Bayesian Networks Practice
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1 ayesian Networks Practice Part young-hee Kim Seong-Ho Son iointelligence ab CSE Seoul National University
2 Agenda Probabilistic Inference in ayesian networks Probability basics D-searation Probabilistic inference in olytrees Exercise Inference by hand (self) Inference by GeNIe (self) earning from data using Weka Aendix AI & Uncertainty SNU CSE iointelligence ab. htt://bi.snu.ac.kr 2
3 (DAG) SNU CSE iointelligence ab. htt://bi.snu.ac.kr 3
4 ayesian Networks The joint distribution defined by a grah is given by the roduct of a conditional distribution of each node conditioned on their arent nodes. (x) K k 1 ( x Pa( k x k )) (Pa(x k ) denotes the set of arents of x k ) ex) x 1 x 2 x 7 = * Without given DAG structure usual chain rule can be alied to get the joint distribution. ut comutational cost is much higher SNU CSE iointelligence ab. htt://bi.snu.ac.kr 4
5 Probability Probability lays a central role in modern attern recognition. The main tool to deal uncertainties All of the robabilistic inference and learning amount to reeated alication of the sum rule and the roduct rule Random Variables: variables + robability SNU CSE iointelligence ab. htt://bi.snu.ac.kr 5
6 19.1 Review of Probability Theory (1/4) Random variables Joint robability Ex. ( (AT_OK) M (MOVES) (IFTAE) G (GUAGE)) Joint Probability (True True True True) (True True True False) (True True False True) (True True False False) (C) SNU CSE iointelligence ab 6
7 19.1 Review of Probability Theory (2/4) Marginal robability Ex. Conditional robability Ex. The robability that the battery is charged given that the arm does not move (C) SNU CSE iointelligence ab 7
8 ayes Theorem ( X Y ) ( Y ) ( Y X ) X ( ) Posterior ikelihood Prior Normalizing constant ( X ) ( X Y ) ( Y ) Y osterior likelihood rior SNU CSE iointelligence ab. htt://bi.snu.ac.kr 8
9 ayes Theorem Figure from Figure 1. in (Adams et all 2013) obtained from htt://journal.frontiersin.org/article/ /fsyt /full SNU CSE iointelligence ab. htt://bi.snu.ac.kr 9
10 ayesian Probabilities -Frequentist vs. ayesian ikelihood: Frequentist w: a fixed arameter determined by estimator Maximum likelihood: Error function = log ( D w) Error bars: Obtained by the distribution of ossible data sets ootstra Cross-validation ayesian ( w D) ( D w) ( D w) ( w) ( D) a robability distribution w: the uncertainty in the arameters Prior knowledge Noninformative (uniform) rior alace correction in estimating riors Monte Carlo methods variational ayes EP Thomas ayes D (See an article WHERE Do PROAIITIES COME FROM? on age 491 in the textbook (Russell and Norvig 2010) for more discussion) SNU CSE iointelligence ab. htt://bi.snu.ac.kr 10
11 Conditional Indeendence Conditional indeendence simlifies both the structure of a model and the comutations An imortant feature of grahical models is that conditional indeendence roerties of the joint distribution can be read directly from the grah without having to erform any analytical maniulations The general framework for this is called d-searation SNU CSE iointelligence ab. htt://bi.snu.ac.kr 11
12 19.3 ayes Networks (1/2) Directed acyclic grah (DAG) whose nodes are labeled by random variables. Characteristics of ayesian networks Node V i is conditionally indeendent of any subset of nodes that are not descendents of V i. V... V V Pa( V V 1 2 k i i ) i1 Prior robability k Conditional robability table (CPT) (C) SNU CSE iointelligence ab 12
13 19.3 ayes Networks (2/2) (C) SNU CSE iointelligence ab 13
14 (C) SNU CSE iointelligence ab Patterns of Inference in ayes Networks (1/3) Causal or to-down inference Ex. The robability that the arm moves given that the block is liftable M M M M M M M
15 19.4 Patterns of Inference in ayes Networks (2/3) Diagnostic or bottom-u inference Using an effect (or symtom) to infer a cause Ex. The robability that the block is not liftable given that the arm does not move. M (using a causal reasoning) M M M M M M M M M M M (ayes rule) (C) SNU CSE iointelligence ab 15
16 (C) SNU CSE iointelligence ab Patterns of Inference in ayes Networks (3/3) Exlaining away exlains M making less certain M M M M M M M (ayes rule) (def. of conditional rob.) (structure of the ayes network)
17 d-searation Tail-to-tail node or head-to-tail node Think of head as arent node and tail as descendant node. The ath is blocked if the node is observed. The ath is unblocked if the node is unobserved. Remember : ath we are talking about here is UNDIRECTED!!! Ex1 : c is tail-to-tail node because both arcs on the ath lead out of c. Ex2 : c is head-to-tail node because one arc on the ath leads in to c while the other leads out SNU CSE iointelligence ab. htt://bi.snu.ac.kr 17
18 d-searation Head-to-head node The ath is blocked when the node is unobserved. The ath is unblocked if the node itself and/or at least one of its descendants is observed. Ex3 : c is head-to-head node because both arcs on the ath leads in to c SNU CSE iointelligence ab. htt://bi.snu.ac.kr 18
19 d-searation d-searation? All aths between two nodes(variables) are blocked. The joint distribution will satisfy conditional indeendence with resect to concerned variables SNU CSE iointelligence ab. htt://bi.snu.ac.kr 19
20 d-searation (Evidence nodes are observed ones.) Ex4 : V_b1 is tail-to-tail node and is observed so it blocks the ath. V_b2 is head-to-tail node and is observed so it blocks the ath. V_b3 is head-to-head node and is unobserved so it blocks the ath. All the aths from V_i to V_j are blocked so they are conditionally indeendent SNU CSE iointelligence ab. htt://bi.snu.ac.kr 20
21 D-Searation: 1 st case None of the variables are observed Node c is tail-to-tail The variable c is observed The conditioned node blocks the ath from a to b causes a and b to become (conditionally) indeendent SNU CSE iointelligence ab. htt://bi.snu.ac.kr 21
22 D-Searation: 2 nd case None of the variables are observed Node c is head-to-tail The variable c is observed The conditioned node blocks the ath from a to b causes a and b to become (conditionally) indeendent SNU CSE iointelligence ab. htt://bi.snu.ac.kr 22
23 D-Searation: 3 rd case None of the variables are observed The variable c is observed Node c is head-to-head When node c is unobserved it blocks the ath and the variables a and b are indeendent. Conditioning on c unblocks the ath and render a and b deendent SNU CSE iointelligence ab. htt://bi.snu.ac.kr 23
24 Fuel gauge examle attery F-fuel G-electric fuel gauge (rather unreliable fuel gauge) Checking the fuel gauge ( Makes it more likely ) Checking the battery also has the meaning? Makes it less likely than observation of fuel gauge only. (exlaining away) SNU CSE iointelligence ab. htt://bi.snu.ac.kr 24
25 d-searation (a) a is deendent to b given c Head-to-head node e is unblocked because a descendant c is in the conditioning set. Tail-to-tail node f is unblocked (b) a is indeendent to b given f Head-to-head node e is blocked Tail-to-tail node f is blocked SNU CSE iointelligence ab. htt://bi.snu.ac.kr 25
26 19.7 Probabilistic Inference in Polytrees (1/2) Polytree A DAG for which there is just one ath along arcs in either direction between any two nodes in the DAG. (C) SNU CSE iointelligence ab 26
27 19.7 Probabilistic Inference in Polytrees (2/2) A node is above The node is connected to only through s arents A node is below The node is connected to only through s immediate successors. Three tyes of evidence. All evidence nodes are above. All evidence nodes are below. There are evidence nodes both above and below. (C) SNU CSE iointelligence ab 27
28 (C) SNU CSE iointelligence ab 28 Evidence Above and elow 2 2 k k 11} }{ 5 { P P P P P P + -
29 (C) SNU CSE iointelligence ab 29 A Numerical Examle (1/2) ku U R R P R R P R R P P R R R P R R P R R P P R
30 A Numerical Examle (2/2) U P0.8 U P U U P0.019 U P U U k U k k k 0.03 k 0.20 U Other techniques for aroximate inference ucket elimination Monte Carlo method Clustering (C) SNU CSE iointelligence ab 30
31 Exercise
32 Exercise 1 (inference) What is the robability that it is raining given the grass is wet? SNU CSE iointelligence ab. htt://bi.snu.ac.kr 32
33 Exercise 2 (inference) 1) (URS) =? 2) (P) =? 3) (P) =? First you may try to calculate by hand Next you can check the answer with GeNIe SNU CSE iointelligence ab. htt://bi.snu.ac.kr 33
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