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Mastering Probabilistic Graphical Models with Python

By : Ankur Ankan
Book Image

Mastering Probabilistic Graphical Models with Python

By: Ankur Ankan

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Mastering Probabilistic Graphical Models with Python
Ankur Ankan
Aug, 2015 | 284 pages
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Table of Contents (14 chapters)
Mastering Probabilistic Graphical Models Using Python
Mastering Probabilistic Graphical Models Using Python
Credits
Credits
About the Authors
About the Authors
About the Reviewers
About the Reviewers
www.PacktPub.com
www.PacktPub.com
Preface
Preface
Free Chapter
1
Bayesian Network Fundamentals
Bayesian Network Fundamentals
Probability theory
Installing tools
Representing independencies using pgmpy
Representing joint probability distributions using pgmpy
Conditional probability distribution
Graph theory
Bayesian models
Relating graphs and distributions
CPD representations
Summary
2
Markov Network Fundamentals
Markov Network Fundamentals
Introducing the Markov network
The factor graph
Independencies in Markov networks
Constructing graphs from distributions
Bayesian and Markov networks
Summary
3
Inference – Asking Questions to Models
Inference – Asking Questions to Models
Inference
Variable elimination
Belief propagation
MAP using variable elimination
Factor maximization
MAP using belief propagation
Finding the most probable assignment
Predictions from the model using pgmpy
A comparison of variable elimination and belief propagation
Summary
4
Approximate Inference
Approximate Inference
The optimization problem
The energy function
Exact inference as an optimization
The propagation-based approximation algorithm
Propagation with approximate messages
Sampling-based approximate methods
Forward sampling
Conditional probability distribution
Likelihood weighting and importance sampling
Importance sampling
Importance sampling in Bayesian networks
Markov chain Monte Carlo methods
Gibbs sampling
The multiple transitioning model
Using a Markov chain
Collapsed particles
Collapsed importance sampling
Summary
5
Model Learning – Parameter Estimation in Bayesian Networks
Model Learning – Parameter Estimation in Bayesian Networks
General ideas in learning
Learning as an optimization
Discriminative versus generative training
Parameter learning
Bayesian parameter estimation
Structure learning in Bayesian networks
The Bayesian score for Bayesian networks
Summary
6
Model Learning – Parameter Estimation in Markov Networks
Model Learning – Parameter Estimation in Markov Networks
Maximum likelihood parameter estimation
Summary
7
Specialized Models
Specialized Models
The Naive Bayes model
Dynamic Bayesian networks
The Hidden Markov model
Applications
Summary
Index
Index

MAP using variable elimination

Let's start with a very basic example of a network A -> B, as shown in the following figure:

Fig 3.13: Basic Bayesian network with two variables

For MAP, we want to compute the following:

If we consider any particular assignment a for the variable A, we have the following:

So, for any given assignment of A, we have to select the assignment of B for which P(b|a) is at maximum. We also have to select the maximum assignment of B as any given assignment of A doesn't guarantee that it would be the global maximum. Therefore, we need to check the values for each assignment of A.

Now, let's try to find the MAP assignment for the network in the Fig 3.13. Assuming the assignment from A to , let's define and similarly, . Now, let's compute the max-marginal over A:

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End of Section 5
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