 1994] Augment stimulus representation with configural unit. eg. XO: X=[A,B,AB]. Which units are active? 7 Observe AB [W& 1972]: All units present. X=[A=1,B=1,AB=1] [Pearce 1994]: Graded activation by generalization rule. X=[A=.5,B=.5,AB=1]
8 Expt. 1 Paired Compounds [escorla, 2003] Training Trials: AB+ CD+ Test Trials: Trained: AB, CD Transfer: AD, BC Elements: A, B, C, D esponses (per min.) Trained Transfer Element Probe Stimulus 8
9 Modeling Paired Compounds esponses (per min.) Training: AB+ CD+ Trained Transfer Element Probe Stimulus escorla-wagner: esponse Strength Pearce: esponse Strength Trained Transfer Element Probe Stimulus 9 0 Trained Transfer Element Probe Stimulus
10 Expt. 2 Asymmetric XO [edhead & Pearce, 1995] 180 Training Trials: A+ BC+ ABC- Test Trials: A BC ABC esponses (per min.) A BC ABC Trial Blocks 10
11 Modeling Asymmetric XO escorla-wagner: 1 esponses (per min.) Training: A+ BC+ ABC Trial Blocks A BC ABC 11 esponse Strength Pearce: esponse Strength A+ BC+ ABC Trial Blocks Trial Blocks A+ BC+ ABC-
12 Issues in Modeling Configural Conditioning How do we choose between the two models? Similarity: How to measure similarity between patterns of stimuli? Discrimination: How do we choose a representation that is flexible enough? A formal Bayesian approach can guide us 12
13 Perspectives on modeling conditioning Discriminative P( A,B,C,D) Generative P(,A,B,C,D) A B C D AB CD x 1 x 2 A B C D 13
14 A latent variable model Generative model: sigmoid belief network. P (S i x) = (1 + exp( w i x)) 1 x 1 x 2 Stimuli and Latent variables are binary (on = 1, off = 0) A B C D Latent variables correlate stimuli configural unit 14
15 Model Inference Learning: P(w,m D) Prediction: P( Stim,D) x 1 x 2 A B C D 15
16 Learning in the L.V. model Learning = Bayesian inference over weights & model structure conditional on training data P (w m, m D) P (D w m, m)p (w m.m) x 1 x 2 A B C D Latent variable is unknown and unwanted so we compute the marginal likelihood: P (D w m, m) = P (S t,i x, w m, m)p (x w m, m) t x i 16
17 Approximate inference Inference is analytically intractable: use reversible-jump MCMC reversible-jump mixes slowly: Exchange MCMC method to help x x x x x x x x x 17
18 L.V. model priors Prior over number of latent variables: Geometric(0.1) Prior over weight magnitudes: Laplace(2.0) num. latent variables weight Additional assumption: Stimuli are a priori rare
19 Prediction x 1 Generalization => inference over latents A B P ( A, B, m, w m ) = x 1 P ( x 1, m, w m )P (x A, B, m, w m ) P (x 1 A, B, m, w m ) P (A x 1, m, w m )P ( B x 1, m, w m )P (x 1 m, w m ) Posterior reinforcement prediction: marginalize over choice of weights and model structure. P ( Stim, m, D) = P ( Stim, m, w m, D)P (w m m, D) dw w P ( Stim, D) = m P ( Stim, m, D)P (m D) 19
20 L.V. Model of Paired Compounds x 1 x 2 MAP Model Structure: 250 Training: AB+ CD+ 1 A B C D esponses (per min.) P( Test, D ) Trained Transfer Element Probe Stimulus 0 Trained Transfer Element Test Stimulus 20
21 L.V. Model of Asymmetric XO MAP Model Structures: 4 trials: x 1 10 trials: x 1 x 2 20 trials: x 1 x 2 x 3 A B C A B C A B C esponses (per min.) Training: A+ BC+ ABC Trial Blocks A BC ABC P( A, ) P( B,C, ) P( A,B,C, ) Trial Blocks
22 What s a configuration? Can account for experiments that are traditionally deemed Configural Conditioning Previous models cast configuration as result of stimuli being trained together. We view it as the result of model complexity pressures to group stimuli. 22
23 Expt: Second-order conditioning versus Conditioned Inhibition [Yin et al, 1994] Group A+ AB- C+ Test esult Test esult No B B _ BC esp. Few B B esp. BC esp. Many B B _ BC _ 23
24 Bayesian Model of Second-Order Conditioning / Conditioned Inhibition Training: A+ AB- C+ MAP Model Structure: 4 trials: x A B C 0.6 P( B, D ) P( C, D ) P( B,C, D ) 18 trials: x 1 x A B C trials: Alternative x 1 x Number of AB- pairings 24 A B C
25 Dealing with einforcement Are reinforcers really just like other stimulus? x 1 Train: A+ B+ A B Do animals do this? 25
26 Acquired elational Equivalence [Honey & Watt, 1999] Biconditional training evaluation Test AY-food AZ-no food BY-food BZ-no food A-shock A vs C CY-no food CZ-food DY-no food DZ-food C-no shock B vs D Activity (%) Activity (%) A C B D
27 Modeling Acq. el. Equiv Activity (%) P( Stim,D) A C 0 1 A C Activity (%) P( Stim,D) B D 27 0 B D
28 Modeling Acq. el. Equiv. Biconditional training evaluation Test AY-food AZ-no food BY-food BZ-no food CY-no food CZ-food DY-no food DZ-food A-shock C-no shock A vs C B vs D x x 1 2 x x x A B C D Y Z Food 28
29 Expt: Food Devaluation [Holland, 1998] Training Trials Phase 1 Phase 2 A-F1 B-F2 B Test F1 F2 Consumption (ml) Food1 12 Food Number of B-Food 2 Trials 29
30 20 L.V. model of Devaluation A-Food1 / B-Food2, B- Consumption (ml) Food1 12 Food Number of B-Food 2 Trials 1 Few B-Food2: A B x 1 x 2 Food 1 Food Many B-Food2: x 1 x 2 x P( Food 1, D) 1-P( Food 2, D) A B Food 1 Food Number of B-Food 2 Trials 30
31 Not the whole story... (Variant on Devaluation) 60 [Holland, 1998] Training Trials Phase 1 Phase 2 A-F1 B-F2 F(1,2)- 16 Test B 160 % time in food cup Food Food Number of B-Food 2 Trials Model Structure? x 1 x 2 A B Food 1 Food 2 31
32 Future Directions Explore the priors: Experimentally manipulatable. emove independent trial assumption. 32
33 Modeling Change Should reflect our understanding of how the world is believed to change. Example: Causal model parameter drift. The marginal distribution of the diffusion process should reflect your prior 33
34 Conclusions Similarity and Discrimination are recognized as the tradeoff between compexity and data fidelity arising in Bayesians inference. A latent variable is a natural (causal) setting for the study of classical conditioning. Account for configural conditionng data and more. 34
Reinforcement learning
einforcement learning How to learn to make decisions in sequential problems (like: chess, a maze) Why is this difficult? Temporal credit assignment Prediction can help Further reading For modeling: Chapter
STA 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 3 Linear
Outline. Limits of Bayesian classification Bayesian concept learning Probabilistic models for unsupervised and semi-supervised category learning
Outline Limits of Bayesian classification Bayesian concept learning Probabilistic models for unsupervised and semi-supervised category learning Limitations Is categorization just discrimination among mutually
Bayesian Learning in Undirected Graphical Models
Bayesian Learning in Undirected Graphical Models Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London, UK http://www.gatsby.ucl.ac.uk/ and Center for Automated Learning and
Every animal is represented by a blue circle. Correlation was measured by Spearman s rank correlation coefficient (ρ).
Supplementary Figure 1 Correlations between tone and context freezing by animal in each of the four groups in experiment 1. Every animal is represented by a blue circle. Correlation was measured by Spearman
Deep Learning Srihari. Deep Belief Nets. Sargur N. Srihari
Deep Belief Nets Sargur N. Srihari srihari@cedar.buffalo.edu Topics 1. Boltzmann machines 2. Restricted Boltzmann machines 3. Deep Belief Networks 4. Deep Boltzmann machines 5. Boltzmann machines for continuous
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/
Probabilistic Graphical Models: MRFs and CRFs. CSE628: Natural Language Processing Guest Lecturer: Veselin Stoyanov
Probabilistic Graphical Models: MRFs and CRFs CSE628: Natural Language Processing Guest Lecturer: Veselin Stoyanov Why PGMs? PGMs can model joint probabilities of many events. many techniques commonly
Cheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 305 Part VII
Large-Scale Feature Learning with Spike-and-Slab Sparse Coding
Large-Scale Feature Learning with Spike-and-Slab Sparse Coding Ian J. Goodfellow, Aaron Courville, Yoshua Bengio ICML 2012 Presented by Xin Yuan January 17, 2013 1 Outline Contributions Spike-and-Slab
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
Structure learning in human causal induction
Structure learning in human causal induction Joshua B. Tenenbaum & Thomas L. Griffiths Department of Psychology Stanford University, Stanford, CA 94305 jbt,gruffydd @psych.stanford.edu Abstract We use
Augmented Rescorla-Wagner and Maximum Likelihood Estimation.
Augmented Rescorla-Wagner and Maximum Likelihood Estimation. Alan Yuille Department of Statistics University of California at Los Angeles Los Angeles, CA 90095 yuille@stat.ucla.edu In Advances in Neural
Sequential Causal Learning in Humans and Rats
Sequential Causal Learning in Humans and Rats Hongjing Lu Department of Psychology University of California, Los Angeles Los Angeles, CA 995 hongjing@ucla.edu Randall R. Rojas Department of Statistics
σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =
Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately,
Bayesian Inference in GLMs. Frequentists typically base inferences on MLEs, asymptotic confidence
Bayesian Inference in GLMs Frequentists typically base inferences on MLEs, asymptotic confidence limits, and log-likelihood ratio tests Bayesians base inferences on the posterior distribution of the unknowns
Lecture 16 Deep Neural Generative Models
Lecture 16 Deep Neural Generative Models CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago May 22, 2017 Approach so far: We have considered simple models and then constructed
Bayesian Learning in Undirected Graphical Models
Bayesian Learning in Undirected Graphical Models Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London, UK http://www.gatsby.ucl.ac.uk/ Work with: Iain Murray and Hyun-Chul
Ch 4. Linear Models for Classification
Ch 4. Linear Models for Classification Pattern Recognition and Machine Learning, C. M. Bishop, 2006. Department of Computer Science and Engineering Pohang University of Science and echnology 77 Cheongam-ro,
A graph contains a set of nodes (vertices) connected by links (edges or arcs)
BOLTZMANN MACHINES Generative Models Graphical Models A graph contains a set of nodes (vertices) connected by links (edges or arcs) In a probabilistic graphical model, each node represents a random variable,
Midterm Review CS 6375: Machine Learning. Vibhav Gogate The University of Texas at Dallas
Midterm Review CS 6375: Machine Learning Vibhav Gogate The University of Texas at Dallas Machine Learning Supervised Learning Unsupervised Learning Reinforcement Learning Parametric Y Continuous Non-parametric
LINEAR MODELS FOR CLASSIFICATION. J. Elder CSE 6390/PSYC 6225 Computational Modeling of Visual Perception
LINEAR MODELS FOR CLASSIFICATION Classification: Problem Statement 2 In regression, we are modeling the relationship between a continuous input variable x and a continuous target variable t. In classification,
Latent state estimation using control theory
Latent state estimation using control theory Bert Kappen SNN Donders Institute, Radboud University, Nijmegen Gatsby Unit, UCL London August 3, 7 with Hans Christian Ruiz Bert Kappen Smoothing problem Given
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
Chapter 20. Deep Generative Models
Peng et al.: Deep Learning and Practice 1 Chapter 20 Deep Generative Models Peng et al.: Deep Learning and Practice 2 Generative Models Models that are able to Provide an estimate of the probability distribution
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
Probabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 295-P, Spring 213 Prof. Erik Sudderth Lecture 11: Inference & Learning Overview, Gaussian Graphical Models Some figures courtesy Michael Jordan s draft
Bayesian Learning. HT2015: SC4 Statistical Data Mining and Machine Learning. Maximum Likelihood Principle. The Bayesian Learning Framework
HT5: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford http://www.stats.ox.ac.uk/~sejdinov/sdmml.html Maximum Likelihood Principle A generative model for
FREQUENTIST BEHAVIOR OF FORMAL BAYESIAN INFERENCE
FREQUENTIST BEHAVIOR OF FORMAL BAYESIAN INFERENCE Donald A. Pierce Oregon State Univ (Emeritus), RERF Hiroshima (Retired), Oregon Health Sciences Univ (Adjunct) Ruggero Bellio Univ of Udine For Perugia
Will Penny. SPM short course for M/EEG, London 2015
SPM short course for M/EEG, London 2015 Ten Simple Rules Stephan et al. Neuroimage, 2010 Model Structure The model evidence is given by integrating out the dependence on model parameters p(y m) = p(y,
Modeling Causal Generalization with Particle Filters
Modeling Causal Generalization with Particle Filters Randall R. Rojas Department of Statistics University of California, Los Angeles Los Angeles, CA 995 rrojas@stat.ucla.edu Hongjing Lu Department of Psychology
Introduction to Gaussian Process
Introduction to Gaussian Process CS 778 Chris Tensmeyer CS 478 INTRODUCTION 1 What Topic? Machine Learning Regression Bayesian ML Bayesian Regression Bayesian Non-parametric Gaussian Process (GP) GP Regression
Overview. Probabilistic Interpretation of Linear Regression Maximum Likelihood Estimation Bayesian Estimation MAP Estimation
Overview Probabilistic Interpretation of Linear Regression Maximum Likelihood Estimation Bayesian Estimation MAP Estimation Probabilistic Interpretation: Linear Regression Assume output y is generated
Bayesian Analysis. Bayesian Analysis: Bayesian methods concern one s belief about θ. [Current Belief (Posterior)] (Prior Belief) x (Data) Outline
![Bayesian Analysis. Bayesian Analysis: Bayesian methods concern one s belief about θ. [Current Belief (Posterior)] (Prior Belief) x (Data) Outline](/thumbs/87/97378301.jpg "Bayesian Analysis. Bayesian Analysis: Bayesian methods concern one s belief about θ. [Current Belief (Posterior)] (Prior Belief) x (Data) Outline")
Bayesian Analysis DuBois Bowman, Ph.D. Gordana Derado, M. S. Shuo Chen, M. S. Department of Biostatistics and Bioinformatics Center for Biomedical Imaging Statistics Emory University Outline I. Introduction
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
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
Reading Group on Deep Learning Session 4 Unsupervised Neural Networks
Reading Group on Deep Learning Session 4 Unsupervised Neural Networks Jakob Verbeek & Daan Wynen 206-09-22 Jakob Verbeek & Daan Wynen Unsupervised Neural Networks Outline Autoencoders Restricted) Boltzmann
Variational Inference via Stochastic Backpropagation
Variational Inference via Stochastic Backpropagation Kai Fan February 27, 2016 Preliminaries Stochastic Backpropagation Variational Auto-Encoding Related Work Summary Outline Preliminaries Stochastic Backpropagation
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
Unsupervised Learning
CS 3750 Advanced Machine Learning hkc6@pitt.edu Unsupervised Learning Data: Just data, no labels Goal: Learn some underlying hidden structure of the data P(, ) P( ) Principle Component Analysis (Dimensionality
Gaussian Process Approximations of Stochastic Differential Equations
Gaussian Process Approximations of Stochastic Differential Equations Cédric Archambeau Centre for Computational Statistics and Machine Learning University College London c.archambeau@cs.ucl.ac.uk CSML
Gaussian Processes in Machine Learning
Gaussian Processes in Machine Learning November 17, 2011 CharmGil Hong Agenda Motivation GP : How does it make sense? Prior : Defining a GP More about Mean and Covariance Functions Posterior : Conditioning
Variational Bayesian Logistic Regression
Variational Bayesian Logistic Regression Sargur N. University at Buffalo, State University of New York USA Topics in Linear Models for Classification Overview 1. Discriminant Functions 2. Probabilistic
Bayesian Quadrature: Model-based Approximate Integration. David Duvenaud University of Cambridge
Bayesian Quadrature: Model-based Approimate Integration David Duvenaud University of Cambridge The Quadrature Problem ˆ We want to estimate an integral Z = f ()p()d ˆ Most computational problems in inference
Logistic Regression. Machine Learning Fall 2018
Logistic Regression Machine Learning Fall 2018 1 Where are e? We have seen the folloing ideas Linear models Learning as loss minimization Bayesian learning criteria (MAP and MLE estimation) The Naïve Bayes
Related Concepts: Lecture 9 SEM, Statistical Modeling, AI, and Data Mining. I. Terminology of SEM
Lecture 9 SEM, Statistical Modeling, AI, and Data Mining I. Terminology of SEM Related Concepts: Causal Modeling Path Analysis Structural Equation Modeling Latent variables (Factors measurable, but thru
The Laplace Approximation
The Laplace Approximation Sargur N. University at Buffalo, State University of New York USA Topics in Linear Models for Classification Overview 1. Discriminant Functions 2. Probabilistic Generative Models
Interpretable Latent Variable Models
Interpretable Latent Variable Models Fernando Perez-Cruz Bell Labs (Nokia) Department of Signal Theory and Communications, University Carlos III in Madrid 1 / 24 Outline 1 Introduction to Machine Learning
Deep unsupervised learning
Deep unsupervised learning Advanced data-mining Yongdai Kim Department of Statistics, Seoul National University, South Korea Unsupervised learning In machine learning, there are 3 kinds of learning paradigm.
Introduction to Machine Learning. Lecture 2
Introduction to Machine Learning Lecturer: Eran Halperin Lecture 2 Fall Semester Scribe: Yishay Mansour Some of the material was not presented in class (and is marked with a side line) and is given for
Dynamic Causal Models
Dynamic Causal Models V1 SPC Will Penny V1 SPC V5 V5 Olivier David, Karl Friston, Lee Harrison, Andrea Mechelli, Klaas Stephan Wellcome Department of Imaging Neuroscience, ION, UCL, UK. Mathematics in
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
A Bayesian Approach to Phylogenetics
A Bayesian Approach to Phylogenetics Niklas Wahlberg Based largely on slides by Paul Lewis (www.eeb.uconn.edu) An Introduction to Bayesian Phylogenetics Bayesian inference in general Markov chain Monte
Linear Classification
Linear Classification Lili MOU moull12@sei.pku.edu.cn http://sei.pku.edu.cn/ moull12 23 April 2015 Outline Introduction Discriminant Functions Probabilistic Generative Models Probabilistic Discriminative
Logistic Regression & Neural Networks
Logistic Regression & Neural Networks CMSC 723 / LING 723 / INST 725 Marine Carpuat Slides credit: Graham Neubig, Jacob Eisenstein Logistic Regression Perceptron & Probabilities What if we want a probability
The Diffusion Model of Speeded Choice, from a Rational Perspective
The Diffusion Model of Speeded Choice, from a Rational Perspective Matt Jones, University of Colorado August 7, 017 1 Binary Decision Tasks This chapter considers tasks in which an experimental subject
Multivariate Bayesian Linear Regression MLAI Lecture 11
Multivariate Bayesian Linear Regression MLAI Lecture 11 Neil D. Lawrence Department of Computer Science Sheffield University 21st October 2012 Outline Univariate Bayesian Linear Regression Multivariate
Bayesian Models in Machine Learning
Bayesian Models in Machine Learning Lukáš Burget Escuela de Ciencias Informáticas 2017 Buenos Aires, July 24-29 2017 Frequentist vs. Bayesian Frequentist point of view: Probability is the frequency of
Bayesian Concept Learning
Learning from positive and negative examples Bayesian Concept Learning Chen Yu Indiana University With both positive and negative examples, it is easy to define a boundary to separate these two. Just with
Parametric Models. Dr. Shuang LIANG. School of Software Engineering TongJi University Fall, 2012
Parametric Models Dr. Shuang LIANG School of Software Engineering TongJi University Fall, 2012 Today s Topics Maximum Likelihood Estimation Bayesian Density Estimation Today s Topics Maximum Likelihood
Learning Parameters of Undirected Models. Sargur Srihari
Learning Parameters of Undirected Models Sargur srihari@cedar.buffalo.edu 1 Topics Difficulties due to Global Normalization Likelihood Function Maximum Likelihood Parameter Estimation Simple and Conjugate
Introduction to Machine Learning
Introduction to Machine Learning Logistic Regression Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574
Midterm Review CS 7301: Advanced Machine Learning. Vibhav Gogate The University of Texas at Dallas
Midterm Review CS 7301: Advanced Machine Learning Vibhav Gogate The University of Texas at Dallas Supervised Learning Issues in supervised learning What makes learning hard Point Estimation: MLE vs Bayesian
Using Graphs to Describe Model Structure. Sargur N. Srihari
Using Graphs to Describe Model Structure Sargur N. srihari@cedar.buffalo.edu 1 Topics in Structured PGMs for Deep Learning 0. Overview 1. Challenge of Unstructured Modeling 2. Using graphs to describe
CSC321 Lecture 18: Learning Probabilistic Models
CSC321 Lecture 18: Learning Probabilistic Models Roger Grosse Roger Grosse CSC321 Lecture 18: Learning Probabilistic Models 1 / 25 Overview So far in this course: mainly supervised learning Language modeling
Modeling human function learning with Gaussian processes
Modeling human function learning with Gaussian processes Thomas L. Griffiths Christopher G. Lucas Joseph J. Williams Department of Psychology University of California, Berkeley Berkeley, CA 94720-1650
Introduction to Systems Analysis and Decision Making Prepared by: Jakub Tomczak
Introduction to Systems Analysis and Decision Making Prepared by: Jakub Tomczak 1 Introduction. Random variables During the course we are interested in reasoning about considered phenomenon. In other words,
Learning Gaussian Process Models from Uncertain Data
Learning Gaussian Process Models from Uncertain Data Patrick Dallaire, Camille Besse, and Brahim Chaib-draa DAMAS Laboratory, Computer Science & Software Engineering Department, Laval University, Canada
Machine Learning - Waseda University Logistic Regression
Machine Learning - Waseda University Logistic Regression AD June AD ) June / 9 Introduction Assume you are given some training data { x i, y i } i= where xi R d and y i can take C different values. Given
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
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
p L yi z n m x N n xi
y i z n x n N x i Overview Directed and undirected graphs Conditional independence Exact inference Latent variables and EM Variational inference Books statistical perspective Graphical Models, S. Lauritzen
Semi-rational Models of Conditioning: The Case of Trial Order
Semi-rational Models of Conditioning: The Case of Trial Order Nathaniel D. Daw, Aaron C. Courville, and Peter Dayan June 16, 27 1 Introduction Bayesian treatments of animal conditioning start from a generative
Lecture 3: Pattern Classification
EE E6820: Speech & Audio Processing & Recognition Lecture 3: Pattern Classification 1 2 3 4 5 The problem of classification Linear and nonlinear classifiers Probabilistic classification Gaussians, mixtures
Online Bayesian Passive-Agressive Learning
Online Bayesian Passive-Agressive Learning International Conference on Machine Learning, 2014 Tianlin Shi Jun Zhu Tsinghua University, China 21 August 2015 Presented by: Kyle Ulrich Introduction Online
Outline Lecture 2 2(32)
Outline Lecture (3), Lecture Linear Regression and Classification it is our firm belief that an understanding of linear models is essential for understanding nonlinear ones Thomas Schön Division of Automatic
COMP90051 Statistical Machine Learning
COMP90051 Statistical Machine Learning Semester 2, 2017 Lecturer: Trevor Cohn 2. Statistical Schools Adapted from slides by Ben Rubinstein Statistical Schools of Thought Remainder of lecture is to provide
CS540 Machine learning L9 Bayesian statistics
CS540 Machine learning L9 Bayesian statistics 1 Last time Naïve Bayes Beta-Bernoulli 2 Outline Bayesian concept learning Beta-Bernoulli model (review) Dirichlet-multinomial model Credible intervals 3 Bayesian
Parameter learning in CRF s
Parameter learning in CRF s June 01, 2009 Structured output learning We ish to learn a discriminant (or compatability) function: F : X Y R (1) here X is the space of inputs and Y is the space of outputs.
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
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
Posterior Regularization
Posterior Regularization 1 Introduction One of the key challenges in probabilistic structured learning, is the intractability of the posterior distribution, for fast inference. There are numerous methods
STA 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
Marginal versus conditional effects: does it make a difference? Mireille Schnitzer, PhD Université de Montréal
Marginal versus conditional effects: does it make a difference? Mireille Schnitzer, PhD Université de Montréal Overview In observational and experimental studies, the goal may be to estimate the effect
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
The Origin of Deep Learning. Lili Mou Jan, 2015
The Origin of Deep Learning Lili Mou Jan, 2015 Acknowledgment Most of the materials come from G. E. Hinton s online course. Outline Introduction Preliminary Boltzmann Machines and RBMs Deep Belief Nets
Chapter 16. Structured Probabilistic Models for Deep Learning
Peng et al.: Deep Learning and Practice 1 Chapter 16 Structured Probabilistic Models for Deep Learning Peng et al.: Deep Learning and Practice 2 Structured Probabilistic Models way of using graphs to describe
Bayesian model selection: methodology, computation and applications
Bayesian model selection: methodology, computation and applications David Nott Department of Statistics and Applied Probability National University of Singapore Statistical Genomics Summer School Program
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
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
Introduction to Gaussian Processes
Introduction to Gaussian Processes Iain Murray murray@cs.toronto.edu CSC255, Introduction to Machine Learning, Fall 28 Dept. Computer Science, University of Toronto The problem Learn scalar function of
Unsupervised Learning
Unsupervised Learning Bayesian Model Comparison Zoubin Ghahramani zoubin@gatsby.ucl.ac.uk Gatsby Computational Neuroscience Unit, and MSc in Intelligent Systems, Dept Computer Science University College
GAUSSIAN PROCESS REGRESSION
GAUSSIAN PROCESS REGRESSION CSE 515T Spring 2015 1. BACKGROUND The kernel trick again... The Kernel Trick Consider again the linear regression model: y(x) = φ(x) w + ε, with prior p(w) = N (w; 0, Σ). The
Collapsed Variational Inference for Sum-Product Networks
for Sum-Product Networks Han Zhao 1, Tameem Adel 2, Geoff Gordon 1, Brandon Amos 1 Presented by: Han Zhao Carnegie Mellon University 1, University of Amsterdam 2 June. 20th, 2016 1 / 26 Outline Background
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
Study Notes on the Latent Dirichlet Allocation
Study Notes on the Latent Dirichlet Allocation Xugang Ye 1. Model Framework A word is an element of dictionary {1,,}. A document is represented by a sequence of words: =(,, ), {1,,}. A corpus is a collection
Chapter 3: Maximum-Likelihood & Bayesian Parameter Estimation (part 1)
HW 1 due today Parameter Estimation Biometrics CSE 190 Lecture 7 Today s lecture was on the blackboard. These slides are an alternative presentation of the material. CSE190, Winter10 CSE190, Winter10 Chapter
Evaluating the Variance of
Evaluating the Variance of Likelihood-Ratio Gradient Estimators Seiya Tokui 2 Issei Sato 2 3 Preferred Networks 2 The University of Tokyo 3 RIKEN ICML 27 @ Sydney Task: Gradient estimation for stochastic
Mathematics I. Exercises with solutions. 1 Linear Algebra. Vectors and Matrices Let , C = , B = A = Determine the following matrices:
Mathematics I Exercises with solutions Linear Algebra Vectors and Matrices.. Let A = 5, B = Determine the following matrices: 4 5, C = a) A + B; b) A B; c) AB; d) BA; e) (AB)C; f) A(BC) Solution: 4 5 a)
Machine Learning Basics Lecture 7: Multiclass Classification. Princeton University COS 495 Instructor: Yingyu Liang
Machine Learning Basics Lecture 7: Multiclass Classification Princeton University COS 495 Instructor: Yingyu Liang Example: image classification indoor Indoor outdoor Example: image classification (multiclass)