Bayesian Networks. Course: CS40022 Instructor: Dr. Pallab Dasgupta
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1 Bayesan Networks Course: CS40022 Instructor: Dr. Pallab Dasgupta Department of Computer Scence & Engneerng Indan Insttute of Technology Kharagpur
2 Example Burglar alarm at home Farly relable at detectng a burglary Responds at tmes to mnor earthquakes Two neghbors, on hearng alarm, calls polce John always calls when he hears the alarm, but sometmes confuses the telephone rngng wth the alarm and calls then, too. Mary lkes loud musc and sometmes msses the alarm altogether
3 Belef Network Example Burglary P(B Earthquake B E P(A Alarm T T T F F T 0.29 A P(J F F JohnCalls T F MaryCalls A T P(M 0.70 F 0.01
4 The jont probablty dstrbuton A generc entry n the jont probablty dstrbuton P(x 1,, x n s gven by: P(x 1,...,x n = n = 1 P(x Parents(
5 The jont probablty dstrbuton Probablty of the event that the alarm has sounded but nether a burglary nor an earthquake has occurred, and both Mary and John call: P(J M A B E = P(J A P(M A P(A B E P( B P( E = =
6 Condtonal ndependence P( x = = = 1,..., P( x P( x n = 1 x n n n P( x x x n 1 n 1 x,...,,..., 1 x x,..., 1 1 P( x P( x x 1 n 1 n 1,..., x...p( x x n 2 2 1,..., The belef network represents condtonal ndependence: P(,..., = 1 P( x 1 x 1 P( x Parents( 1
7 Incremental Network Constructon 1. Choose the set of relevant varables that descrbe the doman 2. Choose an orderng for the varables (very( mportant step 3. Whle there are varables left: a Pck a varable and add a node for t b Set Parents( to some mnmal set of exstng nodes such that the condtonal ndependence property s satsfed c Defne the condtonal prob table for
8 Condtonal Independence Relatons If every undrected path from a node n to a node n Y s d-separated d by a gven set of evdence nodes E, then and Y are condtonally ndependent gven E. A set of nodes E d-separates two sets of nodes and Y f every undrected path from a node n to a node n Y s blocked gven E.
9 Condtonal Independence Relatons A path s blocked gven a set of nodes E f there s a node Z on the path for whch one of three condtons holds: 1. Z s n E and Z has one arrow on the path leadng n and one arrow out 2. Z s n E and Z has both path arrows leadng out 3. Nether Z nor any descendant of Z s n E, and both path arrows lead n to Z
10 Cond Independence n belef networks Battery Rado Ignton Petrol Starts Whether there s petrol and whether the rado plays are ndependent gven evdence about whether the gnton takes place Petrol and Rado are ndependent f t s known whether the battery works
11 Cond Independence n belef networks Battery Rado Ignton Petrol Starts Petrol and Rado are ndependent gven no evdence at all. But they are dependent gven evdence about whether the car starts. If the car does not start, then the rado playng s ncreased evdence that we are out of petrol.
12 Inferences usng belef networks Dagnostc nferences (from effects to causes Gven that JohnCalls,, nfer that P(Burglary JohnCalls = Causal nferences (from causes to effects Gven Burglary, nfer that P(JohnCalls Burglary = 0.86 and P(MaryCalls Burglary = 0.67
13 Inferences usng belef networks Intercausal nferences (between causes of a common effect Gven Alarm, we have P(Burglary Alarm = If we add evdence that Earthquake s true, then P(Burglary Alarm Earthquake goes down to Mxed nferences Settng the effect JohnCalls to true and the cause Earthquake to false gves P(Alarm JohnCalls Earthquake = 0.003
14 The four patterns Q E Q E E Q E Q E Dagnostc Causal InterCausal Mxed
15 Answerng queres We consder cases where the belef network s a poly-tree There s at most one undrected path between any two nodes
16 Answerng queres E U 1 U m Z 1j Z nj E Y 1 Y n
17 Answerng queres U = U 1 U m are parents of node Y = Y 1 Y n are chldren of node s the query varable E s a set of evdence varables The am s to compute P( E
18 Defntons E s the causal support for The evdence varables above above that are connected to through ts parents E s the evdental support for The evdence varables below below that are connected to through ts chldren U \ refers to all the evdence connected to node U except va the path from E U Y \ refers to all the evdence connected to node Y through ts parents for E Y
19 The computaton of P( E = P( = E,E,E P( Snce d-separates E from E, we can use condtonal ndependence to smplfy the frst term n the numerator We can treat the denomnator as a constant E E P( E = α P( E
20 The computaton of P( E We consder all possble confguratons of the parents of and how lkely they are gven E. Let U be the vector of parents U 1,, U m, and let u be an assgnment of values to them. P( E = P( u u,e P(u E
21 The computaton of P( E P( E = P( u u,e P(u U d-separates from E, so the frst term smplfes to P( u We can smplfy the second term by notng E d-separates each U from the others, the probablty of a conjuncton of ndependent varables s equal to the product of ther ndvdual probabltes P( E = P( u u P(u E E
22 The computaton of P( E P( E = P( u u P(u The last term can be smplfed by parttonng E nto E U1\,, E Um\ and notng that E U\ d-separates U from all the other evdence n E = P( u u P(u E E U\ P( u s a lookup n the cond prob table of P(u E U\ s a recursve (smaller sub-problem
23 The computaton of Let Z be the parents of Y other than, and let z be an assgnment of values to the parents The evdence n each Y box s condtonally ndependent of the others gven = Y\
24 The computaton of = Y\ Averagng over Y and z yelds: = y z Y\,y,z P (y,z
25 The computaton of = y z Y\,y,z P (y,z Breakng E Y\ nto the two ndependent components E Y and E Y\ = y z Y,y,z Y\,y,z P(y,z
26 The computaton of = y z Y,y,z Y\,y,z P(y,z E Y s ndependent of and z gven y, and E Y\ s ndependent of and y = y Y Y\ y z z P(y,z
27 The computaton of = y Y Y\ y z z P(y,z Apply Bayes rule to Y\ z : = y P(z EY \ P(z Y y z Y\ P(y,z
28 The computaton of = y P(z EY \ P(z Y y z Y\ P(y,z Rewrtng the conjuncton of Y and z : = z P(z y EY\ P(z Y y Y\ P(y,z P(z
29 The computaton of z = P(z y EY\ P(z Y y Y\ P(y,z P(z P(z = P(z because Z and are d-separated. Also Y\ s a constant y = Y y z βp(z E Y\ P(y,z
30 The computaton of = y Y y z βp(z E Y\ P(y,z The parents of Y (the Z j are ndependent of each other. We also combne the β nto one sngle β
31 The computaton of = β y P(y,z Y y z j P(z j E Z j \Y Y y s a recursve nstance of P(y, z s a cond prob table entry for Y P(z j E Zj\Y s a recursve sub-nstance of the P( E calculaton
32 Inference n multply connected belef networks Clusterng methods Transform the net nto a probablstcally equvalent (but topologcally dfferent poly- tree by mergng offendng nodes Condtonng methods Instantate varables to defnte values, and then evaluate a poly-tree for each possble nstantaton
33 Inference n multply connected belef networks Stochastc smulaton methods Use the network to generate a large number of concrete models of the doman that are consstent wth the network dstrbuton. They gve an approxmaton of the exact evaluaton.
34 Default reasonng Some conclusons are made by default unless a counter-evdence evdence s obtaned Non-monotonc reasonng Ponts to ponder Whats the semantc status of default rules? What happens when the evdence matches the premses of two default rules wth conflctng conclusons? If a belef s retracted later, how can a system keep track of whch conclusons need to be retracted as a consequence?
35 Issues n Rule-based methods for Uncertan Reasonng Localty In logcal reasonng systems, f we have A B, then we can conclude B gven evdence A, wthout worryng about any other rules.. In probablstc systems, we need to consder all avalable evdence.
36 Issues n Rule-based methods for Uncertan Reasonng Detachment Once a logcal proof s found for proposton B, we can use t regardless of how t was derved (t( can be detached from ts justfcaton. In probablstc reasonng, the source of the evdence s mportant for subsequent reasonng.
37 Issues n Rule-based methods for Uncertan Reasonng Truth functonalty In logc, the truth of complex sentences can be computed from the truth of the components. Probablty combnaton does not work ths way, except under strong ndependence assumptons. A famous example of a truth functonal system for uncertan reasonng s the certanty factors model,, developed for the Mycn medcal dagnostc program
38 Dempster-Shafer Theory Desgned to deal wth the dstncton between uncertanty and gnorance. We use a belef functon Bel( probablty that the evdence supports the proposton When we do not have any evdence about, we assgn Bel( = 0 as well as Bel( = 0
39 Dempster-Shafer Theory For example, f we do not know whether a con s far, then: Bel( ( Heads = Bel( Heads = 0 If we are gven that the con s far wth 90% certanty, then: Bel( ( Heads = = 0.45 Bel( Heads = = 0.45 Note that we stll have a gap of 0.1 that s not accounted for by the evdence
40 Fuzzy Logc Fuzzy set theory s a means of specfyng how well an object satsfes a vague descrpton Truth s a value between 0 and 1 Uncertanty stems from lack of evdence, but gven the dmensons of a man concludng whether he s fat has no uncertanty nvolved
41 Fuzzy Logc The rules for evaluatng the fuzzy truth, T, of a complex sentence are T(A B = mn( T(A, T(B T(A B = max( T(A, T(B T( A = 1 T(A
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