Data Mining 2018 Bayesian Networks (1)

Transcription

1 Data Mining 2018 Bayesian Networks (1) Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Data Mining 1 / 49

2 Do you like noodles? Do you like noodles? Race Gender Yes No Black Male Female White Male Female Ad Feelders ( Universiteit Utrecht ) Data Mining 2 / 49

3 Do you like noodles? Undirected G R A G R A Strange: Gender and Race are prior to Answer, but this model says they are independent given Answer! Ad Feelders ( Universiteit Utrecht ) Data Mining 3 / 49

4 Do you like noodles? Marginal table for Gender and Race: Race Gender Black White Male Female From this table we conclude that Race and Gender are independent in the data. Ad Feelders ( Universiteit Utrecht ) Data Mining 4 / 49

5 Do you like noodles? Directed G R A G R, G R A Gender and Race are marginally independent (but dependent given Answer). Ad Feelders ( Universiteit Utrecht ) Data Mining 5 / 49

6 Do you like noodles? Table for Gender and Race given Answer=yes: Race Gender Black White Male Female Table for Gender and Race given Answer=no: Race Gender Black White Male Female From these tables we conclude that Race and Gender are dependent given Answer. Ad Feelders ( Universiteit Utrecht ) Data Mining 6 / 49

7 Directed Independence Graphs G = (K, E), K is a set of vertices and E is a set of edges with ordered pairs of vertices. No directed cycles (DAG) parent/child ancestor/descendant ancestral set Because G is a DAG, there exists a complete ordering of the vertices that is respected in the graph (edges point from lower ordered to higher ordered nodes). Ad Feelders ( Universiteit Utrecht ) Data Mining 7 / 49

8 Parents Of Node i: pa(i) i Ad Feelders ( Universiteit Utrecht ) Data Mining 8 / 49

9 Ancestors Of Node i: an(i) i Ad Feelders ( Universiteit Utrecht ) Data Mining 9 / 49

10 Ancestral Set Of Node i: an + (i) i Ad Feelders ( Universiteit Utrecht ) Data Mining 10 / 49

11 Children Of Node i: ch(i) i Ad Feelders ( Universiteit Utrecht ) Data Mining 11 / 49

12 Descendants Of Node i: de(i) i Ad Feelders ( Universiteit Utrecht ) Data Mining 12 / 49

13 Construction of DAG Suppose that prior knowledge tells us the variables can be labeled X 1, X 2,..., X k such that X i is prior to X i+1. (for example: causal or temporal ordering) Corresponding to this ordering we can use the product rule to factorize the joint distribution of X 1, X 2,..., X k as P(X ) = P(X 1 )P(X 2 X 1 ) P(X k X k 1, X k 2,..., X 1 ) This is an identity of probability theory, no independence assumptions have been made yet! Ad Feelders ( Universiteit Utrecht ) Data Mining 13 / 49

14 Constructing a DAG from pairwise independencies In constructing a DAG, an arrow is drawn from i to j, where i < j, unless P(X j X j 1,..., X 1 ) does not depend on X i, in other words, unless More loosely j i {1,..., j} \ {i, j} j i prior variables Compare this to pairwise independence in undirected independence graphs. j i rest Ad Feelders ( Universiteit Utrecht ) Data Mining 14 / 49

15 Construction Of DAG P(X ) = P(X 1 )P(X 2 X 1 )P(X 3 X 1, X 2 )P(X 4 X 1, X 2, X 3 ) Suppose the following independencies are given: 1 X 1 X 2 2 X 4 X 3 (X 1, X 2 ) 3 X 1 X 3 X 2 Ad Feelders ( Universiteit Utrecht ) Data Mining 15 / 49

16 Construction Of DAG P(X ) = P(X 1 ) P(X 2 X 1 ) P(X 3 X 1, X 2 )P(X 4 X 1, X 2, X 3 ) }{{} P(X 2 ) 1 If X 1 X 2, then P(X 2 X 1 ) = P(X 2 ). The edge 1 2 is removed. Ad Feelders ( Universiteit Utrecht ) Data Mining 16 / 49

17 Construction Of DAG P(X ) = P(X 1 )P(X 2 )P(X 3 X 1, X 2 )P(X 4 X 1, X 2, X 3 ) Ad Feelders ( Universiteit Utrecht ) Data Mining 17 / 49

18 Construction Of DAG P(X ) = P(X 1 )P(X 2 )P(X 3 X 1, X 2 ) P(X 4 X 1, X 2, X 3 ) }{{} P(X 4 X 1,X 2 ) 2 If X 4 X 3 (X 1, X 2 ), then P(X 4 X 1, X 2, X 3 ) = P(X 4 X 1, X 2 ). The edge 3 4 is removed. Ad Feelders ( Universiteit Utrecht ) Data Mining 18 / 49

19 Construction Of DAG P(X ) = P(X 1 )P(X 2 )P(X 3 X 1, X 2 )P(X 4 X 1, X 2 ) Ad Feelders ( Universiteit Utrecht ) Data Mining 19 / 49

20 Construction Of DAG P(X ) = P(X 1 )P(X 2 ) P(X 3 X 1, X 2 ) P(X 4 X 1, X 2 ) }{{} P(X 3 X 2 ) 3 If X 1 X 3 X 2, then P(X 3 X 1, X 2 ) = P(X 3 X 2 ) The edge 1 3 is removed. Ad Feelders ( Universiteit Utrecht ) Data Mining 20 / 49

21 Construction Of DAG P(X ) = P(X 1 )P(X 2 )P(X 3 X 2 )P(X 4 X 1, X 2 ) Ad Feelders ( Universiteit Utrecht ) Data Mining 21 / 49

22 Joint density of Bayesian Network We can write the joint density more elegantly as k P(X 1,..., X k ) = P(X i X pa(i) ) i=1 Ad Feelders ( Universiteit Utrecht ) Data Mining 22 / 49

23 Independence Properties of DAGs: d-separation and Moral Graphs Can we infer other/stronger independence statements from the directed graph like we did using separation in the undirected graphical models? Yes, the relevant concept is called d-separation. establishing d-separation directly (Pearl) establishing d-separation via the moral graph and normal separation We discuss each in turn. Ad Feelders ( Universiteit Utrecht ) Data Mining 23 / 49

24 Independence Properties of DAGs: d-separation A path p is blocked by a set of nodes Z if and only if: 1 p contains a chain of nodes A B C, or a fork A B C, such that the middle node B is in Z; or 2 p contains a collider A B C such that the collision node B is not in Z, and no descendant of B is in Z either. If Z blocks every path between two nodes X and Y, then X and Y are d-separated by Z, and thus X and Y are independent given Z. Ad Feelders ( Universiteit Utrecht ) Data Mining 24 / 49

25 Independence Properties of DAGs: Moral Graph Given a DAG G = (K, E) we construct the moral graph G m by marrying parents, and deleting directions, that is, 1 For each i K, we connect all vertices in pa(i) with undirected edges. 2 We replace all directed edges in E with undirected ones. Ad Feelders ( Universiteit Utrecht ) Data Mining 25 / 49

26 Independence Properties of DAGs: Moral Graph The directed independence graph G possesses the conditional independence properties of its associated moral graph G m. Why? We have the factorisation: P(X ) = = k P(X i X pa(i) ) i=1 k g i (X i, X pa(i) ) i=1 by setting g i (X i, X pa(i) ) = P(X i X pa(i) ). Ad Feelders ( Universiteit Utrecht ) Data Mining 26 / 49

27 Independence Properties of DAGs: Moral Graph We have the factorisation: k P(X ) = g i (X i, X pa(i) ) (1) i=1 We thus have a factorisation of the joint probability distribution in terms of functions g i (X ai ) where a i = {i} pa(i). By application of the factorisation criterion the sets a i become cliques in the undirected independence graph. Such cliques are formed by moralization. Ad Feelders ( Universiteit Utrecht ) Data Mining 27 / 49

28 Moralisation: Example X 1 X2 X 4 X 3 X 5 Ad Feelders ( Universiteit Utrecht ) Data Mining 28 / 49

29 Moralisation: Example X 1 X2 X 4 X 3 X 5 {i} pa(i) becomes a complete subgraph in the moral graph (by marrying all unmarried parents). Ad Feelders ( Universiteit Utrecht ) Data Mining 29 / 49

30 Moralisation Continued Warning: the complete moral graph can obscure independencies! To verify construct the moral graph on i j S that is i, j, S and all their ancestors. A = an + ({i, j} S), Ad Feelders ( Universiteit Utrecht ) Data Mining 30 / 49

31 Moralisation Continued Since for l A, pa(l) A, we know that the joint distribution of X A is given by P(X A ) = l A P(X l X pa(l) ) which corresponds to the subgraph G A of G. 1 This is a product of factors P(X l X pa(l) ), involving the variables X {l} pa(l) only. 2 So it factorizes according to GA m, and thus the independence properties for undirected graphs apply. 3 So, if S separates i from j in GA m, then i j S. Ad Feelders ( Universiteit Utrecht ) Data Mining 31 / 49

32 Full moral graph may obscure independencies: example G R A P(G, R, A) = P(G)P(R)P(A G, R) Does G R hold? Summing out A we obtain: P(G, R) = a = a P(G, R, A = a) P(G)P(R)P(A = a G, R) (sum rule) (BN factorisation) = P(G)P(R) a P(A = a G, R) (rule of summation) = P(G)P(R) ( a P(A = a G, R) = 1) Ad Feelders ( Universiteit Utrecht ) Data Mining 32 / 49

33 Moralisation Continued: example X 1 X2 X 4 X 3 X 5 Are X 3 and X 4 independent? Are X 1 and X 3 independent? Are X 3 and X 4 independent given X 5? Ad Feelders ( Universiteit Utrecht ) Data Mining 33 / 49

34 Equivalence When no marrying of parents is required (there are no immoralities or v-structures ), then the independence properties of the directed graph are identical to those of its undirected version. These three graphs express the same independence properties: Ad Feelders ( Universiteit Utrecht ) Data Mining 34 / 49

35 Learning Bayesian Networks 1 Parameter learning: structure known/given; we only need to estimate the conditional probabilities from the data. 2 Structure learning: structure unknown; we need to learn the networks structure as well as the corresponding conditional probabilities from the data. Ad Feelders ( Universiteit Utrecht ) Data Mining 35 / 49

36 Maximum Likelihood Estimation Find value of unknown parameter(s) that maximize the probability of the observed data. n independent observations on binary variable X {1, 2}. We observe n(1) outcomes X = 1 and n(2) = n n(1) outcomes X = 2. What is the maximum likelihood estimate of p(1)? The likelihood function (probability of the data) is given by: Taking the log we get L = p(1) n(1) (1 p(1)) n n(1) L = n(1) log p(1) + (n n(1)) log(1 p(1)) Ad Feelders ( Universiteit Utrecht ) Data Mining 36 / 49

37 Maximum Likelihood Estimation Take derivative with respect to p(1), equate to zero, and solve for p(1). dl dp(1) = n(1) p(1) n n(1) 1 p(1) = 0, since d log x dx = 1 x (where log is the natural logarithm). Solving for p(1), we get p(1) = n(1) n, i.e., the fraction of one s in the sample! Ad Feelders ( Universiteit Utrecht ) Data Mining 37 / 49

38 ML Estimation of Multinomial Distribution Let X {1, 2,..., J}. Estimate the probabilities p(1), p(2),..., p(j) of getting outcomes 1, 2,..., J. If in n trials, we observe n(1) outcomes of 1, n(2) of 2,..., n(j) of J, then the obvious guess is to estimate p(j) = n(j) n, This is also the maximum likelihood estimate. j = 1, 2,..., J Ad Feelders ( Universiteit Utrecht ) Data Mining 38 / 49

39 BN-Factorisation For a given BN-DAG, the joint distribution factorises according to k P(X ) = p(x i X pa(i) ) So to specify the distribution we have to estimate the probabilities i=1 p(x i X pa(i) ) i = 1, 2,..., k for the conditional distribution of each variable given its parents. Ad Feelders ( Universiteit Utrecht ) Data Mining 39 / 49

40 ML Estimation of BN The joint probability for n independent observations is P(X (1),..., X (n) ) = = n P(X (j) ) j=1 n k p(x (j) i j=1 i=1 X (j) pa(i) ), where X (j) denotes the j-th row in the data table. The likelihood function is therefore given by L = k i=1 x i,x pa(i) p(x i x pa(i) ) n(x i,x pa(i)) where n(x i, x pa(i) ) is a count of the number of records with X i = x i, and X pa(i) = x pa(i). Ad Feelders ( Universiteit Utrecht ) Data Mining 40 / 49

41 ML Estimation of BN Taking the log of the likelihood, we get L = k i=1 x i,x pa(i) n(x i, x pa(i) ) log p(x i x pa(i) ) Maximize the log-likelihood function with respect to the unknown parameters p(x i x pa(i) ). This decomposes into a collection of independent multinomial estimation problems. Separate estimation problem for each X i and configuration of X pa(i). Ad Feelders ( Universiteit Utrecht ) Data Mining 41 / 49

42 ML Estimation of BN The maximum likelihood estimate of p(x i x pa(i) ) is given by: ˆp(x i x pa(i) ) = n(x i, x pa(i) ) n(x pa(i) ), where n(x i, x pa(i) ) is the number of records in the data with X i = x i and X pa(i) = x pa(i), and n(x pa(i) ) is the number of records in the data with X pa(i) = x pa(i). Ad Feelders ( Universiteit Utrecht ) Data Mining 42 / 49

43 Example BN and Factorisation P(X 1, X 2, X 3, X 4 ) = p 1 (X 1 )p 2 (X 2 )p 3 12 (X 3 X 1, X 2 )p 4 3 (X 4 X 3 ) Ad Feelders ( Universiteit Utrecht ) Data Mining 43 / 49

44 Example BN: Parameters P(X 1, X 2, X 3, X 4 ) = p 1 (X 1 )p 2 (X 2 )p 3 12 (X 3 X 1, X 2 )p 4 3 (X 4 X 3 ) Now we have to estimate the following parameters (X 4 ternary, rest binary): p 1 (1) p 1 (2) = 1 p 1 (1) p 2 (1) p 2 (2) = 1 p 2 (1) p 3 1,2 (1 1, 1) p 3 1,2 (2 1, 1) = 1 p 3 1,2 (1 1, 1) p 3 1,2 (1 1, 2) p 3 1,2 (2 1, 2) = 1 p 3 1,2 (1 1, 2) p 3 1,2 (1 2, 1) p 3 1,2 (2 2, 1) = 1 p 3 1,2 (1 2, 1) p 3 1,2 (1 2, 2) p 3 1,2 (2 2, 2) = 1 p 3 1,2 (1 2, 2) p 4 3 (1 1) p 4 3 (2 1) p 4 3 (3 1) = 1 p 4 3 (1 1) p 4 3 (2 1) p 4 3 (1 2) p 4 3 (2 2) p 4 3 (3 2) = 1 p 4 3 (1 2) p 4 3 (2 2) Ad Feelders ( Universiteit Utrecht ) Data Mining 44 / 49

45 Example Data Set obs X 1 X 2 X 3 X Ad Feelders ( Universiteit Utrecht ) Data Mining 45 / 49

46 Maximum Likelihood Estimation obs X 1 X 2 X 3 X ˆp 1 (1) = n(x 1 = 1) n = 5 10 = 1 2 Ad Feelders ( Universiteit Utrecht ) Data Mining 46 / 49

47 Maximum Likelihood Estimation obs X 1 X 2 X 3 X ˆp 2 (1) = n(x 2 = 1) n = 6 10 Ad Feelders ( Universiteit Utrecht ) Data Mining 47 / 49

48 Maximum Likelihood Estimation obs X 1 X 2 X 3 X ˆp 3 1,2 (1 1, 1) = n(x 1 = 1, x 2 = 1, x 3 = 1) n(x 1 = 1, x 2 = 1) = 2 3 Ad Feelders ( Universiteit Utrecht ) Data Mining 48 / 49

49 Maximum Likelihood Estimation obs X 1 X 2 X 3 X ˆp 3 1,2 (1 1, 1) = n(x 1 = 1, x 2 = 1, x 3 = 1) n(x 1 = 1, x 2 = 1) = 2 3 Ad Feelders ( Universiteit Utrecht ) Data Mining 49 / 49