15-381/781 Bayesian Nets & Probabilistic Inference

Transcription

1 15-381/781 Bayesian Nets & Prbabilistic Inference Emma Brunskill (this time) Ariel Prcaccia With thanks t Dan Klein (Berkeley), Percy Liang (Stanfrd) and Past Instructrs fr sme slide cntent, and Russell & Nrvig

2 What Yu Shuld Knw Define prbabilistic inference Hw t define a Bayes Net given a real example Hw jint can be used t answer any query Cmplexity f exact inference Apprximatin inference (direct, likelihd, Gibbs) Be able t implement and run algrithm Cmpare benefits and limitatins f each 2

3 Bayesian Netwrk Cmpact representatin f the jint distributin Cnditinal independence relatinships explicit Each var cnditinally independent f all its nndescendants in the graph given the value f its parents 3

4 Jint Distributin Ex. Variables: Cludy, Sprinkler, Rain, Wet Grass Dmain f each variable: 2 (true r false) Jint encdes prbability f all cmbs f variables & values P(Cludy=false & Sprinkler = true & Rain = false & WetGrass = True) +c +s +r +w.01 +c +s +r -w.01 +c +s -r +w.05 +c +s -r -w.1 +c -s +r +w # +c -s +r -w # +c -s -r +w # +c -s -r -w # -c +s +r +w # -c +s +r -w # -c +s -r +w # -c +s -r -w # -c -s +r +w # -c -s +r -w # -c -s -r +w # -c -s -r -w # 4

5 Jint as Prduct f Cnditinals (Chain rule) +c +s +r +w.01 +c +s +r -w.01 +c +s -r +w.05 +c +s -r -w.1 +c -s +r +w # +c -s +r -w # +c -s -r +w # +c -s -r -w # -c +s +r +w # -c +s +r -w # -c +s -r +w # -c +s -r -w # -c -s +r +w # -c -s +r -w # -c -s -r +w # -c -s -r -w # = P(WetGrass Cludy,Sprinkler,Rain)* P(Rain Cludy,Sprinkler)* P(Sprinkler Cludy)* P(Cludy) 5

6 Jint as Prduct f Cnditinals +c +s +r +w.01 +c +s +r -w.01 +c +s -r +w.05 +c +s -r -w.1 +c -s +r +w # +c -s +r -w # +c -s -r +w # +c -s -r -w # -c +s +r +w # -c +s +r -w # = P(WetGrass Cludy,Sprinkler,Rain)* P(Rain Cludy,Sprinkler)* P(Sprinkler Cludy)* P(Cludy) Cludy -c +s -r +w # -c +s -r -w # -c -s +r +w # -c -s +r -w # Sprinkler Rain -c -s -r +w # -c -s -r -w # Wet Grass but there may be additinal cnditinal independencies 6

7 What if sme variables are cnditinally indep? Cludy Cludy Sprinkler Rain Sprinkler Rain Wet Grass Wet Grass Explicitly shws any cnditinal independencies 7

8 Cnditinal Independencies +c +s +r +w.01 +c c 0.5 +c +s +r -w.01 +c +s -r +w.05 Cludy +c +s -r -w.1 +c -s +r +w # +c -s +r -w # +c -s -r +w # +c -s -r -w # -c +s +r +w # -c +s +r -w # -c +s -r +w # -c +s -r -w # -c -s +r +w # -c -s +r -w # -c -s -r +w # -c -s -r -w # à +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Wet Grass Rain +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w 1.0 +c +r.8 +c - r.2 - c +r.2 - c - r.8 8

9 Bayesian Netwrk Cmpact representatin f the jint distributin Cnditinal independence relatinships explicit Still represents jint s can be used t answer any prbabilistic query 9

10 Prbabilistic Inference Cmpute prbability f a query variable (r variables) taking n a value (r set f values) given sme evidence Pr[Q E 1 =e 1,...,E k =e k ] 10

11 Using the Jint T Answer Queries Jint distributin is sufficient t answer any prbabilistic inference questin invlving variables described in jint Can take Bayes Net, cnstruct full jint, and then lk up entries where evidence variables take n specified values 11

12 But Cnstructing Jint Expensive & Exact Inference is NP-Hard 12

13 Sln: Apprximate Inference Use samples t apprximate psterir distributin Pr[Q E 1 =e 1,...,E k =e k ] Last time Tday Direct sampling Likelihd weighting Gibbs 13

14 Pll: Which Algrithm? Evidence: Cludy=+c, Rain=+r Query variable: Sprinkler P(Sprinkler Cludy=+c,Rain=+r) Samples Cludy +c,+s,+r,-w +c,-s,-r,-w Sprinkler Rain +c,+s,-r,+w +c,-s,+r,-w What algrithm culd ve generated these samples? 1) Direct sampling Wet Grass 2) Likelihd weighting 3) Bth 4) N clue 14

15 Direct Sampling Recap Algrithm: 1. Create a tplgical rder f the variables in the Bayes Net 15

16 Tplgical Order Any rdering in directed acyclic graph where a nde can nly appear after all f its ancestrs in the graph Cludy E.g. Cludy, Sprinkler, Rain, WetGrass Sprinkler Rain Cludy, Rain, Sprinkler, WetGrass Wet Grass 16

17 Direct Sampling Recap Algrithm: 1. Create a tplgical rder f the variables in the Bayes Net 2. Sample each variable cnditined n the values f its parents 3. Use samples which match evidence variable values t estimate prbability f query variable e.g. P(Sprinkler=+s Cludy=+c,Rain=+r) ~ # samples with +s,+c, +r / # samples with +c, +r Cnsistent in limit f infinite samples Inefficient (why?) 17

18 Cnsistency In the limit f infinite samples, estimated Pr[Q E 1 =e 1,...,E k =e k ] will cnverge t true psterir prbability Desirable prperty (therwise always have sme errr) 18

19 Likelihd Weighting Recap 1. Create array TtalWeights 1. Initialize value f each array element t 0 2. Fr j=1:n 1. w tmp = 1 2. Set evidence variables in sample z=<z 1, z n > t bserved values 3. Fr each variable z i in tplgical rder 1. If x i is an evidence variable 1. w tmp = w tmp *P(Z i = e i Parents(Z) = x(parents(z i ))) 2. Else 1. Sample x i cnditined n the values f its parents 4. Update weight f resulting sample 1. TtalWeights[z]=TtalWeights[z]+w tmp 3. Use weights t cmpute prbability f query variable P(Sprinkler=+s Cludy=+c,Rain=+r) ~ Sum c,r,w TtalWeight(+s,c,r,w)/Sum s,c,r,w TtalWeight(s,c,r,w) 19

20 LW Cnsistency Prbability f getting a sample (z,e) where z is a set f values fr the nn-evidence variables and e is the vals f evidence vars Sampling distributin fr a weighted sample (WS) Is this the true psterir distributin P(z e)? N, why? Desn t cnsider evidence that is nt an ancestr Weights fix this! 20

21 Weighted Prbability Samples each nn-evidence variable z accrding t Weight f a sample is Weighted prbability f a sample is Frm chain rule & cnditinal indep 21

22 Des Likelihd Weighting Prduce Cnsistent Estimates? Yes P(X = x e) = P(X = x,e) P(e) P(X = x,e) X is query var(s) E is evidence var(s) Y is nn-query vars P(X = x e) P(X = x,e) = N WS (x, y, e)w(x, y, e) y n * S WS (x, y,e)w(x, y, e) y # f samples where query variables=x, nn-query=y, Evidence=e = P(x, y, e) y as # samples n à infinity = P(x, e) 22

23 Example When sampling S and R the evidence W=t is ignred Samples with S=f and R=f althugh evidence rules this ut Weight makes up fr this difference abve weight wuld be 0 If we have 100 samples with R=t and ttal weight 1, and 400 samples with R=f and ttal weight 2, what is estimate f R=t? = 1/ 3 23

24 Limitatins f Likelihd Weighting Pr perfrmance if evidence vars ccur later in rdering Why? Nt being used t influence samples! Yields samples with lw weights 24

25 Markv Chain Mnte Carl Methds Prir methds generate each new sample frm scratch MCMC generate each new sample by making a randm change t preceding sample Can view algrithm as being in a particular state (assignment f values t each variable) 25

26 Review: Markv Blanket Markv blanket Parents Children Children s parents Variable cnditinally independent f all ther ndes given its Markv Blanket 26

27 Gibbs Sampling: Cmpute P(X e) mb(z i ) = Markv Blanket f Z i frm Russell & Nrvig 27

28 Gibbs Sampling Example Want Pr(R S=t,W=t) Nn-evidence variables are C & R Initialize randmly: C= t and R=f Initial state (C,S,R,W)= [t,t,f,t] +c c 0.5 Cludy Sample C given current values f its Markv Blanket +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Rain +c +r.8 +c - r.2 - c +r.2 - c - r.8 Wet Grass +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w

29 Gibbs Sampling Example Want Pr(R S=t,W=t) Nn-evidence variables are C & R Initialize randmly: C= t and R=f Initial state (C,S,R,W)= [t,t,f,t] +c c 0.5 Cludy Sample C given current values f its Markv Blanket Markv blanket is parents, children and children s parents: fr C=S & R Sample C given P(C S=t,R=f) First have t cmpute P(C S=t,R=f) Use exact inference t d this +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Wet Grass Rain +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w 1.0 +c +r.8 +c - r.2 - c +r.2 - c - r.8 29

30 Exercise: cmpute P(C=t S=t,R=f)? Quick refresher Sum rule +c c 0.5 Cludy Prduct/Chain rule +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Rain +c +r.8 +c - r.2 - c +r.2 - c - r.8 Bayes rule Wet Grass +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w

31 Exact Inference Exercise P(C S=t,R=f) What is the prbability P(C=t S=t, R= f)? = P(C=t, S=t, R=f) / (P(S=t,R=f)) Prprtinal t P(C=t, S=t, R=f) Use nrmalizatin trick, & cmpute the abve fr C=t and C=f P(C=t, S=t, R=f) = P(C=t) P(S=t C=t) P (R=f C=t, S=t) prduct rule = P(C=t) P(S=t C=t) P (R=f C=t) (BN independencies) = 0.5 * 0.1 * 0.2 = 0.01 P(C=f, S=t, R=f) = P(C=f) P (S=t C=f) P(R=f C=f) = 0.5 * 0.5 * 0.8 = 0.2 (P(S=t,R=f)) use sum rule = P(C=f, S=t, R=f) + P(C=t, S=t, R=f) P (C = t S=t, R= f) = 0.21 P (C=t S=t, R = f) = 0.01 / 0.21 ~ c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler +c c 0.5 Cludy Wet Grass Rain +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w 1.0 +c +r.8 +c - r.2 - c +r.2 - c - r.8 31

32 Gibbs Sampling Example Want Pr(R S=t,W=t) Nn-evidence variables are C & R Initialize randmly: C= t and R=f Initial state (C,S,R,W)= [t,t,f,t] +c c 0.5 Cludy Sample C given current values f its Markv Blanket Markv blanket is parents, children and children s parents: fr C=S & R Exactly cmpute P(C S=t,R=f) Sample C given P(C S=t,R=f) Get C = f New state (f,t,f,t) +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Wet Grass Rain +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w 1.0 +c +r.8 +c - r.2 - c +r.2 - c - r.8 32

33 Gibbs Sampling Example Want Pr(R S=t,W=t) Initialize nn-evidence variables (C and R) randmly t t and f Initial state (C,S,R,W)= [t,t,f,t] +c c 0.5 Cludy Sample C given current values f its Markv Blanket, p(c S=t,R=f) Suppse result is C=f +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Rain +c +r.8 +c - r.2 - c +r.2 - c - r.8 New state (f,t,f,t) Sample Rain given its MB What is its Markv blanket? Wet Grass +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w

34 Gibbs Sampling Example Want Pr(R S=t,W=t) Initialize nn-evidence variables (C and R) randmly t t and f Initial state (C,S,R,W)= [t,t,f,t] +c c 0.5 Cludy Sample C given current values f its Markv Blanket, p(c S=t,R=f) Suppse result is C=f +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Rain +c +r.8 +c - r.2 - c +r.2 - c - r.8 New state (f,t,f,t) Sample Rain given its MB, p(r C=f,S=t,W=t) Suppse result is R=t New state (f,t,t,t) Wet Grass +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w

35 Pll: Gibbs Sampling Ex. Want Pr(R S=t,W=t) Initialize nn-evidence variables (C and R) randmly t t and f Initial state (C,S,R,W)= [t,t,f,t] +c c 0.5 Cludy Current state (f,t,t,t) What is nt a pssible next state +c +s.1 +c - s.9 - c +s.5 - c - s.5 Sprinkler Rain +c +r.8 +c - r.2 - c +r.2 - c - r.8 1. (f,t,t,t) 2. (t,t,t,t) 3. (f,t,f,t) 4. (f,f,t,t) (incnsistent w/evid) 5. Nt sure Wet Grass +s +r +w.99 +s +r - w.01 +s - r +w.90 +s - r - w.10 - s +r +w.90 - s +r - w.10 - s - r +w 0 - s - r - w

36 Gibbs Sampling This invlve inference! mb(z i ) = Markv Blanket f Z i frm Russell & Nrvig 36

37 Pll Are Gibbs samples independent? 1. Yes 2. N 3. Nt sure mb(z i ) = Markv Blanket f Z i frm Russell & Nrvig 37

38 Markv Blanket Sampling Want t shw P(Z i mb(z i ) ) is same as P(Z i all ther variables) Implies cnditinal independence f Z i frm rest f netwrk given its Markv Blanket Derive equatin fr cmputing P(Z i mb(z i ) ) 38

39 Prbability Given Markv Blanket 39

40 Why is Gibbs Cnsistent? Sampling prcess settles int a statinary distributin where lng-term fractin f time spent in each state is exactly equal t psterir prbability à Implies that if draw enugh samples frm this statinary distributin, will get cnsistent estimate because sampling frm true psterir 40

41 Markv Chain Let P(x à x ) be prbability the sampling prcess makes a transitin frm x (sme state) t x (sme ther state) E.g. (t,t,f,t) à (t,f,f,t) Run sampling fr t steps P t (x) is prbability system is in state x at time t Next state P t+1 (x ) = Sum x P t (x) P(x à x ) 41

42 Statinary Distributin Let P(x à x ) be prbability the prcess makes a transitin frm x t x P t (x) is prbability system is in state x at time t P t+1 (x ) = Sum x P t (x) P(x à x ) Reached statinary distributin if P t+1 (x )=P t (x) Call statinary distributin π Must satisfy π(x ) = \sum_{x} π(x) P(x à x ) fr all x If P(x à x ) is ergdic, exactly ne such π fr any given P(x à x ) 42

43 Detailed Balance Let P(x à x ) be prbability the prcess makes a transitin frm x t x P t (x) is prbability system is in state x at time t Statinary distributin π Satisfies π(x ) = \sum_{x} π(x) P(x à x ) fr all x Detailed balance: inflw = utflw π(x) P(x à x ) = π(x ) P(x à x) fr all x, x π (x')p(x > x') = π (x') P(x' > x) = π (x') x x 43

44 Prf n bard 44

45 Prving Gibbs Samples frm True Psterir General Gibbs: sample the value f a new variable cnditined n all the ther variables Can prve this versin f Gibbs satisfies detailed balance equatin with statinary distributin f P(X e) Then use prir result that sampling cnditined n all variables is equivalent t sampling given Markv Blanket fr Bayes Nets See text fr recap 45

46 Gibbs Sampling Samples are valid nce reach statinary distributin When d we reach statinary distributin? Unclear 46

47 What Yu Shuld Knw Define prbabilistic inference Hw t define a Bayes Net given a real example Hw jint can be used t answer any query Cmplexity f exact inference Apprximatin inference (direct, likelihd, Gibbs) Be able t implement and run algrithm Cmpare benefits and limitatins f each 47