Introduction to probability theory and graphical models Translational Neuroimaging Seminar on Bayesian Inference Spring 2013 Jakob Heinzle Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering (IBT) University and ETH Zürich

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. Bayesian Inference - Introduction to probability theory2

Probability distribution Bayesian Inference - Introduction to probability theory3 Bishop, Fig. 1.11

Probability theory: Basic rules Bayesian Inference - Introduction to probability theory4 * According to Bishop

Conditional and marginal probability Bayesian Inference - Introduction to probability theory5

Conditional and marginal probability Bayesian Inference - Introduction to probability theory6 Bishop, Fig. 1.11

Independent variables Bayesian Inference - Introduction to probability theory7 Question for later: What does this mean for Bayes?

Probability theory: Bayes’ theorem Bayesian Inference - Introduction to probability theory8

Rephrasing and naming of Bayes’ rule Bayesian Inference - Introduction to probability theory9 MacKay D: data,  : parameters, H: hypothesis we put into the model.

Example: Bishop Fig. 1.9 Bayesian Inference - Introduction to probability theory10 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

Probability density Bayesian Inference - Introduction to probability theory11

PDF and CDF Bayesian Inference - Introduction to probability theory12 Bishop, Fig. 1.12

Cumulative distribution Bayesian Inference - Introduction to probability theory13 Short example: How to use the cumulative distribution to transform a uniform distribution!

Marginal densities Bayesian Inference - Introduction to probability theory14

Two views on probability Bayesian Inference - Introduction to probability theory15 ● 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

Expectation of a function Bayesian Inference - Introduction to probability theory16

Graphical models Bayesian Inference - Introduction to probability theory17 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

Graphical models overview Directed Graph Bayesian Inference - Introduction to probability theory18 For summary of definitions see Barber, Chapter 2 Undirected Graph Names: nodes (vertices), edges (links), paths, cycles, loops, neighbours

Graphical models overview Bayesian Inference - Introduction to probability theory19 Barber, Introduction

Graphical models Bayesian Inference - Introduction to probability theory20 Bishop, Fig. 8.1

Graphical models: parents and children Bayesian Inference - Introduction to probability theory21 Node a is a parent of node b, node b is a child of node a. Bishop, Fig. 8.1

Belief networks = Bayesian belief networks = Bayesian Networks Bayesian Inference - Introduction to probability theory22 Bishop, Fig. 8.2 In general: Every probability distribution can be expressed as a Directed acyclic graph (DAG) Important: No directed cycles!

Conditional independence Bayesian Inference - Introduction to probability theory23

Conditional independence – tail-to- tail path Bayesian Inference - Introduction to probability theory24 Is a independent of b? No!Yes! Bishop, Chapter 8.2

Conditional independence – head- to-tail path Bayesian Inference - Introduction to probability theory25 No!Yes! Bishop, Chapter 8.2 Is a independent of b?

Conditional independence – head- to-head path Bayesian Inference - Introduction to probability theory26 Yes!No! Bishop, Chapter 8.2 Is a independent of b?

Conditional independence – notation Bayesian Inference - Introduction to probability theory27 Bishop, Chapter 8.2

Conditional independence – three basic structures Bayesian Inference - Introduction to probability theory28 Bishop, Chapter 8.2.2

More conventions in graphical notations Bayesian Inference - Introduction to probability theory29 Bishop, Chapter 8 = = Regression modelShort formParameters explicit

More conventions in graphical notations Bayesian Inference - Introduction to probability theory30 Bishop, Chapter 8 Trained on data t n  Complete model used for prediction

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. Bayesian Inference - Introduction to probability theory31 Bishop, Chapter 8.2.2