Inference in Bayesian networks

Transcription

1 Inference in ayesian networks Devika uramanian omp 440 Lecture 7 xact inference in ayesian networks Inference y enumeration The variale elimination algorithm c Devika uramanian

2 roailistic inference using ayesian networks single mechanism can account for a wide range of inferences under uncertainty. Diagnostic inference from effects to causes. xample: ausal inference from causes to effects. xample: c Devika uramanian roailistic inference Inter-causal inference etween causes of a common effect. xample: and ven though and are independent the presence of one makes the other unlikely. This phenomenon is called explaining away. annot e captured in logic which is monotonic. c Devika uramanian

3 roailistic inference Mixed inferences comining two or more of the aove. xample: not 0.03 is a simultaneous use of diagnostic and causal inference. c Devika uramanian Inference y enumeration To compute the proaility of variale Q given evidence 1 k we use the rule of conditional independence: Q Q ach of these terms can e computed y summing terms from the full joint distriution. c Devika uramanian

4 4 c Devika uramanian xample / *0.05] * * [ *0.9 ] [ M M M M M M c Devika uramanian The expression tree ottom up computation

5 5 c Devika uramanian xample contd *0.05] * * [ *0.9 ny query can e answered y computing sums of products of conditional proailities from the network. c Devika uramanian Inter-causal inference *0.95] 0.001[0.002*0.95 ] [ / M M

6 6 c Devika uramanian Inter-causal inference / *0.001] 0.999[0.002*0.29 ] [ c Devika uramanian Inter-causal inference / *0.002* *0.002*0.95 / M M

7 7 c Devika uramanian Variale elimination asic inference: a 3-chain riors at T at : T at : a a c c a If there are k values for and what is the complexity of this calculation? c Devika uramanian Variale elimination asic inference: a 3-chain riors at T at : T at : a a a c c tore and do not recompute! a a a c c factor

8 8 c Devika uramanian Variale elimination asic inference: an n-chain takes Onk 2 computation c f c d D d f f c c f a a f a a c c d D d D c a a c c Devika uramanian ingly connected networks ingly connected networks exactly one undirected path etween any two nodes in network X Y n Y 1 Z What is XZY 1 Y n? ausal support for X vidential support for X

9 Multiply connected networks cloudy not 0.2 rain sprinkler 0.1 not 0.5 Wet grass ompute Wet grass using this network. W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian Variale elimination again W f W f 1 2 W W f 1 We exploit the structure of the elief network and reak up the computation into two pieces as aove. We use a form of dynamic programming to rememer values of the inner sum to prevent re-computing it. c Devika uramanian

10 10 c Devika uramanian The variale elimination algorithm To evaluate For i m to 1 do Group the terms in which X i occurs and construct a factor f that sums over X i. f will e indexed y all other variales that occur in those terms. eplace the sum in the original expression y the factor constructed aove j X j j X X X arents X m c Devika uramanian xample then. liminate and first then and liminate first f W W f f W W f W W W

11 omplexity of variale elimination For singly connected networks time and space complexity of inference is linear in the size of the network. For multiply connected networks time and space complexity is exponential in the size of the network in the worst case. xact inference is #-hard. ractically choosing a good order to eliminate variales in makes process tractale. c Devika uramanian lustering Idea: transform network to a singly connected network y comining nodes. not cloudy not not 0.1 not not 0.4 not not not rain sprinkler not0.72 not 0.02 not not 0.18 Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian

12 roperties of clustering xact method for evaluating elief networks that are not singly connected. hoosing good nodes to cluster is difficult; the same issues as in determining good variale ordering. Further Ts at clustered nodes are exponential in size in the numer of nodes clustered there. c Devika uramanian ummary elief networks are a compact encoding of the full joint proaility distriution over n variales that makes conditional independence assumptions etween these variales explicit. We can use elief networks to exactly compute any proaility of interest over the given variales. xact inference is intractale for multiply connected networks. c Devika uramanian

13 pproximate inference Direct sampling ejection sampling Likelihood weighting Markov hain Monte arlo MM c Devika uramanian Direct sampling Idea: generate samples x 1 x n from the joint distriution specified y the network. For each x i draw a sample according to x i parentsx i. D x 1... xn x i arents x The proaility of x 1 x n is simply the numer of times of x 1 x n is generated divided y the total numer of samples. i i c Devika uramanian

14 Direct sampling method hoose a value for the root nodes weighted y their priors. Use Ts for direct descendants of roots given values of root nodes to choose values for them. Do previous step all the way down to the leaves. c Devika uramanian Direct ampling cloudy not 0.2 rain sprinkler 0.1 not 0.5 Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian

15 Direct ampling cloudy not 0.2 rain sprinkler 0.1 not 0.5 Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian Direct ampling cloudy not 0.2 rain sprinkler 0.1 not 0.5 Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian

16 Direct ampling cloudy not 0.2 rain sprinkler 0.1 not 0.5 Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian Direct ampling cloudy not 0.2 rain sprinkler 0.1 not 0.5 Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian

17 xample To compute W in sprinkler network hoose a value for with prior 0.5; assume we pick false. hoose a value for : not 0.5; assume we pick true. hoose a value for : not 0.2; assume we pick false. hoose a value for W drawn according to W not 0.9; assume we pick W true. We have generated the event not not W. epeat this process and calculate the fraction of events in which W is true. c Devika uramanian ejection sampling Used to estimate Xe in ayesian networks. Generate samples from the full joint distriution using direct sampling. liminate all samples that don t match e. stimate Xe as the fraction of samples with X x from the remaining samples. c Devika uramanian

18 ejection sampling example To find ainprinkler true generate 100 samples y direct sampling of the network. ay 27 of them have prinkler true. 8 of the 27 have ain true. o we estimate aove conditional proaility as 8/ c Devika uramanian Likelihood weighting voids inefficiency of rejection sampling y only generating events that are consistent with evidence e. To find Xe the algorithm fixes e and samples X and the remaining variales Y in the network. ach event is weighted y the proaility of its occurrence. c Devika uramanian

19 Likelihood weighting cloudy not 0.2 rain sprinkler 0.1 not 0.5 w1.0 W? Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian Likelihood weighting cloudy not 0.2 rain sprinkler 0.1 not 0.5 w1.0 W? Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian

20 Likelihood weighting cloudy not 0.2 rain sprinkler 0.1 not 0.5 w1.0*0.1 W? Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian Likelihood weighting cloudy not 0.2 rain sprinkler 0.1 not 0.5 w1.0*0.1 W? Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian

21 Likelihood weighting cloudy not 0.2 rain sprinkler 0.1 not 0.5 w1.0*0.1*0.99 W? Wet grass W0.99 Wnot 0.90 Wnot 0.90 Wnot not 0.00 c Devika uramanian Likelihood weighting example ainpriklerwetgrass? w 1.0 ample from loudy <0.50.5>; suppose it returns loudy true. prinkler is an evidence variale with value true. We set w w x prinklerloudy 0.1 ample from ainloudy <0.80.2>. Let say we get ain true. WetGrass is an evidence variale so we set w w x WetGrassprinklerain0.099 We have generated the event loudyprinklerainwetgrass with weight c Devika uramanian

22 Why likelihood weighting works ampling distriution W is W z e zi arents Zi i Likelihood weighting wze w z e ei parents i i Note that W ze x wze ze c Devika uramanian Markov processes: a quick intro We are interested in predicting weather and for the purposes of this example weather can take on one of three values: {sunny rainycloudy}. The weather on a given day is dependent only on the weather on the previous day. wt wt 1... w1 wt wt 1 This is the Markov property. c Devika uramanian

23 Markov process example We have knowledge of the transition proailities etween the various states: qss. q is called the transition kernel. s r c s r c c Devika uramanian rediction uppose day 1 is rainy. We will represent this as a vector of proailities over the three values. π 1 [0 1 0]; How do we predict the weather for day 2 given pi1 and the transition kernel q? From the transition kernel we can see that the proaility of day 2 eing sunny is.5 and that the proailities for eing cloudy or rainy are 0.25 each. π 2 π 1* q [ ]; c Devika uramanian

24 rediction contd. We can calculate the distriution of weather at time t1 given the distriution for time t. π t 1 π t* q π t-1* q* q π 1* q t c Devika uramanian rediction What s the weather going to e like on the 3 rd 5 th 7 th and 9 th days? π 3 π 1* q π 5 π 1* q π 7 π 1* q [0.375 [ [ π 9 π 1* q [ ] π t [ ] for all t ] ] ] c Devika uramanian

25 new start state Let the weather on day 1 e sunny. How does the distriution of weather change with time? 1 [1 0 0] π π 3 π 1* q π 5 π 1* q π 7 π 1* q [ [ [ π 9 π 1* q [ ] π t [ ] for all t ] ] ] c Devika uramanian tationary distriution Independent of the start state this Markov process converges to a stationary distriution [ ] in the limit. The stationary distriution p* is the solution to the equation p* q p*. c Devika uramanian

26 ampling from a Markov chain We can sample from the discrete distriution [ ] as follows tart the Markov chain at a random state at time 1. Use the transition kernel q to generate a state at time t1 given the value of the state at time t. Keep repeating aove step to generate a long chain. fter eliminating an initial prefix of the chain urn-in use the rest as samples from the aove distriution. c Devika uramanian When does this work? s t infinity a Markov chain converges to a unique stationary distriution if it is Irreducile every state in the state space is reachale from every other state. Has no periodic cycles The stationary distriution pi satisfies π s ' π s q s s' s uch a Markov chain is called ergodic and the aove theorem is called the ergodicity theorem. c Devika uramanian

27 Detailed alance equation π s q s s' π s' q s' s The detailed alance equation implies stationarity. π q s s' π s' q s' s s s π s' s c Devika uramanian Designing an MM sampler Need to find qss that satisfies the detailed alance equation with respect to the proaility of interest say xe. c Devika uramanian

28 Gis sampler ach variale is sampled conditionally on the current values of all the other variales. xample: sampling from a 2d distriution y sampling first coordinate from a 1d conditional distriution and then sampling the second coordinate from another 1d conditional distriution. c Devika uramanian Gis sampling and ayesian networks Let X i e the variale to e sampled and let Y e all the hidden variales other than X i. Let their current values e x i and y respectively. We will sample a new value for X i conditioned on all the other variales including the evidence. qxx qx i yx i yx i ye c Devika uramanian

29 29 c Devika uramanian Gis sampling works! ' ' ' ' ' ' ' x x q x e y x e y x e y x e y e y x e y x e y x e y x e x x x q x i i i i i i i π π Detailed alance equation is satisfied with xe as the stationary distriution. c Devika uramanian Inference y MM Instead of generating events from scratch MM generates events y making a random change to the preceding event. tart the network at a random state assignment of values to all its nodes. Generate the next state y randomly sampling a value for one of the non-evidence variales X conditioned on the current values of the variales in the Markov lanket of X. fter a urn-in period each visited state is a sample from the desired distriution.

30 xample W Query: W ample from not W ample from not W not W W ample from c Devika uramanian MM example Query:ainprinklerWetGrass tart at: loudynot ainprinklerwetgrass ample from loudyprinklernot ain suppose loudy false. not loudynot ainprinklerwetgrass ample from ainnot loudyprinklerwetgrass suppose we get ain true. epeat not loudyainprinklerwetgrass these two c Devika uramanian steps. 30

31 31 c Devika uramanian ampling step How do we sample loudy according to loudyprinkler not ain? Use the network! c Devika uramanian MM not not not not Query: W onstruct the transition kernel explicitly and verify that its fix point is in fact: W

32 Why does MM work? The sampling process settles into a dynamic equilirium in which the long run fraction of time spent in each state is exactly proportional to the proaility Xe. s s Markov process qs s sns/l c Devika uramanian L Gis sampling in ayesian networks Query: Xe; Z is the set of all variales in network. tart with a random value z for Z. ick a variale X i in Z-e; generate new value v according to X i Mx i Move to new state which is z with value of X i replaced y v. c Devika uramanian

33 Knowledge engineering for uilding ayesian networks Decide what to talk aout Identify relevant factors Identify relevant relationships etween them. Decide on a vocaulary of random variales What are the variales that represent the factors? What are the values they take on? hould a variale e treated as continuous or should it e discretized? c Devika uramanian Knowledge engineering contd. ncode knowledge aout dependence etween variales Qualitative part: identify topology of elief network. Quantitative part: specify the Ts ncode description of specific prolem instance Translate prolem givens into values of nodes in the elief network ose queries to inference procedure and get answers Formulate quantity of interest as a conditional proaility. c Devika uramanian

34 athfinder ystem Designed y David Heckerman roailistic similarity networks MIT ress Diagnostic system for lymph node diseases. 60 diseases and 100 symptoms and test results proailities in ayesian network. ffort to uild system: 8 hours to determine nodes. 35 hours to determine topology 45 hours to determine proaility values erforms etter than expert doctors. c Devika uramanian