Learning Probabilistic Graphical Models in R

Table of Contents

Learning Probabilistic Graphical Models in R

Credits

About the Author

About the Reviewers

www.PacktPub.com

eBooks, discount offers, and more

Why subscribe?

Preface

What this book covers

What you need for this book

Who this book is for

Conventions

Reader feedback

Customer support

Downloading the example code

Errata

Piracy

Questions

1. Probabilistic Reasoning

Machine learning

Representing uncertainty with probabilities

Beliefs and uncertainty as probabilities

Conditional probability

Probability calculus and random variables

Sample space, events, and probability

Random variables and probability calculus

Joint probability distributions

Bayes' rule

Interpreting the Bayes' formula

A first example of Bayes' rule

A first example of Bayes' rule in R

Probabilistic graphical models

Probabilistic models

Graphs and conditional independence

Factorizing a distribution

Directed models

Undirected models

Examples and applications

Summary

2. Exact Inference

Building graphical models

Types of random variable

Building graphs

Probabilistic expert system

Basic structures in probabilistic graphical models

Variable elimination

Sum-product and belief updates

The junction tree algorithm

Examples of probabilistic graphical models

The sprinkler example

The medical expert system

Models with more than two layers

Tree structure

Summary

3. Learning Parameters

Introduction

Learning by inference

Maximum likelihood

How are empirical and model distribution related?

The ML algorithm and its implementation in R

Application

Learning with hidden variables – the EM algorithm

Latent variables

Principles of the EM algorithm

Derivation of the EM algorithm

Applying EM to graphical models

Summary

4. Bayesian Modeling – Basic Models

The Naive Bayes model

Representation

Learning the Naive Bayes model

Bayesian Naive Bayes

Beta-Binomial

The prior distribution

The posterior distribution with the conjugacy property

Which values should we choose for the Beta parameters?

The Gaussian mixture model

Definition

Summary

5. Approximate Inference

Sampling from a distribution

Basic sampling algorithms

Standard distributions

Rejection sampling

An implementation in R

Importance sampling

An implementation in R

Markov Chain Monte-Carlo

General idea of the method

The Metropolis-Hastings algorithm

MCMC for probabilistic graphical models in R

Installing Stan and RStan

A simple example in RStan

Summary

6. Bayesian Modeling – Linear Models

Linear regression

Estimating the parameters

Bayesian linear models

Over-fitting a model

Graphical model of a linear model

Posterior distribution

Implementation in R

A stable implementation

More packages in R

Summary

7. Probabilistic Mixture Models

Mixture models

EM for mixture models

Mixture of Bernoulli

Mixture of experts

Latent Dirichlet Allocation

The LDA model

Variational inference

Examples

Summary

A. Appendix

References

Books on the Bayesian theory

Books on machine learning

Papers

Index

Learning Probabilistic Graphical Models in R

Learning Probabilistic Graphical Models in R

Copyright © 2016 Packt Publishing

All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior written permission of the publisher, except in the case of brief quotations embedded in critical articles or reviews.

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First published: April 2016

Production reference: 1270416

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ISBN 978-1-78439-205-5

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Credits

Author

David Bellot

Reviewers

Mzabalazo Z. Ngwenya

Prabhanjan Tattar

Acquisition Editor

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Content Development Editor

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Cover Work

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About the Author

David Bellot is a PhD graduate in computer science from INRIA, France, with a focus on Bayesian machine learning. He was a postdoctoral fellow at the University of California, Berkeley, and worked for companies such as Intel, Orange, and Barclays Bank. He currently works in the financial industry, where he develops financial market prediction algorithms using machine learning. He is also a contributor to open source projects such as the Boost C++ library.

About the Reviewers

Mzabalazo Z. Ngwenya holds a postgraduate degree in mathematical statistics from the University of Cape Town. He has worked extensively in the field of statistical consulting and has considerable experience working with R. Areas of interest to him are primarily centered around statistical computing. Previously, he has been involved in reviewing the following Packt Publishing titles: Learning RStudio for R Statistical Computing, Mark P.J. van der Loo and Edwin de Jonge; R Statistical Application Development by Example Beginner's Guide, Prabhanjan Narayanachar Tattar; Machine Learning with R, Brett Lantz; R Graph Essentials, David Alexandra Lillis; R Object-oriented Programming, Kelly Black; Mastering Scientific Computing with R, Paul Gerrard and Radia Johnson; and Mastering Data Analysis with R, Gergely Darócz.

Prabhanjan Tattar is currently working as a senior data scientist at Fractal Analytics, Inc. He has 8 years of experience as a statistical analyst. Survival analysis and statistical inference are his main areas of research/interest. He has published several research papers in peer-reviewed journals and authored two books on R: R Statistical Application Development by Example, Packt Publishing; and A Course in Statistics with R, Wiley. The R packages gpk, RSADBE, and ACSWR are also maintained by him.

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Preface

Probabilistic graphical models is one of the most advanced techniques in machine learning to represent data and models in the real world with probabilities. In many instances, it uses the Bayesian paradigm to describe algorithms that can draw conclusions from noisy and uncertain real-world data.

The book covers topics such as inference (automated reasoning and learning), which is automatically building models from raw data. It explains how all the algorithms work step by step and presents readily usable solutions in R with many examples. After covering the basic principles of probabilities and the Bayes formula, it presents Probabilistic Graphical Models(PGMs) and several types of inference and learning algorithms. The reader will go from the design to the automatic fitting of the model.

Then, the books focuses on useful models that have proven track records in solving many data science problems, such as Bayesian classifiers, Mixtures models, Bayesian Linear Regression, and also simpler models that are used as basic components to build more complex models.

What this book covers

Chapter 1, Probabilistic Reasoning, covers topics from the basic concepts of probabilities to PGMs as a generic framework to do tractable, efficient, and easy modeling with probabilistic models, through the presentation of the Bayes formula.

Chapter 2, Exact Inference, shows you how to build PGMs by combining simple graphs and perform queries on the model using an exact inference algorithm called the junction tree algorithm.

Chapter 3, Learning Parameters, includes fitting and learning the PGM models from data sets with the Maximum Likelihood approach.

Chapter 4, Bayesian Modeling – Basic Models, covers simple and powerful Bayesian models that can be used as building blocks for more advanced models and shows you how to fit and query them with adapted algorithms.

Chapter 5, Approximate Inference, covers the second way to perform an inference in PGM using sampling algorithms and a presentation of the main sampling algorithms such as MCMC.

Chapter 6, Bayesian Modeling – Linear Models, shows you a more Bayesian view of the standard linear regression algorithm and a solution to the problem of over-fitting.

Chapter 7, Probabilistic Mixture Models, goes over more advanced probabilistic models in which the data comes from a mixture of several simple models.

Appendix, References, includes all the books and articles which have been used to write this book.

What you need for this book

All the examples in this book can be used with R version 3 or above on any platform and operating system supporting R.

Who this book is for

This book is for anyone who has to deal with lots of data and draw conclusions from it, especially when the data is noisy or uncertain. Data scientists, machine learning enthusiasts, engineers, and those who are curious about the latest advances in machine learning will find PGM interesting.

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Code words in text, database table names, folder names, filenames, file extensions, pathnames, dummy URLs, user input, and Twitter handles are shown as follows: We can also mention the arm package, which provides Bayesian versions of glm() and polr() and implements hierarchical models.

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pred_sigma <- sqrt(sigma^2 + apply((T%*%posterior_sigma)*T, MARGIN=1, FUN=sum)) upper_bound <- T%*%posterior_beta + qnorm(0.95)*pred_sigma lower_bound <- T%*%posterior_beta - qnorm(0.95)*pred_sigma

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Chapter 1. Probabilistic Reasoning

Among all the predictions that were made about the 21st century, maybe the most unexpected one was that we would collect such a formidable amount of data about everything, everyday, and everywhere in the world. Recent years have seen an incredible explosion of data collection about our world, our lives, and technology; this is the main driver of what we can certainly call a revolution. We live in the Age of Information. But collecting data is nothing if we don't exploit it and try to extract knowledge out of it.

At the beginning of the 20th century, with the birth of statistics, the world was all about collecting data and making statistics. In that time, the only reliable tools were pencils and paper and of course, the eyes and ears of the observers. Scientific observation was still in its infancy, despite the prodigious development of the 19th century.

More than a hundred years later, we have computers, we have electronic sensors, we have massive data storage and we are able to store huge amounts of data continuously about, not only our physical world, but also our lives, mainly through the use of social networks, the Internet, and mobile phones. Moreover, the density of our storage technology has increased so much that we can, nowadays, store months if not years of data into a very small volume that can fit in the palm of our hand.

But storing data is not acquiring knowledge. Storing data is just keeping it somewhere for future use. At the same time as our storage capacity dramatically evolved, the capacity of modern computers increased too, at a pace that is sometimes hard to believe. When I was a doctoral student, I remember how proud I was when in the laboratory I received that brand-new, shiny, all-powerful PC for carrying my research work. Today, my old smart phone, which fits in my pocket, is more than 20 times faster.

Therefore in this book, you will learn one of the most advanced techniques to transform data into knowledge: machine learning. This technology is used in every aspect of modern life now, from search engines, to stock market predictions, from speech recognition to autonomous vehicles. Moreover it is used in many fields where one would not suspect it at all, from quality assurance in product chains to optimizing the placement of antennas for mobile phone networks.

Machine learning is the marriage between computer science and probabilities and statistics. A central theme in machine learning is the problem of inference or how to produce knowledge or predictions using an algorithm fed with data and examples. And this brings us to the two fundamental aspects of machine learning: the design of algorithms that can extract patterns and high-level knowledge from vast amounts of data and also the design of algorithms that can use this knowledge—or, in scientific terms: learning and inference.

Pierre-Simon Laplace (1749-1827) a French mathematician and one of the greatest scientists of all time, was presumably among the first to understand an important aspect of data collection: data is unreliable, uncertain and, as we say today, noisy. He was also the first to develop the use of probabilities to deal with such aspects of uncertainty and