Introduction to probability theory and graphical models

Translational Neuroimaging Seminar on Bayesian InferenceSpring 2013

Jakob HeinzleTranslational Neuromodeling Unit (TNU) Institute for Biomedical Engineering (IBT)University and ETH Zürich

Bayesian Inference - Introduction to probability theory

Literature and References

• Literature:• Bishop (Chapters 1.2, 1.3, 8.1, 8.2)• MacKay (Chapter 2)• Barber (Chapters 1, 2, 3, 4)

• Many images in this lecture are taken from the above references.

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Bayesian Inference - Introduction to probability theory

Probability distribution

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A probability P(x=true) is defined on a sample space (domain) and defines (for every possible) event in the sample space the certainty

of it to occur.

Sample space: dom(X)={0,1},

Probabilities sum to one.

Bishop, Fig. 1.11

Bayesian Inference - Introduction to probability theory

Probability theory: Basic rules

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.

Sum rule* - P(X) is also called the marginal distribution

Product rule -

* According to Bishop

Bayesian Inference - Introduction to probability theory

Conditional and marginal probability

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Bayesian Inference - Introduction to probability theory

Conditional and marginal probability

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Bishop, Fig. 1.11

Bayesian Inference - Introduction to probability theory

Independent variables

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Question for later: What does this mean for Bayes?

Bayesian Inference - Introduction to probability theory

Probability theory: Bayes’ theorem

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is derived from the product rule

Bayesian Inference - Introduction to probability theory

Rephrasing and naming of Bayes’ rule

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MacKay

D: data, q: parameters, H: hypothesis we put into the model.

Bayesian Inference - Introduction to probability theory

Example: Bishop Fig. 1.9

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Box (B): blue (b) or red (r)Fruit (F): apple (a) or orange (o)

p(B=r) = 0.4, p(B=b) = 0.6.

What is the probability of having a red box if one has drawn an orange?

Bishop, Fig. 1.9

Bayesian Inference - Introduction to probability theory

Probability density

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Bayesian Inference - Introduction to probability theory 12

PDF and CDF

Bishop, Fig. 1.12

Bayesian Inference - Introduction to probability theory

Cumulative distribution

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𝑃 (𝑧)=∫−∞

𝑧

𝑝 (𝑥 )𝑑𝑥

Short example: How to use the cumulative distribution to transform a uniform distribution!

Bayesian Inference - Introduction to probability theory

Marginal densities

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p

Integration instead of summing

Bayesian Inference - Introduction to probability theory

Two views on probability

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● Probability can …– … describe the frequency of outcomes in random

experiments classical interpretation.

– … describe the degree of belief about a particular event Bayesian viewpoint or subjective interpretation of probability.

MacKay, Chapter 2

Bayesian Inference - Introduction to probability theory

Expectation of a function

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Or

Bayesian Inference - Introduction to probability theory

Graphical models

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1. They provide a simple way to visualize the structure of a probabilistic model and can be used to design and motivate new models.

2. Insights into the properties of the model, including conditional independence properties, can be obtained by inspection of the graph.

3. Complex computations, required to perform inference and learning in sophisticated models, can be expressed in terms of graphical manipulations, in which underlying mathematical expressions are carried along implicitly.

Bishop, Chap. 8

Bayesian Inference - Introduction to probability theory 18

Graphical models overview

Directed Graph

For summary of definitions see Barber, Chapter 2

Undirected Graph

Names: nodes (vertices), edges (links), paths, cycles, loops, neighbours

Bayesian Inference - Introduction to probability theory 19

Graphical models overview

Barber, Introduction

Bayesian Inference - Introduction to probability theory

Graphical models

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Bishop, Fig. 8.1

Bayesian Inference - Introduction to probability theory

Graphical models: parents and children

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Node a is a parent of node b, node b is a child of node a.

Bishop, Fig. 8.1

Bayesian Inference - Introduction to probability theory

Belief networks = Bayesian belief networks = Bayesian Networks

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Bishop, Fig. 8.2

In general:

Every probability distributioncan be expressed as a Directed acyclic graph (DAG)

Important: No directed cycles!

Bayesian Inference - Introduction to probability theory

Conditional independence

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A variable a is conditionally independent of b given c, if

In bayesian networks conditional independence can betested by following some simple rules

Bayesian Inference - Introduction to probability theory

Conditional independence – tail-to-tail path

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Is a independent of b?

No! Yes!Bishop, Chapter 8.2

Bayesian Inference - Introduction to probability theory

Conditional independence – head-to-tail path

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No! Yes!Bishop, Chapter 8.2

Is a independent of b?

Bayesian Inference - Introduction to probability theory

Conditional independence – head-to-head path

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Yes! No!Bishop, Chapter 8.2

Is a independent of b?

Bayesian Inference - Introduction to probability theory

Conditional independence – notation

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Bishop, Chapter 8.2

Bayesian Inference - Introduction to probability theory

Conditional independence – three basic structures

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Bishop, Chapter 8.2.2

Bayesian Inference - Introduction to probability theory

More conventions in graphical notations

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Bishop, Chapter 8

= =

Regression model Short form Parameters explicit

Bayesian Inference - Introduction to probability theory

More conventions in graphical notations

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Bishop, Chapter 8

Trained on data tn

Complete model usedfor prediction

Bayesian Inference - Introduction to probability theory

Summary – things to remember

• Probabilities and how to compute with the Product rule, Bayes’ Rule, Sum rule

• Probability densities PDF, CDF

• Conditional and Marginal distributions

• Basic concepts of graphical models Directed vs. Undirected, nodes and edges, parents and children.

• Conditional independence in graphs and how to check it.

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Bishop, Chapter 8.2.2