Introduction to Bayesian Networks Instructor: Dan Geiger על מה המהומה ? נישואים מאושרים בין תורת ההסתברות ותורת הגרפים. הילדים המוצלחים: אלגוריתמים לגילוי תקלות, קודים לגילוי שגיאות, מודלים למערכות מורכבות. שימושים במגוון רחב של תחומים. קורס ממבט אישי למדי על נושא מחקריי לאורך שנים רבות. Web page: www.cs.technion.ac.il/~dang/courseBN http://webcourse.cs.technion.ac.il/236372 Email: dang@cs.technion.ac.il Phone: 829 4339 Office: Taub 616. .

What is it all about ? How to use graphs to represent probability distributions over thousands of random variables ? How to encode conditional independence in directed and undirected graphs ? How to use such representations for efficient computations of the probability of events of interest ? How to learn such models from data ?

Course Information Meetings: Lecture: Mondays 10:30 –12:30 Tutorial: Mondays 13:30 – 14:30 Grade: 40% in 4 question sets. These questions sets are obligatory. Each contains mostly theoretical problems. Submit in pairs before due time (three weeks). 50% Bochan on January 9, based on first 9 weeks including Chanuka, and on understanding Chapter 3 & 4 of Pearl’s book. Obligatory to PASS the test and get a grade, but grade is fully replaceable with a programming project using BNs software. Miluim has to send me an advanced notice two weeks before the Bochan. 10%. Obligatory for a **passing grade**. Attending all lectures and recitation classes except one. (In very special circumstances 2% per missing lecture or recitation). Prerequisites: Data structure 1 (cs234218) Algorithms 1 (cs234247) Probability (any course) Information and handouts: http://webcourse.cs.technion.ac.il/236372 http://www.cs.technion.ac.il/~dang/courseBN/ (Only lecture slides)

Relations to Some Other Courses אמור לי מי חבריך ואומר לך מי אתה. Introduction to Artificial Intelligence (cs236501) Introduction to Machine Learning (cs236756) Introduction to Neural Networks (cs236950) Algorithms in Computational Biology (cs236522) Error correcting codes Data mining

Mathematical Foundations Inference with Bayesian Networks Learning Bayesian Networks Applications

Homework HMW #1. Read Chapter 3.1 & 3.2.1. Answer Questions 3.1, 3.2, Prove Eq 3.5b, and fully expand the proof details of Theorem 2. Submit in pairs no later than noon of 14/11/12 (Two weeks). Pearl’s book contains all the notations that I happen not to define in these slides – consult it often – it is also a very unique and interesting classic text book.

The Traditional View of Probability in Text Books Probability theory provides the impression that we need to literally represent a joint distribution explicitly as P(x1,…,xn) on all propositions and their combinations. It is consistent and exhaustive. This representation stands in sharp contrast to human reasoning: It requires exponential computations to compute marginal probabilities like P(x1) or conditionals like P(x1|x2). Humans judge pairwise conditionals swiftly while conjunctions are judged hesitantly. Numerical calculations do not reflect simple reasoning tasks.

The Qualitative Notion of Dependence Marginal independence is defined numerically as P(x,y)=P(x) P(y). The truth of this equation is hard to judge by humans, while judging whether X and Y are dependent is often easy. “Burglary within a day” and “nuclear war within five years” Likewise, the three place relationship (X influences Y, given Z) is judjed easily. For example: X = time of last pickup from a bus station and Y= time for next bus are dependent, but are conditionally independent given Z= whereabouts of the next bus

The notions of relevance and dependence are far more basic than the numerical values. In a resonating system it should be asserted once and not be sensitive to numerical changes. Acquisition of new facts may destroy existing dependencies as well as creating new once. Learning child’s age Z destroys the dependency between height X and reading ability Y. Learning symptoms Z of a patient creates dependencies between the diseases X and Y that could account for it. Probability theory provides in principle such a device via P(X | Y, K) = P(X |K) But can we model the dynamics of dependency changes based on logic, without reference to numerical quantities ?

Definition of Marginal Independence Definition: IP(X,Y) iff for all xDX and yDy Pr(X=x, Y=y) = Pr(X=x) Pr(Y=y) Comments: Each Variable X has a domain DX with value (or state) x in DX. We often abbreviate via P(x, y) = P(x) P(y). When Y is the emptyset, we get Pr(X=x) = Pr(X=x). Alternative notations to IP(X,Y) such as: I(X,Y) or XY Next few slides on properties of marginal independence are based on “Axioms and algorithms for inferences involving probabilistic independence.”

Properties of Marginal Independence   Proof (Soundness). Trivial independence and Symmetry follow from the definition. Decomposition: Given P(x,y,w) = P(x) P(y,w), simply sum over w on both sides of the given equation. Mixing: Given: P(x,y) = P(x) P(y) and P(x,y,w) = P(x,y) P(w). Hence, P(x,y,w) = P(x) P(y) P(w) = P(x) P(y,w).

Properties of Marginal Independence Are there more such independent properties of independence ? No. There are none ! No more independent axioms of the form 1 & … & n   where each statement  stands for an independence statement. 2. We use the symbol  for a set of independence statements. In this notation:  is derivable from  via these properties if and only if  is entailed by  (i.e.,  holds in all probability distributions that satisfy ). 3. For every set  and a statement  not derivable from , there exists a probability distribution P that satisfies  and not .

Properties of Marginal Independence Can we use these properties to infer a new independence statements  from a set of given independence statements  in polynomial time ? YES. The “membership algorithm” and completeness proof in Recitation class today (Paper P2). Comment. The question “does  entail ” could in principle be undecidable, drops to being decidable via a complete set of axioms, and then drops to polynomial with the above claim.

Definitions of Conditional Independence Ip(X,Z,Y) if and only if whenever P(Y=y,Z=z) >0 (3.1) Pr(X=x | Y=y , Z=z) = Pr(X=x |Z=z)

Properties of Conditional Independence  

Important Properties of Conditional Independence Recall there are some more notations.

Graphical Interpretation

Use graphs and not pure logic Variables represented by nodes and dependencies by edges. Common in our language: “threads of thoughts”, “lines of reasoning”, “connected ideas”, “far-fetched arguments”. Still, capturing the essence of dependence is not an easy task. When modeling causation, association, and relevance, it is hard to distinguish between direct and indirect neighbors. If we just connect “dependent variables” we will get cliques.

M = { IG(M1,{F1,F2},M2), IG(F1,{M1,M2},F2) + symmetry } Markov Network Example Other semantics. The color of each pixel depends on its neighbor. M = { IG(M1,{F1,F2},M2), IG(F1,{M1,M2},F2) + symmetry }

Markov Networks 1. Define for each (maximal) clique Ci a non-negative function g(Ci) called the compatibility function. 2. Take the product i g(Ci) over all cliques. 3. Define P(X1,…,Xn) = K· i g(Ci) where K is a normalizing factor (inverse sum of the product). Mp = { IP(M1,{F1,F2},M2), IP(F1,{M1,M2},F2) + symmetry } SOUNDNESS: IG(X, Z,Y)  IP(X, Z,Y) For all, sets of nodes X,Y,Z

The two males and females example P(M1,M2,F1,F2)= K g(M1,F1) g(M1,F2) g(M2,F1) g(M2,F2) Where K is a normalizing constant (to 1).

Bayesian Networks (Directed Acyclic Graphical Models) Coin1 Bell Coin2 A situation of a bell that rings whenever the outcome of two coins are equal can not be well represented by undirected graphical models. A clique will be formed because of induced dependency of the two coins given the bell.

Bayesian Networks (BNs) Examples of models for diseases & symptoms & risk factors One variable for all diseases (values are diseases) One variable per disease (values are True/False) Naïve Bayesian Networks versus Bipartite BNs

Natural Direction of Information Even simple tasks are not natural to phrase in terms of joint distributions. Consider the example of a hypothesis H=h indicating a rare disease and A set of symptoms such as fever, blood preasure, pain, etc, marked by E1=e1, E2=e2, E3=e3 or in short E=e. We need P(h | e). Can we naturally assume P(H,E) is given and compute P(h | e) ?