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Learning Probabilistic Graphical Models in R

Book Image

Learning Probabilistic Graphical Models in R

Overview of this book

Probabilistic graphical models (PGM, also known as graphical models) are a marriage between probability theory and graph theory. Generally, PGMs use a graph-based representation. Two branches of graphical representations of distributions are commonly used, namely Bayesian networks and Markov networks. R has many packages to implement graphical models. We’ll start by showing you how to transform a classical statistical model into a modern PGM and then look at how to do exact inference in graphical models. Proceeding, we’ll introduce you to many modern R packages that will help you to perform inference on the models. We will then run a Bayesian linear regression and you’ll see the advantage of going probabilistic when you want to do prediction. Next, you’ll master using R packages and implementing its techniques. Finally, you’ll be presented with machine learning applications that have a direct impact in many fields. Here, we’ll cover clustering and the discovery of hidden information in big data, as well as two important methods, PCA and ICA, to reduce the size of big problems.

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Learning Probabilistic Graphical Models in R
Apr, 2016 | 250 pages
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Table of Contents (15 chapters)
Learning Probabilistic Graphical Models in R
Learning Probabilistic Graphical Models in R
Credits
Credits
About the Author
About the Author
About the Reviewers
About the Reviewers
www.PacktPub.com
www.PacktPub.com
Preface
Preface
Free Chapter
1
Probabilistic Reasoning
Probabilistic Reasoning
Machine learning
Representing uncertainty with probabilities
Probabilistic graphical models
Summary
2
Exact Inference
Exact Inference
Building graphical models
Variable elimination
Sum-product and belief updates
The junction tree algorithm
Examples of probabilistic graphical models
Summary
3
Learning Parameters
Learning Parameters
Introduction
Learning by inference
Maximum likelihood
Learning with hidden variables – the EM algorithm
Principles of the EM algorithm
Summary
4
Bayesian Modeling – Basic Models
Bayesian Modeling – Basic Models
The Naive Bayes model
Beta-Binomial
The Gaussian mixture model
Summary
5
Approximate Inference
Approximate Inference
Sampling from a distribution
Basic sampling algorithms
Rejection sampling
Importance sampling
Markov Chain Monte-Carlo
MCMC for probabilistic graphical models in R
Summary
6
Bayesian Modeling – Linear Models
Bayesian Modeling – Linear Models
Linear regression
Bayesian linear models
Summary
7
Probabilistic Mixture Models
Probabilistic Mixture Models
Mixture models
EM for mixture models
Mixture of Bernoulli
Mixture of experts
Latent Dirichlet Allocation
Summary
Appendix
Appendix
References
Index
Index

Linear regression

We start by looking at the most simple and most used model in statistics, which consists of fitting a straight line to a dataset. We assume we have a data set of pairs (xi, yi) that are i.i.d and we want to fit a model such that:

y = βx +β0 + ϵ

Here, ϵ is a Gaussian noise. If we assume that xi ϵ ℝn then the expected value can also be written as:

Or, in matrix notation, we can also include the intercept β0 into the vector of parameters and add a column on 1 in X, such that X = (1, x1, …, xn) to finally obtain:

ŷ = XTβ

The following figure shows an example (in one dimension) of a data set with its corresponding regression line:

In R, fitting a linear model is an easy task, as we will see now. Here, we produce a small data set with an artificial number, in order to reproduce the previous figure. In R, the function to fit a linear model is lm() and it is the workhorse of this language in many situations. Of course, later in this chapter we will see more advanced algorithms:

N=30...
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