Learning Bayesian Networks (part 2) Goals for the lecture

Transcription

1 Learning ayesian Networks (part 2) Mark raven and avid Page omputer Sciences 0 Spring 20 Some of the slides in these lectures have been adapted/borrowed from materials developed by Tom ietterich, Pedro omingos, Tom Mitchell, avid Page, and Jude Shavlik oals for the lecture you should understand the following concepts the how-liu algorithm for structure search structure learning as search Kullback-Leibler divergence the Sparse andidate algorithm

2 Learning structure + parameters number of structures is superexponential in the number of variables finding optimal structure is NP-complete problem two common options: search very restricted space of possible structures (e.g. networks with tree s) use heuristic search (e.g. sparse candidate) The how-liu algorithm learns a N with a tree structure that maximizes the likelihood of the training data algorithm. compute weight I(X i, X j ) of each possible edge (X i, X j ) 2. find maximum weight spanning tree (MST) 3. assign edge directions in MST 2

3 The how-liu algorithm. use mutual information to calculate edge weights I(X,Y ) = P(x, y)log 2 x values( X ) y values(y ) P(x, y) P(x)P(y) The how-liu algorithm 2. find maximum weight spanning tree: a maximal-weight tree that connects all vertices in a graph 3

4 Prim s algorithm for finding an MST given: graph with vertices V and edges V new { v } where v is an arbitrary vertex from V new { } repeat until V new = V { choose an edge (u, v) in with max weight where u is in V new and v is not add v to V new and (u, v) to new } return V new and new which represent an MST Kruskal s algorithm for finding an MST given: graph with vertices V and edges new { } for each (u, v) in ordered by weight (from high to low) { remove (u, v) from if adding (u, v) to new does not create a cycle add (u, v) to new } return V and new which represent an MST 4

5 inding MST in how-liu i. ii. iii. iv. inding MST in how-liu v. vi.

6 Returning directed graph in how-liu 3. pick a node for the root, and assign edge directions The how-liu algorithm How do we know that how-liu will find a tree that maximizes the data likelihood? Two key questions: Why can we represent data likelihood as sum of I(X;Y) over edges? Why can we pick any direction for edges in the tree?

7 Why how-liu maximizes likelihood (for a tree) data likelihood given directed edges log 2 P(,θ ) = log 2 P(x i (d ) Parents(X i )) d i ( ) = I(X i,parents(x i )) H (X i ) we re interested in finding the graph that maximizes this i argmax log 2 P(,θ ) = argmax I(X i,parents(x i )) if we assume a tree, each node has at most one parent argmax log 2 P(,θ ) = argmax I(X i, X j ) i ( X i,x j ) edges edge directions don t matter for likelihood, because MI is symmetric I(X i, X j ) = I(X j, X i ) Heuristic search for structure learning each state in the search space represents a ayes net structure to instantiate a search approach, we need to specify scoring function state transition operators search algorithm

8 Scoring function decomposability when the appropriate priors are used, and all instances in are complete, the scoring function can be decomposed as follows score(, ) = score(x i,parents(x i ) : ) i thus we can score a network by summing terms over the nodes in the network efficiently score changes in a local search procedure Scoring functions for structure learning an we find a good structure just by trying to maximize the likelihood of the data? argmax, θ log P(,θ ) If we have a strong restriction on the the structures allowed (e.g. a tree), then maybe. Otherwise, no! dding an edge will never decrease likelihood. Overfitting likely.

9 Scoring functions for structure learning there are many different scoring functions for N structure search one general approach argmax, θ log P(,θ ) f (m) θ complexity penalty kaike Information riterion (I): f (m) = ayesian Information riterion (I): f (m) = 2 log(m) Structure search operators given the current network at some stage of the search, we can add an edge delete an edge reverse an edge

10 ayesian network search: hill-climbing given: data set, initial network 0 i = 0 best 0 while stopping criteria not met { for each possible operator application a { new apply(a, i ) if score( new ) > score( best ) best new } ++i i best } return i ayesian network search: the Sparse andidate algorithm [riedman et al., UI ] given: data set, initial network 0, parameter k i = 0 repeat { ++i // restrict step select for each variable X j a set ji of candidate parents ( ji k) // maximize step find network i maximizing score among networks where X j, Parents(X j ) j i } until convergence return i 0

11 The restrict step in Sparse andidate to identify candidate parents in the first iteration, can compute the mutual information between pairs of variables P(x, y) I(X,Y ) = P( x, y)log P(x)P(y) x,y The restrict step in Sparse andidate Suppose: true distribution current network we re selecting two candidate parents for, and I(, ) > I(, ) > I(, ) with mutual information, the candidate parents for would be and how could we get as a candidate parent?

12 The restrict step in Sparse andidate Kullback-Leibler (KL) divergence provides a distance measure between two distributions, P and Q KL ( P( X ) Q( X )) = x P( x) P( x)log Q( x) mutual information can be thought of as the KL divergence between the distributions P(X,Y ) P(X)P(Y ) (assumes X and Y are independent) The restrict step in Sparse andidate we can use KL to assess the discrepancy between the network s P net (X, Y) and the empirical P(X, Y) M (X,Y ) = KL (P(X,Y )) P net (X,Y )) true distribution current ayes net KL (P(, )) P net (, )) can estimate P net (X, Y) by sampling from the network (i.e. using it to generate instances) 2

13 The restrict step in Sparse andidate given: data set, current network i, parameter k for each variable X j { calculate M(X j, X l ) for all X j X l such that X l Parents(X j ) choose highest ranking X... X k-s where s= Parents(X j ) // include current parents in candidate set to ensure monotonic // improvement in scoring function ji =Parents(X j ) X... X k-s } return { i j } for all X j The maximize step in Sparse andidate hill-climbing search with add-edge, delete-edge, reverse-edge operators test to ensure that cycles aren t introduced into the graph 3

14 fficiency of Sparse andidate n = number of variables ordinary greedy search possible parent sets for each node changes scored on first iteration of search O( 2 n ) O( n 2 ) changes scored on subsequent iterations O( n) greedy search w/at most k parents O n k O( n 2 ) O( n) Sparse andidate O( 2 k ) O( kn) O( k) after we apply an operator, the scores will change only for edges from the parents of the node with the new impinging edge 4