Latent Gaussian Processes for Distribution Estimation of Multivariate Categorical Data
|
|
- Eugenia Warren
- 5 years ago
- Views:
Transcription
1 Latent Gaussian Processes for Distribution Estimation of Multivariate Categorical Data Yarin Gal Yutian Chen Zoubin Ghahramani
2 Distribution Estimation Distribution estimation of categorical data {y n n = 1,..., N} with y n {1,..., K } is easy. But learning a distribution P(y) from vectors of categorical data, {y n = (y n1,..., y nd ) n = 1,..., N} with y nd {1,..., K }, is much more difficult. 2 of 19
3 Multivariate Categorical Data 3 of 19
4 Multivariate Categorical Data 3 of 19
5 Example: Wisconsin Breast Cancer Sample code Thickness Unif. Cell Size Unif. Cell Shape Marginal Adhesion Epithelial Cell Size Bare Nuclei Bland Chromatin Normal Nucleoli Mitoses Class Benign Benign Benign Benign Benign Malignant Benign Benign Benign Benign Benign Benign Malignant Benign Malignant Malignant Benign Benign Malignant Benign Malignant Malignant Benign ? Malignant Benign Malignant Benign Benign Benign Benign Benign Benign Malignant 683 patients, possible configurations 4 of 19
6 Categorical Latent Gaussian Process We define our model as Continuous latent space of patients x n iid N (0, σ 2 xi) Dist. over K dim. functions generating weights f d GP( ) iid y nd Softmax(fd (x n )) test result y n = (y n1,..., y nd ) medical assessment N patients x n, D categorical tests, each with K possible test results 5 of 19
7 Sparse Gaussian Processes A small number of inducing points support a distribution over functions In dark blue are the data points, in red are the inducing points 6 of 19
8 Sparse Gaussian Processes Given sufficient statistics {Z m, u m } M m=1, Define KMM as the kernel evaluated at points (Z m ), Define KMn = K T nm as the kernel evaluated between points (Z m) and point x n, A sparse Gaussian process is defined through a conditional Gaussian distribution by f n N (a T n u, b n ) with or equivalently, a n = K 1 MM K Mn, b n = K nn K nm K 1 MM K Mn, f n = a T n u + b n ε (f ) n ε (f ) n N (0, 1). 7 of 19
9 Generative Model The resulting generative model, with kernel K(, ), is x n iid N (0, σ 2 x I) patient n (u dk ) K k=1 GP(0, K((Z m) M m=1 )) inducing points f ndk x n, u dk N (a T n u dk, b n ) weight of test result k y nd iid Softmax(fnd1,..., f ndk ) y n = (y n1,..., y nd ) examination d medical assessment We want to find the posterior over X and U: p(x, U Y). 8 of 19
10 Variational Inference Our marginal log-likelihood is intractable. We lower bound the log evidence with a variational approximate posterior q(x, F, U) = q(x)q(u)p(f X, U), with Such that... q(x) = q(u) = N n=1 i=1 D d=1 k=1 Q N (x ni m ni, sni 2 ), K N (u dk µ dk, L d L T d ) q(x, F, U) p(x, F, U Y). 9 of 19
11 Re-parametrising the Model This is equivalent to... Then, x ni = m ni + s ni ε (x) ni u dk = µ dk + L d ε (u) dk ε (x) ni N (0, 1) ε (u) dk N (0, I M) f ndk = a T n u dk + b n ε (f ) ndk ε (f ) ndk N (0, 1) log p(y) KL divergence terms N D [ + n=1 d=1 E (x) ε n,ε (u) d,ε(f ) nd ( fnd log Softmax (y nd ε (f ) nd, U d(ε (u) d ), x n(ε (x) n ))) ]. 10 of 19
12 Monte Carlo integration Monte Carlo integration approximates the likelihood obtaining noisy gradients: E (x) ε n,ε (u) d,ε(f ) log Softmax ( ) nd 1 T ( )) fnd log Softmax (y nd ε (f ) T nd,i, U d(ε (u) d,i ), x n(ε (x) n,i ) i=1 with ε (x),i, ε (u),i, ε (f ),i N (0, 1), independent of the parameters. Adaptive learning-rate stochastic optimisation is used to optimise the noisy objective wrt the parameters of the variational distribution. 11 of 19
13 Categorical Latent Gaussian Process Symbolic differentiation gives simple code: 1 import theano.tensor as T 2 X = m + s * randn(n, Q) 3 U = mu + L.dot(randn(M, K)) 4 Kmm, Kmn, Knn = RBF(sf2, l, Z), RBF(sf2, l, Z, X), RBFnn(sf2, l, X) 5 KmmInv = st.matrix_inverse(kmm) 6 A = KmmInv.dot(Kmn) 7 B = Knn - T.sum(Kmn * KmmInv.dot(Kmn), 0) 8 F = A.T.dot(U) + B[:,None]**0.5 * randn(n, K) 9 S = T.nnet.softmax(F) 10 KL_U, KL_X = get_kl_u(), get_kl_x() 11 LS = T.sum(T.log(T.sum(Y * S, 1))) - KL_U - KL_X 12 LS_func = theano.function([ inputs ], LS) 13 dls_dm = theano.function([ inputs ], T.grad(LS, m)) # and others 14 #... and run RMS-PROP 12 of 19
14 Relations to Existing Models Linear regression Gaussian process regression Factor analysis Gaussian process latent variable model Logistic regression Gaussian process classification Linear Non-linear Continuous Observed input Discrete Latent input Latent Gaussian models Categorical Latent Gaussian Process 13 of 19
15 (b) Categorical Latent Gaussian Process p(y d = 1 x) as a function of x, for d = 0, 1, 2 (first, second, and third digits in the XOR triplet left to right) 14 of 19 Multi-modal Distributions (a) (Linear) Latent Gaussian Model
16 Data Visualisation (a) Example alphadigits Latent Dim Latent Dim 1 (b) (Linear) Latent Gaussian Model latent space Latent Dim Latent Dim 1 (c) Categorical Latent Gaussian Process latent space 15 of 19
17 Data Imputation Terror Warning Effects on Political Attitude (START Terrorism Data Archive - 17 categorical variables with 5-6 values) Uniform Multinomial (Linear) Latent Gaussian Model Categorical Latent Gaussian Process Wisconsin Breast cancer (9 categorical variables with 10 values) (Linear) Categorical Uniform Multinomial Latent Latent Gaussian Gaussian Model Process ± ± Test perplexity predicting randomly missing values 16 of 19
18 Inference Robustness x ELBO log iter Lower bound and MC standard deviation per iteration (on log scale) for the Alphadigits dataset. 17 of 19
19 Scaling Up! Why not scale the model up? We recently showed that sparse Gaussian processes can be scaled well to millions of data points [Gal, van der Wilk, Rasmussen, 2014] Mini-batch GP optimisation [Hensman el at. 2014] Recognition models with sparse Gaussian processes Integrate over unobserved variables Handle missing data Mixed data Using link functions alternative to the Softmax Continuous variables, Positives, Ordinals Currently running experiments! 18 of 19
20 Scaling Up! Why not scale the model up? We recently showed that sparse Gaussian processes can be scaled well to millions of data points [Gal, van der Wilk, Rasmussen, 2014] Mini-batch GP optimisation [Hensman el at. 2014] Recognition models with sparse Gaussian processes Integrate over unobserved variables Handle missing data Mixed data Using link functions alternative to the Softmax Continuous variables, Positives, Ordinals Currently running experiments! 18 of 19
21 Scaling Up! Why not scale the model up? We recently showed that sparse Gaussian processes can be scaled well to millions of data points [Gal, van der Wilk, Rasmussen, 2014] Mini-batch GP optimisation [Hensman el at. 2014] Recognition models with sparse Gaussian processes Integrate over unobserved variables Handle missing data Mixed data Using link functions alternative to the Softmax Continuous variables, Positives, Ordinals Currently running experiments! 18 of 19
22 Thank you 19 of 19
arxiv: v1 [stat.ml] 7 Mar 2015
Latent Gaussian Processes for Distribution Estimation of Multivariate Categorical Data arxiv:1503.02182v1 [stat.ml] 7 Mar 2015 Yarin Gal Yutian Chen Zoubin Ghahramani University of Cambridge Abstract Multivariate
More informationStochastic Variational Inference for Gaussian Process Latent Variable Models using Back Constraints
Stochastic Variational Inference for Gaussian Process Latent Variable Models using Back Constraints Thang D. Bui Richard E. Turner tdb40@cam.ac.uk ret26@cam.ac.uk Computational and Biological Learning
More informationNon-Factorised Variational Inference in Dynamical Systems
st Symposium on Advances in Approximate Bayesian Inference, 08 6 Non-Factorised Variational Inference in Dynamical Systems Alessandro D. Ialongo University of Cambridge and Max Planck Institute for Intelligent
More informationLearning Bayesian Networks for Biomedical Data
Learning Bayesian Networks for Biomedical Data Faming Liang (Texas A&M University ) Liang, F. and Zhang, J. (2009) Learning Bayesian Networks for Discrete Data. Computational Statistics and Data Analysis,
More informationThe connection of dropout and Bayesian statistics
The connection of dropout and Bayesian statistics Interpretation of dropout as approximate Bayesian modelling of NN http://mlg.eng.cam.ac.uk/yarin/thesis/thesis.pdf Dropout Geoffrey Hinton Google, University
More informationNonparametric 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
More informationContents. Part I: Fundamentals of Bayesian Inference 1
Contents Preface xiii Part I: Fundamentals of Bayesian Inference 1 1 Probability and inference 3 1.1 The three steps of Bayesian data analysis 3 1.2 General notation for statistical inference 4 1.3 Bayesian
More informationDropout as a Bayesian Approximation: Insights and Applications
Dropout as a Bayesian Approximation: Insights and Applications Yarin Gal and Zoubin Ghahramani Discussion by: Chunyuan Li Jan. 15, 2016 1 / 16 Main idea In the framework of variational inference, the authors
More informationApproximate Bayesian inference
Approximate Bayesian inference Variational and Monte Carlo methods Christian A. Naesseth 1 Exchange rate data 0 20 40 60 80 100 120 Month Image data 2 1 Bayesian inference 2 Variational inference 3 Stochastic
More informationVariational 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
More informationBayesian 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
More informationIntroduction 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
More informationDeep learning with differential Gaussian process flows
Deep learning with differential Gaussian process flows Pashupati Hegde Markus Heinonen Harri Lähdesmäki Samuel Kaski Helsinki Institute for Information Technology HIIT Department of Computer Science, Aalto
More informationThe Variational Gaussian Approximation Revisited
The Variational Gaussian Approximation Revisited Manfred Opper Cédric Archambeau March 16, 2009 Abstract The variational approximation of posterior distributions by multivariate Gaussians has been much
More informationFaster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions
Faster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions Mohammad Emtiyaz Khan, Reza Babanezhad, Wu Lin, Mark Schmidt, Masashi Sugiyama Conference on Uncertainty
More informationRecent 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
More informationarxiv: v2 [stat.ml] 29 Sep 2014
Variational Inference in Sparse Gaussian Process Regression and Latent Variable Models a Gentle Tutorial Yarin Gal, Mark van der Wilk October 1, 014 1 Introduction arxiv:140.141v [stat.ml] 9 Sep 014 In
More informationNon-Gaussian likelihoods for Gaussian Processes
Non-Gaussian likelihoods for Gaussian Processes Alan Saul University of Sheffield Outline Motivation Laplace approximation KL method Expectation Propagation Comparing approximations GP regression Model
More informationMachine Learning. Regression-Based Classification & Gaussian Discriminant Analysis. Manfred Huber
Machine Learning Regression-Based Classification & Gaussian Discriminant Analysis Manfred Huber 2015 1 Logistic Regression Linear regression provides a nice representation and an efficient solution to
More informationSTAT 518 Intro Student Presentation
STAT 518 Intro Student Presentation Wen Wei Loh April 11, 2013 Title of paper Radford M. Neal [1999] Bayesian Statistics, 6: 475-501, 1999 What the paper is about Regression and Classification Flexible
More informationGaussian Processes for Big Data. James Hensman
Gaussian Processes for Big Data James Hensman Overview Motivation Sparse Gaussian Processes Stochastic Variational Inference Examples Overview Motivation Sparse Gaussian Processes Stochastic Variational
More informationLecture 13 : Variational Inference: Mean Field Approximation
10-708: Probabilistic Graphical Models 10-708, Spring 2017 Lecture 13 : Variational Inference: Mean Field Approximation Lecturer: Willie Neiswanger Scribes: Xupeng Tong, Minxing Liu 1 Problem Setup 1.1
More informationProbabilistic & Bayesian deep learning. Andreas Damianou
Probabilistic & Bayesian deep learning Andreas Damianou Amazon Research Cambridge, UK Talk at University of Sheffield, 19 March 2019 In this talk Not in this talk: CRFs, Boltzmann machines,... In this
More informationApproximate 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
More informationPILCO: 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
More informationClassification CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2012
Classification CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2012 Topics Discriminant functions Logistic regression Perceptron Generative models Generative vs. discriminative
More informationApproximate 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
More informationLecture 6: Graphical Models: Learning
Lecture 6: Graphical Models: Learning 4F13: Machine Learning Zoubin Ghahramani and Carl Edward Rasmussen Department of Engineering, University of Cambridge February 3rd, 2010 Ghahramani & Rasmussen (CUED)
More informationStein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm
Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm Qiang Liu and Dilin Wang NIPS 2016 Discussion by Yunchen Pu March 17, 2017 March 17, 2017 1 / 8 Introduction Let x R d
More informationMachine Learning. Gaussian Mixture Models. Zhiyao Duan & Bryan Pardo, Machine Learning: EECS 349 Fall
Machine Learning Gaussian Mixture Models Zhiyao Duan & Bryan Pardo, Machine Learning: EECS 349 Fall 2012 1 The Generative Model POV We think of the data as being generated from some process. We assume
More informationVariational 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
More informationIntroduction to Gaussian Processes
Introduction to Gaussian Processes Iain Murray School of Informatics, University of Edinburgh The problem Learn scalar function of vector values f(x).5.5 f(x) y i.5.2.4.6.8 x f 5 5.5 x x 2.5 We have (possibly
More informationRestricted 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 informationECE521 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 informationVariational Model Selection for Sparse Gaussian Process Regression
Variational Model Selection for Sparse Gaussian Process Regression Michalis K. Titsias School of Computer Science University of Manchester 7 September 2008 Outline Gaussian process regression and sparse
More informationTree-structured Gaussian Process Approximations
Tree-structured Gaussian Process Approximations Thang Bui joint work with Richard Turner MLG, Cambridge July 1st, 2014 1 / 27 Outline 1 Introduction 2 Tree-structured GP approximation 3 Experiments 4 Summary
More informationLearning with Noisy Labels. Kate Niehaus Reading group 11-Feb-2014
Learning with Noisy Labels Kate Niehaus Reading group 11-Feb-2014 Outline Motivations Generative model approach: Lawrence, N. & Scho lkopf, B. Estimating a Kernel Fisher Discriminant in the Presence of
More informationUniversität 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
More informationSparse Approximations for Non-Conjugate Gaussian Process Regression
Sparse Approximations for Non-Conjugate Gaussian Process Regression Thang Bui and Richard Turner Computational and Biological Learning lab Department of Engineering University of Cambridge November 11,
More informationGaussian 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
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 2: PROBABILITY DISTRIBUTIONS Parametric Distributions Basic building blocks: Need to determine given Representation: or? Recall Curve Fitting Binary Variables
More informationProbabilistic 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 informationMultiple-step Time Series Forecasting with Sparse Gaussian Processes
Multiple-step Time Series Forecasting with Sparse Gaussian Processes Perry Groot ab Peter Lucas a Paul van den Bosch b a Radboud University, Model-Based Systems Development, Heyendaalseweg 135, 6525 AJ
More informationCircular Pseudo-Point Approximations for Scaling Gaussian Processes
Circular Pseudo-Point Approximations for Scaling Gaussian Processes Will Tebbutt Invenia Labs, Cambridge, UK will.tebbutt@invenialabs.co.uk Thang D. Bui University of Cambridge tdb4@cam.ac.uk Richard E.
More informationLecture 7 and 8: Markov Chain Monte Carlo
Lecture 7 and 8: Markov Chain Monte Carlo 4F13: Machine Learning Zoubin Ghahramani and Carl Edward Rasmussen Department of Engineering University of Cambridge http://mlg.eng.cam.ac.uk/teaching/4f13/ Ghahramani
More informationReplicated Softmax: an Undirected Topic Model. Stephen Turner
Replicated Softmax: an Undirected Topic Model Stephen Turner 1. Introduction 2. Replicated Softmax: A Generative Model of Word Counts 3. Evaluating Replicated Softmax as a Generative Model 4. Experimental
More informationPart I. C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS
Part I C. M. Bishop PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 8: GRAPHICAL MODELS Probabilistic Graphical Models Graphical representation of a probabilistic model Each variable corresponds to a
More informationGaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
More informationLecture: 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
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Contents in latter part Linear Dynamical Systems What is different from HMM? Kalman filter Its strength and limitation Particle Filter
More informationAn Infinite Product of Sparse Chinese Restaurant Processes
An Infinite Product of Sparse Chinese Restaurant Processes Yarin Gal Tomoharu Iwata Zoubin Ghahramani yg279@cam.ac.uk CRP quick recap The Chinese restaurant process (CRP) Distribution over partitions of
More informationBayesian Inference: Principles and Practice 3. Sparse Bayesian Models and the Relevance Vector Machine
Bayesian Inference: Principles and Practice 3. Sparse Bayesian Models and the Relevance Vector Machine Mike Tipping Gaussian prior Marginal prior: single α Independent α Cambridge, UK Lecture 3: Overview
More informationECE521 Lectures 9 Fully Connected Neural Networks
ECE521 Lectures 9 Fully Connected Neural Networks Outline Multi-class classification Learning multi-layer neural networks 2 Measuring distance in probability space We learnt that the squared L2 distance
More informationProbabilistic Graphical Models for Image Analysis - Lecture 4
Probabilistic Graphical Models for Image Analysis - Lecture 4 Stefan Bauer 12 October 2018 Max Planck ETH Center for Learning Systems Overview 1. Repetition 2. α-divergence 3. Variational Inference 4.
More informationIntroduction 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 informationECE521 lecture 4: 19 January Optimization, MLE, regularization
ECE521 lecture 4: 19 January 2017 Optimization, MLE, regularization First four lectures Lectures 1 and 2: Intro to ML Probability review Types of loss functions and algorithms Lecture 3: KNN Convexity
More informationExplicit Rates of Convergence for Sparse Variational Inference in Gaussian Process Regression
JMLR: Workshop and Conference Proceedings : 9, 08. Symposium on Advances in Approximate Bayesian Inference Explicit Rates of Convergence for Sparse Variational Inference in Gaussian Process Regression
More informationStochastic Backpropagation, Variational Inference, and Semi-Supervised Learning
Stochastic Backpropagation, Variational Inference, and Semi-Supervised Learning Diederik (Durk) Kingma Danilo J. Rezende (*) Max Welling Shakir Mohamed (**) Stochastic Gradient Variational Inference Bayesian
More informationA variational radial basis function approximation for diffusion processes
A variational radial basis function approximation for diffusion processes Michail D. Vrettas, Dan Cornford and Yuan Shen Aston University - Neural Computing Research Group Aston Triangle, Birmingham B4
More informationMachine Learning Summer School
Machine Learning Summer School Lecture 3: Learning parameters and structure Zoubin Ghahramani zoubin@eng.cam.ac.uk http://learning.eng.cam.ac.uk/zoubin/ Department of Engineering University of Cambridge,
More informationLatent Variable Models
Latent Variable Models Stefano Ermon, Aditya Grover Stanford University Lecture 5 Stefano Ermon, Aditya Grover (AI Lab) Deep Generative Models Lecture 5 1 / 31 Recap of last lecture 1 Autoregressive models:
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.utstat.utoronto.ca/~rsalakhu/ Sidney Smith Hall, Room 6002 Lecture 7 Approximate
More informationGAUSSIAN 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
More informationVariational Autoencoder
Variational Autoencoder Göker Erdo gan August 8, 2017 The variational autoencoder (VA) [1] is a nonlinear latent variable model with an efficient gradient-based training procedure based on variational
More informationThe Expectation Maximization or EM algorithm
The Expectation Maximization or EM algorithm Carl Edward Rasmussen November 15th, 2017 Carl Edward Rasmussen The EM algorithm November 15th, 2017 1 / 11 Contents notation, objective the lower bound functional,
More informationCh 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 informationKernel 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
More informationHomework 6. Due: 10am Thursday 11/30/17
Homework 6 Due: 10am Thursday 11/30/17 1. Hinge loss vs. logistic loss. In class we defined hinge loss l hinge (x, y; w) = (1 yw T x) + and logistic loss l logistic (x, y; w) = log(1 + exp ( yw T x ) ).
More informationKernel Sequential Monte Carlo
Kernel Sequential Monte Carlo Ingmar Schuster (Paris Dauphine) Heiko Strathmann (University College London) Brooks Paige (Oxford) Dino Sejdinovic (Oxford) * equal contribution April 25, 2016 1 / 37 Section
More informationVariational Autoencoders (VAEs)
September 26 & October 3, 2017 Section 1 Preliminaries Kullback-Leibler divergence KL divergence (continuous case) p(x) andq(x) are two density distributions. Then the KL-divergence is defined as Z KL(p
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationGWAS V: Gaussian processes
GWAS V: Gaussian processes Dr. Oliver Stegle Christoh Lippert Prof. Dr. Karsten Borgwardt Max-Planck-Institutes Tübingen, Germany Tübingen Summer 2011 Oliver Stegle GWAS V: Gaussian processes Summer 2011
More informationStochastic Gradient Descent
Stochastic Gradient Descent Machine Learning CSE546 Carlos Guestrin University of Washington October 9, 2013 1 Logistic Regression Logistic function (or Sigmoid): Learn P(Y X) directly Assume a particular
More informationNonparametric Inference for Auto-Encoding Variational Bayes
Nonparametric Inference for Auto-Encoding Variational Bayes Erik Bodin * Iman Malik * Carl Henrik Ek * Neill D. F. Campbell * University of Bristol University of Bath Variational approximations are an
More informationCSci 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
More informationUnsupervised 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
More informationCheng Soon Ong & Christian Walder. Canberra February June 2017
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2017 (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 679 Part XIX
More informationA 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 informationMarkov Chain Monte Carlo Algorithms for Gaussian Processes
Markov Chain Monte Carlo Algorithms for Gaussian Processes Michalis K. Titsias, Neil Lawrence and Magnus Rattray School of Computer Science University of Manchester June 8 Outline Gaussian Processes Sampling
More informationRecurrent Latent Variable Networks for Session-Based Recommendation
Recurrent Latent Variable Networks for Session-Based Recommendation Panayiotis Christodoulou Cyprus University of Technology paa.christodoulou@edu.cut.ac.cy 27/8/2017 Panayiotis Christodoulou (C.U.T.)
More informationAfternoon Meeting on Bayesian Computation 2018 University of Reading
Gabriele Abbati 1, Alessra Tosi 2, Seth Flaxman 3, Michael A Osborne 1 1 University of Oxford, 2 Mind Foundry Ltd, 3 Imperial College London Afternoon Meeting on Bayesian Computation 2018 University of
More informationCSC2535: Computation in Neural Networks Lecture 7: Variational Bayesian Learning & Model Selection
CSC2535: Computation in Neural Networks Lecture 7: Variational Bayesian Learning & Model Selection (non-examinable material) Matthew J. Beal February 27, 2004 www.variational-bayes.org Bayesian Model Selection
More informationExpectation Propagation in Dynamical Systems
Expectation Propagation in Dynamical Systems Marc Peter Deisenroth Joint Work with Shakir Mohamed (UBC) August 10, 2012 Marc Deisenroth (TU Darmstadt) EP in Dynamical Systems 1 Motivation Figure : Complex
More informationSandwiching the marginal likelihood using bidirectional Monte Carlo. Roger Grosse
Sandwiching the marginal likelihood using bidirectional Monte Carlo Roger Grosse Ryan Adams Zoubin Ghahramani Introduction When comparing different statistical models, we d like a quantitative criterion
More informationVariational Inference in TensorFlow. Danijar Hafner Stanford CS University College London, Google Brain
Variational Inference in TensorFlow Danijar Hafner Stanford CS 20 2018-02-16 University College London, Google Brain Outline Variational Inference Tensorflow Distributions VAE in TensorFlow Variational
More informationarxiv: v5 [stat.ml] 25 May 2016
Dropout as a Bayesian Approximation: Appendix arxiv:506.0257v5 [stat.ml] 25 May 206 Yarin Gal University of Cambridge {yg279,zg20}@cam.ac.uk Abstract Zoubin Ghahramani We show that a neural network with
More informationLatent Variable Models and EM algorithm
Latent Variable Models and EM algorithm SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic 3.1 Clustering and Mixture Modelling K-means and hierarchical clustering are non-probabilistic
More informationHow to build an automatic statistician
How to build an automatic statistician James Robert Lloyd 1, David Duvenaud 1, Roger Grosse 2, Joshua Tenenbaum 2, Zoubin Ghahramani 1 1: Department of Engineering, University of Cambridge, UK 2: Massachusetts
More informationCS839: Probabilistic Graphical Models. Lecture 7: Learning Fully Observed BNs. Theo Rekatsinas
CS839: Probabilistic Graphical Models Lecture 7: Learning Fully Observed BNs Theo Rekatsinas 1 Exponential family: a basic building block For a numeric random variable X p(x ) =h(x)exp T T (x) A( ) = 1
More informationVariational Autoencoders
Variational Autoencoders Recap: Story so far A classification MLP actually comprises two components A feature extraction network that converts the inputs into linearly separable features Or nearly linearly
More informationIEOR E4570: Machine Learning for OR&FE Spring 2015 c 2015 by Martin Haugh. The EM Algorithm
IEOR E4570: Machine Learning for OR&FE Spring 205 c 205 by Martin Haugh The EM Algorithm The EM algorithm is used for obtaining maximum likelihood estimates of parameters when some of the data is missing.
More informationDeep latent variable models
Deep latent variable models Pierre-Alexandre Mattei IT University of Copenhagen http://pamattei.github.io @pamattei 19 avril 2018 Séminaire de statistique du CNAM 1 Overview of talk A short introduction
More informationDensity Estimation. Seungjin Choi
Density Estimation Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr http://mlg.postech.ac.kr/
More informationProbabilistic Reasoning in Deep Learning
Probabilistic Reasoning in Deep Learning Dr Konstantina Palla, PhD palla@stats.ox.ac.uk September 2017 Deep Learning Indaba, Johannesburgh Konstantina Palla 1 / 39 OVERVIEW OF THE TALK Basics of Bayesian
More informationBayesian Mixtures of Bernoulli Distributions
Bayesian Mixtures of Bernoulli Distributions Laurens van der Maaten Department of Computer Science and Engineering University of California, San Diego Introduction The mixture of Bernoulli distributions
More informationLearning 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
More informationLecture 10. Announcement. Mixture Models II. Topics of This Lecture. This Lecture: Advanced Machine Learning. Recap: GMMs as Latent Variable Models
Advanced Machine Learning Lecture 10 Mixture Models II 30.11.2015 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de/ Announcement Exercise sheet 2 online Sampling Rejection Sampling Importance
More informationExpectation Propagation for Approximate Bayesian Inference
Expectation Propagation for Approximate Bayesian Inference José Miguel Hernández Lobato Universidad Autónoma de Madrid, Computer Science Department February 5, 2007 1/ 24 Bayesian Inference Inference Given
More informationStatistical Data Mining and Machine Learning Hilary Term 2016
Statistical Data Mining and Machine Learning Hilary Term 2016 Dino Sejdinovic Department of Statistics Oxford Slides and other materials available at: http://www.stats.ox.ac.uk/~sejdinov/sdmml Naïve Bayes
More informationSparse Gaussian Markov Random Field Mixtures for Anomaly Detection
Sparse Gaussian Markov Random Field Mixtures for Anomaly Detection Tsuyoshi Idé ( Ide-san ), Ankush Khandelwal*, Jayant Kalagnanam IBM Research, T. J. Watson Research Center (*Currently with University
More information