Introduction to probability theory and graphical models

1. Introductiontoprobabilitytheoryandgraphicalmodels Translational Neuroimaging Seminar on BayesianInference Spring 2013 Jakob Heinzle Translational Neuromodeling Unit (TNU) Institute for Biomedical Engineering (IBT) University and ETH Zürich

2. Literatureand References • Literature: • Bishop (Chapters 1.2, 1.3, 8.1, 8.2) • MacKay (Chapter 2) • Barber (Chapters 1, 2, 3, 4) • Manyimages in thislecturearetakenfromtheabovereferences. Bayesian Inference - Introduction to probability theory

3. Probabilitydistribution A probability P(x=true) isdefined on a sample space (domain) anddefines (foreverypossible) event in the sample spacethecertaintyofittooccur. Sample space: dom(X)={0,1} , Probabilitiessumtoone. Bishop, Fig. 1.11 Bayesian Inference - Introduction to probability theory

4. Probabilitytheory: Basic rules . Sumrule* - P(X) is also calledthe marginal distribution Productrule - * Accordingto Bishop Bayesian Inference - Introduction to probability theory

5. Conditionaland marginal probability Bayesian Inference - Introduction to probability theory

6. Conditionaland marginal probability Bishop, Fig. 1.11 Bayesian Inference - Introduction to probability theory

7. Independent variables Questionforlater: WhatdoesthismeanforBayes? Bayesian Inference - Introduction to probability theory

8. Probabilitytheory: Bayes’ theorem isderivedfromtheproductrule Bayesian Inference - Introduction to probability theory

9. RephrasingandnamingofBayes’ rule D: data, q: parameters, H: hypothesisweputintothemodel. MacKay Bayesian Inference - Introduction to probability theory

10. Example: Bishop Fig. 1.9 Box (B): blue (b) orred (r) Fruit (F): apple (a) or orange (o) p(B=r) = 0.4, p(B=b) = 0.6. Whatistheprobabilityofhaving a red box ifonehasdrawn an orange? Bishop, Fig. 1.9 Bayesian Inference - Introduction to probability theory

11. Probabilitydensity Bayesian Inference - Introduction to probability theory

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

13. Cumulativedistribution Short example: Howtousethecumulativedistribution totransform a uniform distribution! Bayesian Inference - Introduction to probability theory

14. Marginal densities p Integration insteadofsumming Bayesian Inference - Introduction to probability theory

15. Twoviewson probability • Probabilitycan … • … describethefrequencyofoutcomes in randomexperiments classicalinterpretation. • … describethedegreeof belief about a particularevent Bayesianviewpointorsubjectiveinterpretationofprobability. MacKay, Chapter 2 Bayesian Inference - Introduction to probability theory

16. Expectationof a function Or Bayesian Inference - Introduction to probability theory

17. Graphicalmodels They provide a simple way to visualize the structure of a probabilistic model and can be used to design and motivate new models. Insights into the properties of the model, including conditional independence properties, can be obtained by inspection of the graph. 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. Graphicalmodelsoverview Directed Graph Undirected Graph Names: nodes (vertices), edges (links), paths, cycles, loops, neighbours Forsummaryofdefinitionssee Barber, Chapter 2 Bayesian Inference - Introduction to probability theory

19. Graphicalmodelsoverview Barber, Introduction Bayesian Inference - Introduction to probability theory

20. Graphicalmodels Bishop, Fig. 8.1 Bayesian Inference - Introduction to probability theory

21. Graphicalmodels: parentsandchildren Node a is a parentofnode b, node b is a childofnode a. Bishop, Fig. 8.1 Bayesian Inference - Introduction to probability theory

22. Belief networks = Bayesian belief networks = BayesianNetworks In general: • Every probabilitydistribution • canbeexpressedas a • Directedacyclicgraph (DAG) • Important: Nodirectedcycles! Bishop, Fig. 8.2 Bayesian Inference - Introduction to probability theory

23. Conditionalindependence A variable aisconditionallyindependentofbgivenc, if In bayesiannetworksconditionalindependencecanbe testedbyfollowingsome simple rules Bayesian Inference - Introduction to probability theory

24. Conditionalindependence – tail-to-tailpath Is a independentof b? No! Yes! Bishop, Chapter 8.2 Bayesian Inference - Introduction to probability theory

25. Conditionalindependence – head-to-tailpath Is a independentof b? No! Yes! Bishop, Chapter 8.2 Bayesian Inference - Introduction to probability theory

26. Conditionalindependence – head-to-headpath Is a independentof b? Yes! No! Bishop, Chapter 8.2 Bayesian Inference - Introduction to probability theory

27. Conditionalindependence – notation Bishop, Chapter 8.2 Bayesian Inference - Introduction to probability theory

28. Conditionalindependence – threebasicstructures Bishop, Chapter 8.2.2 Bayesian Inference - Introduction to probability theory

29. More conventions in graphicalnotations Regression model Short form Parameters explicit = = Bishop, Chapter 8 Bayesian Inference - Introduction to probability theory

30. More conventions in graphicalnotations Completemodelused forprediction Trained on datatn  Bishop, Chapter 8 Bayesian Inference - Introduction to probability theory

31. Summary – thingstoremember • ProbabilitiesandhowtocomputewiththeProductrule, Bayes’ Rule, Sumrule • Probabilitydensities PDF, CDF • Conditionaland Marginal distributions • Basic conceptsofgraphicalmodels  Directed vs. Undirected, nodesandedges, parentsandchildren. • Conditionalindependence in graphsandhowto check it. Bishop, Chapter 8.2.2 Bayesian Inference - Introduction to probability theory