Notes on Noise Contrastive Estimation (NCE)

Size: px
Start display at page:

Download "Notes on Noise Contrastive Estimation (NCE)"

Transcription

1 Notes on Noise Contrastive Estimation NCE) David Meyer March 0, 207 Introduction In this note we follow the notation used in [2]. Suppose X x, x 2,, x Td ) is a sample of a random vector x R n, where each x i is drawn unknown probability density function p d. Now, one way to describe the properties of the observed data X is to describe its properties relative to the properties of some reference data Y. The basic idea behind Noise Contrastive Estimation NCE) is to draw the reference noise) sample Y y, y 2,, y Tn ) from a known pdf p n and then estimate the ratio p d /p n, which will in turn give us an estimate of p d. 2 Definitions Let U X Y {u, u 2,, u Td +T n }, where is the number of data samples and T n is the number of samples from a noise distribution. We associate with each datapoint u t a class label C t such that C t { if ut X 0 if u t Y Since p d is unknown, we model p. C ) p m.; θ). Note that we re making an assumption here that there exists a θ such that p d.) p m.; θ ). That is, we assume that the empirical distribution p d.) is a member of the parameterized family p m.; θ). Given these definitions, the likelihoods are Some authors characterize the problem as drawing sample from the empirical distribution ) and k samples from the reference noise) distribution T n k), where the total number of samples per turn is k +. See e.g.,

2 pu C ) p m u; θ) # data ) pu C 0) p n u) # noise 2) The priors are The probability of any given u, P u), is thus P C ) + T n 3) P C 0) T n + T n 4) Remembering Bayes Rule: P u) P C ) pu C )) + P C 0) pu C 0)) 5) T ) d T ) n p m u; θ) + p n u) 6) + T n + T n P Θ X ) P X Θ)P Θ) P X ) # posterior likelihood prior evidence we get the posterior probabilities P u C ; θ) P C ) P C u; θ) P u) p m u; θ) ) Td +T n p m u; θ) Td +T n p m u; θ) p m u; θ) p m u; θ) + Tn )p n u) p m u; θ) P C u; θ) p m u; θ) + vp n u) vp n u) P C 0 u; θ) p m u; θ) + vp n u) +T n + Tn +T n p n u) p m u; θ) ) + Tn +T n p n u) 7) ) 8) )) # v T n P C 0) P C ) +T n 9) 0) ) # same analysis 2) 2

3 Now we have p m u; θ) P C u; θ) 3) p m u; θ) + vp n u) p m u; θ) p mu;θ) 4) p m u; θ) + vp n u)) p mu;θ) + v pnu) 5) p mu;θ) + v p ) nu) 6) p m u; θ) This is the first time we see an estimate of ratio we are looking for, namely define Gu; θ) Gu; θ) ln p mu; θ) p n u) pnu) p mu;θ). Now, 7) ln p m u; θ) ln p n u) 8) Gu; θ) is called the log-ratio between p m u; θ) and p n u). If we let P C u; θ) hu; θ) then hu; θ) can be written as hu; θ) r v Gu; θ)) 9) where r v u) + v exp u) that is, r v.) is the sigmoid/logistic function parameterized by v. 20) Note that the class labels C t are assumed to be Bernoulli and iid. The conditional loglikelihood is given by lθ) +T n C t ln P C t u t ; θ) + C t ) ln P C 0 u t ; θ) 2) ln [ hx t ; θ) ] + T n ln [ hy t ; θ) ] 22) Of the many interesting things here: optimizing lθ) with respect to θ leads to an estimate G.; ˆθ) of the log-ratio lnp d /p n ). Said another way, an approximate description of X 3

4 relative to Y can be obtained by optimizing Equation 22. cross-entropy function. Note also that lθ) is the This result shows that density estimation, an unsupervised learning task, can be performed by logistic regression which is supervised learning). 3 The NCE Estimator Recall that the normalized pdf p m.; ˆθ) satisfies the following constraints: p m u; θ)du # normalization constraint 23) p m.; θ) 0 # positivity constraint 24) Note that if the constraints for the model p m.; θ) hold for all θ and not just ˆθ) then the model is said to be normalized and Maximum Likelihood can be used to estimate θ. If only the positivity constraint and not the normalization constraint) is satisfied then we say the model is unnormalized. The main assumption here is that there exists at least one set of parameters, θ, for which the unnormalized model integrations to one 2. That is, p d.) p m.; θ ). The unnormalized model is usually denoted by p 0 m.; α). 3 Now, define the partition function Zα) as Zα) p 0 mu; α)du 25) Zα) can be used to convert an unnormalized model p 0 m.; α) into a normalized one: p 0 m.; α)/zα) which integrates to one for every value of α. Examples of distributions that are typically specified by unnormalized models include Gibbs distributions and Markov networks. The problem, however, is that the mapping α Zα) is defined by an integral Equation 25), so unless p 0 m.; α) has some convenient form, the integral usually cannot be computed analytically and thus Zα) is not available in closed form). For low-dimensional problems numerical integration can provide good accuracy, but for high-dimensional problems computation of Zα) becomes intractable cf the curse of dimensionality). 2 Another way to think about this: as mentioned above, p d.) comes from the family of parameterized distributions, p mu; θ). 3 p 0 m.; α), the unnormalized model, is sometimes written φξ; θ) []; the usual lack of standardized notation in machine learning... 4

5 It is worth noting that unnormalized models are widely used, with examples including models of images Markov Random Fields), models of text neural probabilistic language models), various models in physics Ising models), and more. The advantages of using unnormalized models in that they are often easier to specify than normalized models. The disadvantage is that the likelihood function is generally intractable. Noise Contrastive Estimation NCE) is an estimation method for unnormalized models. The basic idea here is to consider Z, or equivalently c ln /Z as a parameter to the model rather than a partition function). In particular, NCE considers p 0 m.; α) to include a new normalizing parameter c and estimates ln p m ;.; θ) ln p 0 m.; α) + c 26) where the parameter θ α, c). In particular, the estimate ˆθ ˆα, ĉ) then causes the unnormalized model p 0 m.; α)) to match the shape of p d and ĉ provides the appropriate scaling so that the normalization constraint Equation 24) holds. Observe that after training, ĉ provides an estimate for ln /Zα). Of course, if the model is normalized in the first place c is unnecessary. Given these definitions, we can define the estimator ˆθ T as the value of θ which maximizes { J T θ) Td ln [ hx t ; θ) ] + T n ln [ hy t ; θ) ]} 27) where the nonlinearity h.; θ) is defined in Equation 9. Note that the objection function J T is the log-likelihood Equation 22) divided by. J T can be rewritten as J T θ) ln [ hx t ; θ) ] + v T n T n ln [ hy t ; θ) ] 28) Interestingly, note that h.; θ) 0, ) and h.; θ) { 0 lim G.; θ) lim G.; θ) 5

6 As a result, the optimal paratement ˆθ T makes Gu T ; ˆθ T ) as large as possible for u T X and as small as possible for u T Y. Said another way, we determine the parameters θ such that P C ; u; θ) is large for most x i and small for most y i. Amazingly, this means that in this case logistic regression has learned to discriminate classify) between the data and noise sets. That is, what we end up with is unsupervised learning by supervised learning, where successful classification is equivalent to learning the differences between the data and the noise. Here we used nonlinear logistic regression for classification, but other classifiers are possible. 6

7 References [] Noise-Contrastive Estimation and its Generalizations. [2] Aapo Hyvarinen Michael U. Gutmann. Noise-contrastive estimation of unnormalized statistical models, with applications to natural image statistics. Journal of Machine Learning Research, 2:307 36,

CSC321 Lecture 18: Learning Probabilistic Models

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

More information

Support Vector Machines

Support Vector Machines Support Vector Machines Le Song Machine Learning I CSE 6740, Fall 2013 Naïve Bayes classifier Still use Bayes decision rule for classification P y x = P x y P y P x But assume p x y = 1 is fully factorized

More information

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 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

More information

Estimating Unnormalized models. Without Numerical Integration

Estimating Unnormalized models. Without Numerical Integration Estimating Unnormalized Models Without Numerical Integration Dept of Computer Science University of Helsinki, Finland with Michael Gutmann Problem: Estimation of unnormalized models We want to estimate

More information

Qualifying Exam in Machine Learning

Qualifying Exam in Machine Learning Qualifying Exam in Machine Learning October 20, 2009 Instructions: Answer two out of the three questions in Part 1. In addition, answer two out of three questions in two additional parts (choose two parts

More information

Gatsby Theoretical Neuroscience Lectures: Non-Gaussian statistics and natural images Parts III-IV

Gatsby Theoretical Neuroscience Lectures: Non-Gaussian statistics and natural images Parts III-IV Gatsby Theoretical Neuroscience Lectures: Non-Gaussian statistics and natural images Parts III-IV Aapo Hyvärinen Gatsby Unit University College London Part III: Estimation of unnormalized models Often,

More information

Logistic Regression. Machine Learning Fall 2018

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

More information

Parametric Techniques

Parametric Techniques Parametric Techniques Jason J. Corso SUNY at Buffalo J. Corso (SUNY at Buffalo) Parametric Techniques 1 / 39 Introduction When covering Bayesian Decision Theory, we assumed the full probabilistic structure

More information

Noise-contrastive estimation of unnormalized statistical models, and its application to natural image statistics

Noise-contrastive estimation of unnormalized statistical models, and its application to natural image statistics 9/2/2 Noise-contrastive estimation of unnormalized statistical models, and its application to natural image statistics Michael U. Gutmann Aapo Hyvärinen Department of Computer Science Department of Mathematics

More information

Naïve Bayes classification

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

More information

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

Machine Learning. Lecture 4: Regularization and Bayesian Statistics. Feng Li. https://funglee.github.io Machine Learning Lecture 4: Regularization and Bayesian Statistics Feng Li fli@sdu.edu.cn https://funglee.github.io School of Computer Science and Technology Shandong University Fall 207 Overfitting Problem

More information

Learning features by contrasting natural images with noise

Learning features by contrasting natural images with noise Learning features by contrasting natural images with noise Michael Gutmann 1 and Aapo Hyvärinen 12 1 Dept. of Computer Science and HIIT, University of Helsinki, P.O. Box 68, FIN-00014 University of Helsinki,

More information

Parametric Techniques Lecture 3

Parametric Techniques Lecture 3 Parametric Techniques Lecture 3 Jason Corso SUNY at Buffalo 22 January 2009 J. Corso (SUNY at Buffalo) Parametric Techniques Lecture 3 22 January 2009 1 / 39 Introduction In Lecture 2, we learned how to

More information

PATTERN RECOGNITION AND MACHINE LEARNING

PATTERN RECOGNITION AND MACHINE LEARNING PATTERN RECOGNITION AND MACHINE LEARNING Chapter 1. Introduction Shuai Huang April 21, 2014 Outline 1 What is Machine Learning? 2 Curve Fitting 3 Probability Theory 4 Model Selection 5 The curse of dimensionality

More information

Lecture 2 Machine Learning Review

Lecture 2 Machine Learning Review Lecture 2 Machine Learning Review CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago March 29, 2017 Things we will look at today Formal Setup for Supervised Learning Things

More information

Statistical learning. Chapter 20, Sections 1 4 1

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

More information

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 Machine Learning Basics Lecture 7: Multiclass Classification Princeton University COS 495 Instructor: Yingyu Liang Example: image classification indoor Indoor outdoor Example: image classification (multiclass)

More information

Probabilistic Graphical Models

Probabilistic Graphical Models Probabilistic Graphical Models David Sontag New York University Lecture 4, February 16, 2012 David Sontag (NYU) Graphical Models Lecture 4, February 16, 2012 1 / 27 Undirected graphical models Reminder

More information

ECE521 week 3: 23/26 January 2017

ECE521 week 3: 23/26 January 2017 ECE521 week 3: 23/26 January 2017 Outline Probabilistic interpretation of linear regression - Maximum likelihood estimation (MLE) - Maximum a posteriori (MAP) estimation Bias-variance trade-off Linear

More information

CSCI-567: Machine Learning (Spring 2019)

CSCI-567: Machine Learning (Spring 2019) CSCI-567: Machine Learning (Spring 2019) Prof. Victor Adamchik U of Southern California Mar. 19, 2019 March 19, 2019 1 / 43 Administration March 19, 2019 2 / 43 Administration TA3 is due this week March

More information

Naïve Bayes classification. p ij 11/15/16. Probability theory. Probability theory. Probability theory. X P (X = x i )=1 i. Marginal Probability

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

More information

Machine Learning for natural language processing

Machine Learning for natural language processing Machine Learning for natural language processing Classification: Maximum Entropy Models Laura Kallmeyer Heinrich-Heine-Universität Düsseldorf Summer 2016 1 / 24 Introduction Classification = supervised

More information

Chapter 16. Structured Probabilistic Models for Deep Learning

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

More information

Bayesian Learning (II)

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

More information

Generative Learning. INFO-4604, Applied Machine Learning University of Colorado Boulder. November 29, 2018 Prof. Michael Paul

Generative Learning. INFO-4604, Applied Machine Learning University of Colorado Boulder. November 29, 2018 Prof. Michael Paul Generative Learning INFO-4604, Applied Machine Learning University of Colorado Boulder November 29, 2018 Prof. Michael Paul Generative vs Discriminative The classification algorithms we have seen so far

More information

Introduction to Bayesian Learning. Machine Learning Fall 2018

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

More information

A graph contains a set of nodes (vertices) connected by links (edges or arcs)

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,

More information

Maximum Likelihood, Logistic Regression, and Stochastic Gradient Training

Maximum Likelihood, Logistic Regression, and Stochastic Gradient Training Maximum Likelihood, Logistic Regression, and Stochastic Gradient Training Charles Elkan elkan@cs.ucsd.edu January 17, 2013 1 Principle of maximum likelihood Consider a family of probability distributions

More information

Logistic Regression. Will Monroe CS 109. Lecture Notes #22 August 14, 2017

Logistic Regression. Will Monroe CS 109. Lecture Notes #22 August 14, 2017 1 Will Monroe CS 109 Logistic Regression Lecture Notes #22 August 14, 2017 Based on a chapter by Chris Piech Logistic regression is a classification algorithm1 that works by trying to learn a function

More information

Week 5: Logistic Regression & Neural Networks

Week 5: Logistic Regression & Neural Networks Week 5: Logistic Regression & Neural Networks Instructor: Sergey Levine 1 Summary: Logistic Regression In the previous lecture, we covered logistic regression. To recap, logistic regression models and

More information

Lecture 16 Deep Neural Generative Models

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

More information

Machine Learning Practice Page 2 of 2 10/28/13

Machine Learning Practice Page 2 of 2 10/28/13 Machine Learning 10-701 Practice Page 2 of 2 10/28/13 1. True or False Please give an explanation for your answer, this is worth 1 pt/question. (a) (2 points) No classifier can do better than a naive Bayes

More information

Homework 1 Solutions Probability, Maximum Likelihood Estimation (MLE), Bayes Rule, knn

Homework 1 Solutions Probability, Maximum Likelihood Estimation (MLE), Bayes Rule, knn Homework 1 Solutions Probability, Maximum Likelihood Estimation (MLE), Bayes Rule, knn CMU 10-701: Machine Learning (Fall 2016) https://piazza.com/class/is95mzbrvpn63d OUT: September 13th DUE: September

More information

Relationship between Least Squares Approximation and Maximum Likelihood Hypotheses

Relationship between Least Squares Approximation and Maximum Likelihood Hypotheses Relationship between Least Squares Approximation and Maximum Likelihood Hypotheses Steven Bergner, Chris Demwell Lecture notes for Cmpt 882 Machine Learning February 19, 2004 Abstract In these notes, a

More information

Probabilistic Machine Learning. Industrial AI Lab.

Probabilistic Machine Learning. Industrial AI Lab. Probabilistic Machine Learning Industrial AI Lab. Probabilistic Linear Regression Outline Probabilistic Classification Probabilistic Clustering Probabilistic Dimension Reduction 2 Probabilistic Linear

More information

Notes on Machine Learning for and

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

More information

Ch 4. Linear Models for Classification

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,

More information

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE GRAVIS 2016 BASEL. Logistic Regression. Pattern Recognition 2016 Sandro Schönborn University of Basel

> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE GRAVIS 2016 BASEL. Logistic Regression. Pattern Recognition 2016 Sandro Schönborn University of Basel Logistic Regression Pattern Recognition 2016 Sandro Schönborn University of Basel Two Worlds: Probabilistic & Algorithmic We have seen two conceptual approaches to classification: data class density estimation

More information

Linear Models for Classification

Linear Models for Classification Linear Models for Classification Oliver Schulte - CMPT 726 Bishop PRML Ch. 4 Classification: Hand-written Digit Recognition CHINE INTELLIGENCE, VOL. 24, NO. 24, APRIL 2002 x i = t i = (0, 0, 0, 1, 0, 0,

More information

σ(a) = a N (x; 0, 1 2 ) dx. σ(a) = Φ(a) =

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,

More information

CSC321 Lecture 7 Neural language models

CSC321 Lecture 7 Neural language models CSC321 Lecture 7 Neural language models Roger Grosse and Nitish Srivastava February 1, 2015 Roger Grosse and Nitish Srivastava CSC321 Lecture 7 Neural language models February 1, 2015 1 / 19 Overview We

More information

Machine Learning, Fall 2009: Midterm

Machine Learning, Fall 2009: Midterm 10-601 Machine Learning, Fall 009: Midterm Monday, November nd hours 1. Personal info: Name: Andrew account: E-mail address:. You are permitted two pages of notes and a calculator. Please turn off all

More information

EEL 851: Biometrics. An Overview of Statistical Pattern Recognition EEL 851 1

EEL 851: Biometrics. An Overview of Statistical Pattern Recognition EEL 851 1 EEL 851: Biometrics An Overview of Statistical Pattern Recognition EEL 851 1 Outline Introduction Pattern Feature Noise Example Problem Analysis Segmentation Feature Extraction Classification Design Cycle

More information

Bayesian Inference by Density Ratio Estimation

Bayesian Inference by Density Ratio Estimation Bayesian Inference by Density Ratio Estimation Michael Gutmann https://sites.google.com/site/michaelgutmann Institute for Adaptive and Neural Computation School of Informatics, University of Edinburgh

More information

Undirected Graphical Models

Undirected Graphical Models Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Properties Properties 3 Generative vs. Conditional

More information

Density Estimation. Seungjin Choi

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

More information

Introduction: MLE, MAP, Bayesian reasoning (28/8/13)

Introduction: MLE, MAP, Bayesian reasoning (28/8/13) STA561: Probabilistic machine learning Introduction: MLE, MAP, Bayesian reasoning (28/8/13) Lecturer: Barbara Engelhardt Scribes: K. Ulrich, J. Subramanian, N. Raval, J. O Hollaren 1 Classifiers In this

More information

University of Cambridge Engineering Part IIB Module 4F10: Statistical Pattern Processing Handout 2: Multivariate Gaussians

University of Cambridge Engineering Part IIB Module 4F10: Statistical Pattern Processing Handout 2: Multivariate Gaussians Engineering Part IIB: Module F Statistical Pattern Processing University of Cambridge Engineering Part IIB Module F: Statistical Pattern Processing Handout : Multivariate Gaussians. Generative Model Decision

More information

Machine Learning, Fall 2012 Homework 2

Machine Learning, Fall 2012 Homework 2 0-60 Machine Learning, Fall 202 Homework 2 Instructors: Tom Mitchell, Ziv Bar-Joseph TA in charge: Selen Uguroglu email: sugurogl@cs.cmu.edu SOLUTIONS Naive Bayes, 20 points Problem. Basic concepts, 0

More information

Machine Learning 4771

Machine Learning 4771 Machine Learning 4771 Instructor: Tony Jebara Topic 7 Unsupervised Learning Statistical Perspective Probability Models Discrete & Continuous: Gaussian, Bernoulli, Multinomial Maimum Likelihood Logistic

More information

Machine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function.

Machine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function. Bayesian learning: Machine learning comes from Bayesian decision theory in statistics. There we want to minimize the expected value of the loss function. Let y be the true label and y be the predicted

More information

Generalized Linear Models

Generalized Linear Models Generalized Linear Models David Rosenberg New York University April 12, 2015 David Rosenberg (New York University) DS-GA 1003 April 12, 2015 1 / 20 Conditional Gaussian Regression Gaussian Regression Input

More information

Probabilistic Time Series Classification

Probabilistic Time Series Classification Probabilistic Time Series Classification Y. Cem Sübakan Boğaziçi University 25.06.2013 Y. Cem Sübakan (Boğaziçi University) M.Sc. Thesis Defense 25.06.2013 1 / 54 Problem Statement The goal is to assign

More information

CS Machine Learning Qualifying Exam

CS Machine Learning Qualifying Exam CS Machine Learning Qualifying Exam Georgia Institute of Technology March 30, 2017 The exam is divided into four areas: Core, Statistical Methods and Models, Learning Theory, and Decision Processes. There

More information

CPSC 340: Machine Learning and Data Mining. MLE and MAP Fall 2017

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

More information

Restricted Boltzmann Machines

Restricted Boltzmann Machines Restricted Boltzmann Machines Boltzmann Machine(BM) A Boltzmann machine extends a stochastic Hopfield network to include hidden units. It has binary (0 or 1) visible vector unit x and hidden (latent) vector

More information

CMU-Q Lecture 24:

CMU-Q Lecture 24: CMU-Q 15-381 Lecture 24: Supervised Learning 2 Teacher: Gianni A. Di Caro SUPERVISED LEARNING Hypotheses space Hypothesis function Labeled Given Errors Performance criteria Given a collection of input

More information

STA 4273H: Sta-s-cal Machine Learning

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

More information

Mathematical Formulation of Our Example

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

More information

STA414/2104. Lecture 11: Gaussian Processes. Department of Statistics

STA414/2104. Lecture 11: Gaussian Processes. Department of Statistics STA414/2104 Lecture 11: Gaussian Processes Department of Statistics www.utstat.utoronto.ca Delivered by Mark Ebden with thanks to Russ Salakhutdinov Outline Gaussian Processes Exam review Course evaluations

More information

Introduction to SVM and RVM

Introduction to SVM and RVM Introduction to SVM and RVM Machine Learning Seminar HUS HVL UIB Yushu Li, UIB Overview Support vector machine SVM First introduced by Vapnik, et al. 1992 Several literature and wide applications Relevance

More information

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 Overview Probabilistic Interpretation of Linear Regression Maximum Likelihood Estimation Bayesian Estimation MAP Estimation Probabilistic Interpretation: Linear Regression Assume output y is generated

More information

Tutorial on Approximate Bayesian Computation

Tutorial on Approximate Bayesian Computation Tutorial on Approximate Bayesian Computation Michael Gutmann https://sites.google.com/site/michaelgutmann University of Helsinki Aalto University Helsinki Institute for Information Technology 16 May 2016

More information

Intractable Likelihood Functions

Intractable Likelihood Functions Intractable Likelihood Functions Michael Gutmann Probabilistic Modelling and Reasoning (INFR11134) School of Informatics, University of Edinburgh Spring semester 2018 Recap p(x y o ) = z p(x,y o,z) x,z

More information

CS 195-5: Machine Learning Problem Set 1

CS 195-5: Machine Learning Problem Set 1 CS 95-5: Machine Learning Problem Set Douglas Lanman dlanman@brown.edu 7 September Regression Problem Show that the prediction errors y f(x; ŵ) are necessarily uncorrelated with any linear function of

More information

Logistic Regression. COMP 527 Danushka Bollegala

Logistic Regression. COMP 527 Danushka Bollegala Logistic Regression COMP 527 Danushka Bollegala Binary Classification Given an instance x we must classify it to either positive (1) or negative (0) class We can use {1,-1} instead of {1,0} but we will

More information

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 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

More information

Pattern Recognition and Machine Learning

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

More information

Support Vector Machines. CAP 5610: Machine Learning Instructor: Guo-Jun QI

Support Vector Machines. CAP 5610: Machine Learning Instructor: Guo-Jun QI Support Vector Machines CAP 5610: Machine Learning Instructor: Guo-Jun QI 1 Linear Classifier Naive Bayes Assume each attribute is drawn from Gaussian distribution with the same variance Generative model:

More information

A Few Notes on Fisher Information (WIP)

A Few Notes on Fisher Information (WIP) A Few Notes on Fisher Information (WIP) David Meyer dmm@{-4-5.net,uoregon.edu} Last update: April 30, 208 Definitions There are so many interesting things about Fisher Information and its theoretical properties

More information

A fast and simple algorithm for training neural probabilistic language models

A fast and simple algorithm for training neural probabilistic language models A fast and simple algorithm for training neural probabilistic language models Andriy Mnih Joint work with Yee Whye Teh Gatsby Computational Neuroscience Unit University College London 25 January 2013 1

More information

Introduction to Probabilistic Machine Learning

Introduction to Probabilistic Machine Learning Introduction to Probabilistic Machine Learning Piyush Rai Dept. of CSE, IIT Kanpur (Mini-course 1) Nov 03, 2015 Piyush Rai (IIT Kanpur) Introduction to Probabilistic Machine Learning 1 Machine Learning

More information

COMP90051 Statistical Machine Learning

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

More information

Tensor intro 1. SIAM Rev., 51(3), Tensor Decompositions and Applications, Kolda, T.G. and Bader, B.W.,

Tensor intro 1. SIAM Rev., 51(3), Tensor Decompositions and Applications, Kolda, T.G. and Bader, B.W., Overview 1. Brief tensor introduction 2. Stein s lemma 3. Score and score matching for fitting models 4. Bringing it all together for supervised deep learning Tensor intro 1 Tensors are multidimensional

More information

Learning from Data Logistic Regression

Learning from Data Logistic Regression Learning from Data Logistic Regression Copyright David Barber 2-24. Course lecturer: Amos Storkey a.storkey@ed.ac.uk Course page : http://www.anc.ed.ac.uk/ amos/lfd/ 2.9.8.7.6.5.4.3.2...2.3.4.5.6.7.8.9

More information

Probabilistic Graphical Models for Image Analysis - Lecture 1

Probabilistic Graphical Models for Image Analysis - Lecture 1 Probabilistic Graphical Models for Image Analysis - Lecture 1 Alexey Gronskiy, Stefan Bauer 21 September 2018 Max Planck ETH Center for Learning Systems Overview 1. Motivation - Why Graphical Models 2.

More information

Midterm sample questions

Midterm sample questions Midterm sample questions CS 585, Brendan O Connor and David Belanger October 12, 2014 1 Topics on the midterm Language concepts Translation issues: word order, multiword translations Human evaluation Parts

More information

Statistical learning. Chapter 20, Sections 1 3 1

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

More information

Density Estimation: ML, MAP, Bayesian estimation

Density Estimation: ML, MAP, Bayesian estimation Density Estimation: ML, MAP, Bayesian estimation CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Introduction Maximum-Likelihood Estimation Maximum

More information

Classification 2: Linear discriminant analysis (continued); logistic regression

Classification 2: Linear discriminant analysis (continued); logistic regression Classification 2: Linear discriminant analysis (continued); logistic regression Ryan Tibshirani Data Mining: 36-462/36-662 April 4 2013 Optional reading: ISL 4.4, ESL 4.3; ISL 4.3, ESL 4.4 1 Reminder:

More information

Machine Learning

Machine Learning Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University February 1, 2011 Today: Generative discriminative classifiers Linear regression Decomposition of error into

More information

Generative Clustering, Topic Modeling, & Bayesian Inference

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

More information

Notes on Discriminant Functions and Optimal Classification

Notes on Discriminant Functions and Optimal Classification Notes on Discriminant Functions and Optimal Classification Padhraic Smyth, Department of Computer Science University of California, Irvine c 2017 1 Discriminant Functions Consider a classification problem

More information

Notes on Adversarial Examples

Notes on Adversarial Examples Notes on Adversarial Examples David Meyer dmm@{1-4-5.net,uoregon.edu,...} March 14, 2017 1 Introduction The surprising discovery of adversarial examples by Szegedy et al. [6] has led to new ways of thinking

More information

Probabilistic classification CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2016

Probabilistic classification CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2016 Probabilistic classification CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2016 Topics Probabilistic approach Bayes decision theory Generative models Gaussian Bayes classifier

More information

Machine Learning (CSE 446): Multi-Class Classification; Kernel Methods

Machine Learning (CSE 446): Multi-Class Classification; Kernel Methods Machine Learning (CSE 446): Multi-Class Classification; Kernel Methods Sham M Kakade c 2018 University of Washington cse446-staff@cs.washington.edu 1 / 12 Announcements HW3 due date as posted. make sure

More information

Machine learning - HT Maximum Likelihood

Machine learning - HT Maximum Likelihood Machine learning - HT 2016 3. Maximum Likelihood Varun Kanade University of Oxford January 27, 2016 Outline Probabilistic Framework Formulate linear regression in the language of probability Introduce

More information

An Introduction to Statistical and Probabilistic Linear Models

An Introduction to Statistical and Probabilistic Linear Models An Introduction to Statistical and Probabilistic Linear Models Maximilian Mozes Proseminar Data Mining Fakultät für Informatik Technische Universität München June 07, 2017 Introduction In statistical learning

More information

Machine Learning Basics: Maximum Likelihood Estimation

Machine Learning Basics: Maximum Likelihood Estimation Machine Learning Basics: Maximum Likelihood Estimation Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics 1. Learning

More information

CPSC 540: Machine Learning

CPSC 540: Machine Learning CPSC 540: Machine Learning Undirected Graphical Models Mark Schmidt University of British Columbia Winter 2016 Admin Assignment 3: 2 late days to hand it in today, Thursday is final day. Assignment 4:

More information

DEPARTMENT OF COMPUTER SCIENCE Autumn Semester MACHINE LEARNING AND ADAPTIVE INTELLIGENCE

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

More information

Intro. ANN & Fuzzy Systems. Lecture 15. Pattern Classification (I): Statistical Formulation

Intro. ANN & Fuzzy Systems. Lecture 15. Pattern Classification (I): Statistical Formulation Lecture 15. Pattern Classification (I): Statistical Formulation Outline Statistical Pattern Recognition Maximum Posterior Probability (MAP) Classifier Maximum Likelihood (ML) Classifier K-Nearest Neighbor

More information

Least Squares Regression

Least Squares Regression E0 70 Machine Learning Lecture 4 Jan 7, 03) Least Squares Regression Lecturer: Shivani Agarwal Disclaimer: These notes are a brief summary of the topics covered in the lecture. They are not a substitute

More information

Introduction to Machine Learning

Introduction to Machine Learning Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Prof. Erik Sudderth Lecture 20: Expectation Maximization Algorithm EM for Mixture Models Many figures courtesy Kevin Murphy s

More information

Introduction to Machine Learning

Introduction to Machine Learning Outline Introduction to Machine Learning Bayesian Classification Varun Chandola March 8, 017 1. {circular,large,light,smooth,thick}, malignant. {circular,large,light,irregular,thick}, malignant 3. {oval,large,dark,smooth,thin},

More information

Lecture 1: Bayesian Framework Basics

Lecture 1: Bayesian Framework Basics Lecture 1: Bayesian Framework Basics Melih Kandemir melih.kandemir@iwr.uni-heidelberg.de April 21, 2014 What is this course about? Building Bayesian machine learning models Performing the inference of

More information

Machine Learning: Logistic Regression. Lecture 04

Machine Learning: Logistic Regression. Lecture 04 Machine Learning: Logistic Regression Razvan C. Bunescu School of Electrical Engineering and Computer Science bunescu@ohio.edu Supervised Learning Task = learn an (unkon function t : X T that maps input

More information

Loss Functions, Decision Theory, and Linear Models

Loss Functions, Decision Theory, and Linear Models Loss Functions, Decision Theory, and Linear Models CMSC 678 UMBC January 31 st, 2018 Some slides adapted from Hamed Pirsiavash Logistics Recap Piazza (ask & answer questions): https://piazza.com/umbc/spring2018/cmsc678

More information

Chapter 3: Maximum-Likelihood & Bayesian Parameter Estimation (part 1)

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

More information

MIDTERM: CS 6375 INSTRUCTOR: VIBHAV GOGATE October,

MIDTERM: CS 6375 INSTRUCTOR: VIBHAV GOGATE October, MIDTERM: CS 6375 INSTRUCTOR: VIBHAV GOGATE October, 23 2013 The exam is closed book. You are allowed a one-page cheat sheet. Answer the questions in the spaces provided on the question sheets. If you run

More information