Artificial Intelligence Bayesian Networks

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Jayson Jessie Strickland
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1 Artfcal Intellgence Bayesan Networks Adapted from sldes by Tm Fnn and Mare desjardns. Some materal borrowed from Lse Getoor. 1

2 Outlne Bayesan networks Network structure Condtonal probablty tables Condtonal ndependence Inference n Bayesan networks Exact nference Approxmate nference 2

3 Bayesan Belef Networks (BNs) Defnton: BN = (DAG, CPD) DAG: drected acyclc graph (BN s structure) Nodes: random varables (typcally bnary or dscrete, but methods also exst to handle contnuous varables) Arcs: ndcate probablstc dependences between nodes (lack of lnk sgnfes condtonal ndependence) CPD: condtonal probablty dstrbuton (BN s parameters) Condtonal probabltes at each node, usually stored as a table (condtonal probablty table, or CPT) P ( x π ) where π s the set of all parent nodes of x Root nodes are a specal case no parents, so just use prors n CPD: π = so P ( x π ) = P( x ), 3

4 Example BN P(A) = a P(B A) = 0.3 P(B A) = b c P(C A) = 0.2 P(C A) = d e P(D B,C) = 0.1 P (D B, C) = 0.01 P(D B,C) = 0.01 P (D B, C) = P(E C) = 0.4 P(E C) = Note that we only specfy P(A) etc., not P( A), snce they have to add to one 4

5 Condtonal ndependence and channg Condtonal ndependence assumpton P( x π, q) = P( x π ) π where q s any set of varables q (nodes) other than x and ts successors π x blocks nfluence of other nodes on x and ts successors (q nfluences x only through varables n π ) Wth ths assumpton, the complete jont probablty dstrbuton of all varables n the network can be represented by (recovered from) local CPDs by channg these CPDs: n P( x,..., xn) = Π = 1P( x π ) 1 5

6 Channg: Example a b c d e Computng the jont probablty for all varables s easy: P(a, b, c, d, e) = P(e a, b, c, d) P(a, b, c, d) by the product rule = P(e c) P(a, b, c, d) by cond. ndep. assumpton = P(e c) P(d a, b, c) P(a, b, c) = P(e c) P(d b, c) P(c a, b) P(a, b) = P(e c) P(d b, c) P(c a) P(b a) P(a) 6

7 Topologcal semantcs A node s condtonally ndependent of ts nondescendants gven ts parents A node s condtonally ndependent of all other nodes n the network gven ts parents, chldren, and chldren s parents (also known as ts Markov blanket) The method called d-separaton can be appled to decde whether a set of nodes X s ndependent of another set Y, gven a thrd set Z 7

8 Inference tasks Smple queres: Computer posteror margnal P(X E=e) E.g., P(NoGas Gauge=empty, Lghts=on, Starts=false) Conjunctve queres: P(X, X j E=e) = P(X e=e) P(X j X, E=e) Optmal decsons: Decson networks nclude utlty nformaton; probablstc nference s requred to fnd P (outcome acton, evdence) Value of nformaton: Whch evdence should we seek next? Senstvty analyss: Whch probablty values are most crtcal? Explanaton: Why do I need a new starter motor? 8

9 Approaches to nference Exact nference Enumeraton Belef propagaton n polytrees Varable elmnaton Clusterng / jon tree algorthms Approxmate nference Stochastc smulaton / samplng methods Markov chan Monte Carlo methods Genetc algorthms Neural networks Smulated annealng Mean feld theory 9

10 Drect nference wth BNs Instead of computng the jont, suppose we just want the probablty for one varable Exact methods of computaton: Enumeraton Varable elmnaton Jon trees: get the probabltes assocated wth every query varable 10

11 Inference by enumeraton Add all of the terms (atomc event probabltes) from the full jont dstrbuton If E are the evdence (observed) varables and Y are the other (unobserved) varables, then: P(X e) = α P(X, E) = α P(X, E, Y) Each P(X, E, Y) term can be computed usng the chan rule Computatonally expensve! 11

12 Example: Enumeraton a b c d e P(x ) = Σ π P(x π ) P(π ) Suppose we want P(D=true), and only the value of E s gven as true P (d e) = α Σ ABC P(a, b, c, d, e) = α Σ ABC P(a) P(b a) P(c a) P(d b,c) P(e c) Wth smple teraton to compute ths expresson, there s gong to be a lot of repetton (e.g., P(e c) has to be recomputed every tme we terate over C=true) 12

13 Exercse: Enumeraton p(smart)=.8 smart study p(study)=.6 prepared far p(far)=.9 pass smart smart p(pass ) prep prep prep prep far far p(prep ) smart smart study.9.7 study.5.1 Query: What s the probablty that a student studed, gven that they pass the exam? 13

14 Summary Bayes nets Structure Parameters Condtonal ndependence Channg BN nference Enumeraton Varable elmnaton Samplng methods 14

Hidden Markov Models

Hidden Markov Models CM229S: Machne Learnng for Bonformatcs Lecture 12-05/05/2016 Hdden Markov Models Lecturer: Srram Sankararaman Scrbe: Akshay Dattatray Shnde Edted by: TBD 1 Introducton For a drected graph G we can wrte