The Acyclic Bayesian Net Generator (Student Paper)

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1 The Acyclic Bayesian Net Generator (Student Paper) Pankaj B. Gupta and Vicki H. Allan Microsoft Corporation, One Microsoft Way, Redmond, WA 98, USA, Computer Science Department, Utah State University, Logan, UT 8, Abstract. We present the Acyclic Bayesian Net Generator, a new approach to learn the structure of a Bayesian network using genetic algorithms. Due to the encoding mechanism, acyclicity is preserved through mutation and crossover. We present a detailed description of how our method works and explain why it is better than previous approaches. We can efficiently perform crossover on chromosomes with different node orders without the danger of cycle formation. The approach is capable of learning over all variable node orderings and structures. We also present a proof that our technique of choosing the initial population semi-randomly ensures that the genetic algorithm searches over the whole solution space. Tests show that the method is effective. Keywords: Structure Learning, Bayesian Networks, Genetic Algorithms. Introduction Many intelligent systems attempt to discover the relationships between data []. Often data is available for the parameters of a problem area but no information is available about the relations between them. Bayesian networks are graphical models capable of representing relations between the parameters of a problem in a way that is easy for humans to interpret and visualize. Bayesian networks encode relations that are intuitive to humans and are also suitable for statistical analysis. The nodes in a Bayesian network represent the parameters, and the directed edges represent the cause-effect relationships between them. Learning the structure of a Bayesian network is a challenging problem. Genetic algorithms are evolutionary algorithms capable of solving problems with large solution spaces. To use genetic algorithms for a problem area, it requires the following: an Encoding Scheme, a Score function, a Crossover function and a Mutation function. A way of representing a possible solution as a low level data structure, such as an array of bits, is known as the encoding scheme. An encoded solution is called a chromosome. A set of chromosomes which are contestants for producing the optimal solution is called a population. A crossover function mixes two chromosomes (called parent chromosomes) and regenerates two chromosomes (called daughter chromosomes). A mutation function randomly changes a chromosome. A Score function is a measure of the

2 quality of the solution (chromosome). Initially a population of random chromosomes is generated. In each iteration of the genetic algorithm, a set of daughter chromosomes is created by crossing over pairs of parents chosen randomly from the population. The set of daughters chromosomes is added to the population and the population is reduced to the original size by discarding the worst quality chromosomes. This process is repeated until a quality chromosome is generated. Learning the structure of a Bayesian network from data has a large solution space and is known to be NP-Hard [, 6]. Thus, genetic algorithms provide an effective solution for the problem. In this research, we present a new method that learns the structure of a Bayesian network using genetic algorithms. We name our method the Acyclic Bayesian Net Generator. We preserve acyclicity without restricting node order. The initial population has a significant role in the success of a genetic algorithm. We present a technique to generate the initial population, in a semirandom manner, to ensure that the search is performed over the whole solution space. Previous Work Initial attempts at using genetic algorithms for learning the structure of Bayesian networks are credited to Larranaga, et al. [8, 9]. They use a structure of booleans which they term a connectivity matrix {c ij }. In this matrix, c ij = indicates that i is a parent of j. In order to preserve an acyclic Bayesian net, either an existing node ordering is assumed for the learning algorithm (and parents must be selected from nodes earlier in the order), or a repair operator is applied to delete cycle-causing edges from cyclic networks formed during the process. Myers, et. al., use genetic algorithms to learn the structure of Bayesian networks from incomplete data []. Adjacency lists are used to describe the parents of a node of the Bayesian network. Illegal Bayesian networks generated in the process are handled by assigning them a low score. Guo, et al., use genetic algorithm to tune the node ordering of a Bayesian network []. The node ordering serves an input to the K algorithm []. There are several improvements in our approach, the Acyclic Bayesian Net Generator. Our encoding scheme is inherently acyclic, and thus, we do not have to deal with any cyclic networks. Neither do we assume any pre-defined node ordering, nor do we delete arcs (and thus lose information) in maintaining network acyclicity. Our crossover and mutation functions are closed operations. Our method has the advantage that we optimize the complete solution rather than optimizing pieces and attempting a combination, which is typically a less than optimal approach. Definitions, Assumptions For this research, n represents the number of nodes of a Bayesian network. The nodes are labeled from to n. Chromosomes are represented with capital

3 letters such as A, B, C. Sets of chromosomes are represented by bold letters such as U, V. A chromosome C for a Bayesian network of n nodes is represented as C n. We use the term C to refer to the node order associated with a chromosome C. Bayesian Network Node : Encoded Chromosome C 8 C.Gene C.Gene 7 C.Gene C.Gene 6 9 Node Parents Fig.. A Bayesian Network, A Topological Node And The Corresponding Encoded Chromosome The Encoding Scheme A chromosome defines both a Bayesian network and a topological order on nodes of the Bayesian network. Associating a node order with a chromosome has the following advantages: First, by crossing over chromosomes of different node orders, our algorithm learns the optimal node ordering, and second, acyclic nature is preserved over operations like crossover and mutation. Consider a Bayesian network of size n. Let C be the corresponding encoded chromosome. C is an array of n genes, C.Gene, C.Gene,..., C.Gene n, such that there is one gene corresponding to each node in the Bayesian network. Figure shows an encoding for a sample Bayesian network. In the figure, each row of the encoded chromosome represents a gene. Each gene contains the following two pieces of information: Node: The i th gene of the chromosome C corresponds to the i th node in the corresponding topological node order. Node information is stored in the gene and is represented as C i. The sequence, C, C,..., C n, represents the node order (C ). In Figure, the gene C.Gene corresponds to node 6, thus C is 6. Parents: The parent information of chromosome C is a connectivity matrix {C i,j } with i < j and C i,j ɛ{, }. C i,j = if and only if C j is a parent of

4 C i. The i th row of the connectivity matrix represents the parent information associated with i th gene. In Figure, nodes, 8, 7, are the possible parents of node 6 and node 8 is the only actual parent of node 6. Crossover Point = Chromosome A Node : Bayesian Network A Chromosome B Node : Bayesian Network B Fig.. Parent Chromosomes Crossover Function Our O(n ) crossover function ensures that the daughter chromosomes produced are acyclic in nature. Let the two parent chromosomes be A and B, and the two daughter chromosomes be X and Y. The crossover function can be thought of as a two step process: () producing node orders, and () assigning parents.. Step : Daughter s Figure shows two parent chromosomes, the associated Bayesian networks and node orders. Our crossover point, cp, divides the nodes into two pieces and is determined randomly. Based on the crossover point, A is split into A and A. A has nodes A... A cp and A has nodes A cp... A n. In figure, our crossover point is. The node order for chromosome B is divided into two parts B and B, such that B has the same nodes as in A (but the order within B is preserved), and B has the same nodes as in A (but the order within B is preserved). To generate the node ordering of the two daughter chromosomes, we concatenate A with B to get X, and B with A to get Y. Figure shows this process for the parent chromosomes in Figure. This step enables our Acyclic Bayesian Net Generator to build from different node orders to learn the optimal one. Later in this paper, we present a proof that our learning algorithm is capable of iterating over every node order.

5 Division of Node of Parent Chromosomes: New Node s for Daughter Chromosomes: A A For Chromosome A based on Crossover Point A. B For Chromosome X B B For Chromosome B based on A and A B. A For Chromosome Y Fig.. Generating New Topological Node. Step : Parent Information Once we have determined X and Y, we need to generate the parent information for the nodes in X and Y. This step can generate two different sets of parent information for the daughter chromosomes. It is decided randomly which set of daughter chromosomes is actually produced. We refer to the methods as M and M. Let x and x be two nodes such that x precedes x in X. Remember that X is formed by concatenating A and B. In daughter chromosome X, x is a parent of x if and only if one of the following conditions is met.. Either, both x and x belong to A and x is a parent of x in A, or both x and x belong to B, and x is a parent of x in B (solid edges in Figure ).. x belongs to A and x belongs to B, and Method M: x is a parent of x in A (dotted edges in Figure ). Method M: either x is a parent of x in B (large dashed lines in in Figure ), or, x is a parent of x in B (small dashed edges in Figure ). The steps for generating daughter chromosome Y differ slightly. Let y and y be two nodes such that y precedes y in Y. In daughter chromosome Y, y is a parent of y if and only if one of the following conditions is met.. Either, both y and y belong to B and y is a parent of y in B, or both y and y belong to A, and y is a parent of y in A (solid edges in Figure ).. x belongs to A and x belongs to B, and Method M: either y is a parent of y in B (large dashed lines in in Figure ), or, y is a parent of y in B (small dashed edges in Figure ). Method M: y is a parent of y in A (dotted edges in Figure ). In condition, both x and x are initially a part of the same parent chromosome, thus we have transferred the parent relation from that parent chromosome. In condition, node x is from A and x is a part of B. We have to transfer the parent relation between x and x to X either from A or B. This

6 A B (a) Parent Chromosomes X Y (b) Daughters Using M X Y (c) Daughters using M Fig.. Generating Parent Information choice is made at random. If A is chosen, we say that we are using method M. In case of method M, the parent relation from B is used. Though the steps for generating Y appear to be similar, they differ slightly. Condition is very similar to that of chromosome X. In condition, the roles of M and M are reversed. This ensures that in this case we use A (or B) to transfer the parent information to Y, if B (or A) is used for X in a similar situation. Figure shows the behavior of our crossover function when two chromosomes of the same node order are crossed over. A B Using method M X Y A B Using Method M X Y Fig.. Crossover For Chromosomes of Same Node s Previous attempts of structure learning using genetic algorithms have to deal with the cyclic networks. In our method, the edges represented by small dashes in Figure have the potential to form cycles in the daughter chromosomes. To prevent cycles, two approaches can be considered. The first approach is to reverse such edges and the second is to delete them. Steck uses the concept of edge reversal in his learning algorithm [], whereas Larranaga, in his algorithm,

7 deletes the cycle forming edges [8]. We perform a test to investigate which of the two approaches is better. We take a sample data set for ASIA network [] and use it to learn the network using both approaches. Figure 6 shows the percentage of times the optimal ASIA network is learned. From the figure, it is evident that reversing an edge is a better approach than deleting an edge. Though reversing an edge between two nodes may not represent the causal relationship between them as before, often there is little difference if we assume A causes B rather than B causes A. Since we have a node order associated with our daughter chromosomes, cycles are prevented as the direction of potentially dangerous edges gets reversed intrinsically. Thus, it is ensured that we do not lose any parent information by deleting edges. For any two nodes of the Bayesian network, the parent information for the first daughter directly follows from the parent information between the same nodes in one of the two parent chromosomes. The parent information for the second daughter follows directly from the other parent chromosome. Each parent relation between nodes of the parent chromosomes is transferred to exactly one of the two daughter chromosomes. In the crossover operation, no edges are lost, nor do any new edges appear in the daughter chromosomes. During the process, nodes inherited from the same parent preserve the mutual relations as in that parent. This preserves the goodness of the chromosomes and proves particularly beneficial if one of the parent chromosome has captured the dependence between a few nodes and the other parent has captured dependence among the other nodes. Fig. 6. Comparison of two approaches to prevent cycle formation while learning the structure of ASIA network. In one approach, potentially dangerous edges are reversed, whereas in the other approach they are deleted. As explained above, during crossover, one of the two methods (M and M) is randomly chosen to generate daughter chromosomes. It is imperative that both

8 methods be used so that the learning process spans the whole solution space. To confirm this experimentally, we perform a test on a node subset of ALARM []. It is observed that the quality of the networks learned deteriorates when only one of the two methods is used. The Hamming Distance (HD) between two networks is a measure of the number of edges that are not common to both the networks. The best network learned by using only method M has a HD value of 8, whereas when both methods are used (all other parameters are kept constant), we learn a network that has a HD value of. We use the the operator to represent our crossover function. The equation A B cp,m X, Y is read as: A parent chromosome A crossed over with a parent chromosome B, at crossover point cp using method m (M or M), produces daughter chromosomes X and Y. We use the same notation if we want to talk only about node orders. However, it should be noted that generation of node orders for the daughter chromosomes is independent of the method used. A B cp X, Y Note, our crossover function is asymmetric. Thus, the order of inputs and outputs is important. Let S be a set of chromosomes or node orders. S represents the closure of S over the crossover function applied at every crossover point and using both the methods. Crossing over chromosomes with identical node order produces daughter chromosomes with the same order. This can be represented as the following axiom. Axiom : {A } = {A } 6 Generation Of Population We use the term spanning population to mean that through crossover the initial population is capable of producing every chromosome in the solution space. A spanning population is important for the success of a genetic algorithm. If care is not taken, it is easy to generate a population not rich enough to result in an optimal solution. We introduce the following functions to discuss the formation of our initial population: Invert generates a chromosome that has a node order which is the reverse of the input chromosome A. The connectivity matrix associated with the chromosome is not altered. We will show that Invert is required for our algorithm to be able to iterate over all node orders in the solution space. We use the following notation: Invert(A) = Â. Figure 7 shows the behavior of our crossover function when A is crossed over with its inverse order. Please note that we show only the node orders in figure 7. ToggleParents inverts the bits of the parent information of each gene to produce a new chromosome. ToggleParents is important to be able to produce every possible parent configuration for a given node order. For each node n that

9 A Reverse X Y Fig. 7. Crossover Of An With Its Reverse precedes node m in the node order of a chromosome A, n is a parent of m in ToggleParents(A) if and only if n is not a parent of m in Chromosome A. See Figure 8. The following notation is used: T ogglep arents(a) = A A Invert A Toggle Parents Toggle Parents A Invert A = A Fig. 8. Invert And ToggleParent Operations Axiom For any chromosome A: A = A Â = A Â = Â Our initial population is generated according to the following mechanism. Chromosomes generated randomly are added to the population. To make the population rich, for every chromosome A, we add to the population Â, A and Â. This ensures that every chromosome in the solution space is derivable from our initial population. Theorem Let U be the universal set of all chromosomes of n nodes and A is any chromosome of n nodes. { A, Â, A, Â } = U

10 The theorem states that starting with A and its three variants, and by repeatedly applying crossover, it is possible to generate every chromosome in the universal set U. To prove this theorem, we need to define and prove some lemmas. Lemma For any two chromosomes A and B : if A B cp,m X, Y A B cp,m X, Y In our crossover function, the parent relation between two nodes of the first daughter chromosome is derived directly from one of the two parent chromosomes. The relation between the same two nodes of the second daughter chromosome is derived from the other of the two parents. Though the direction of edge may be reversed to preserve the acyclic nature of the daughters, the edge still exists. Thus toggling all the parent bits in the parent chromosomes result in toggling parent bits for the daughter chromosomes. Lemma describes this argument mathematically. Lemma Let A, B and R be three node orders such that: R ɛ { A, B } and x is a new node. Letting be concatenation, x R ɛ { x A, x B } Consider that A and B are two node orders and x is a new node. The following equation is true by the definition of step one of our crossover function. cp A B C, D () cp+ x A x B x C, x D Lemma follows from Equation. If we concatenate x to all the steps taken to derive R from A and B of lemma, it is clear that x.r is derivable from x.a and x.b. Lemma Let U n be the universal set of node orders of n nodes and A be any node order of n nodes. This implies: {A, Â } = U n We prove lemma by induction on n. The lemma is trivially true for n =. Assume that it is true for n=k. Thus, for any chromosome B k : {B k, B k } = U k () Recall that the superscript on the chromosome represents the number of nodes of the chromosome. It will be sufficient to prove that for any A k+ and R k+ : R k+ ɛ {Ak+, Âk+ } ()

11 Let the first node of R k+ be the t+th node of A k+, A t = R. As shown in Figure 7, we perform crossover on A and  with t+ as the crossover point: A  t+ L, M () where M = A t,..., A, A t+,..., A k. We perform crossover on  and A with k t as the crossover point:  A k t N, O () where N = A k,..., A t+, A,..., A t. From equations and, we can see N = M. Crossing over M and N with as the crossover point: M M P, Q (6) where P = A t, A k..., A t+, A,..., A t and Q = A t, A t..., A, A t+,..., A k. Recall that A t = R. Thus P can be represented as R B k where B k = A k,...,a t+,a,...,a t. Consequently, Q can be represented as R B k. Thus, we can say: From lemma and equation : R B k ɛ {A k+, Âk+ } R B k ɛ {A k+, (7) Âk+ } R k+ ɛ {R B k, R B k } (8) Combining equations 7 and 8 proves the equation. Hence lemma is proved. Lemma Let Z be any chromosome and T be a chromosome formed by toggling any one parent bit in Z. This implies T ɛ {Z, Z} and T ɛ {Z, Z} For the following steps, we are crossing over chromosomes with identical node orders. Please refer to the Figure. Consider that T differs from Z only in the following bit, T i,j = Z i,j. To prove lemma, we will use two crossover operations using method M to arrive at a chromosome G that differs with Z in the j th parent bit of every gene. In the next two steps we use method M to swap out the i th gene of Z with the i th gene of G to generate T. The steps are as follows: Z Z j,m E, F and E Z j+,m G, H (9) where E i.p arents = Z i,,..., Z i,j, Z i,j,..., Z i,i and G i.p arents = Z i,,..., Z i,j, Z i,j, Z i,j+,..., Z i,i. Recall that we are working with chromosomes with identical node orders. Thus, G.Gene i = T.Gene i. Z G i,m I, J and I Z i+,m K, L ()

12 where I = Z.Gene,..., Z.Gene i, T.Gene i, G.Gene i+,..., G.Gene n and K = Z.Gene,..., Z.Gene i, T.Gene i, Z.Gene i+,..., Z.Gene n. It is evident that K = T. Thus, from equations 9 and we can say T ɛ{z, Z}. Also, from lemma we can deduce T ɛ {Z, Z}. From axiom, Z = Z. Thus lemma has been proved. Lemma Let Z be any chromosome and V be the universal set of all chromosomes with Z as their node order. This implies: {Z, Z} = V This lemma is an extension of lemma. Any chromosome in V is reachable from S by applying lemma multiple times. Now we can prove theorem by the following argument. Consider S be any chromosome from the universal set U mentioned in theorem. Lemma states that starting with A and Â, it is possible to reach some chromosome Q whose node order is same as S. Mathematically, Q s.t. Q = S and Q ɛ {A, Â}. Also, from lemma, we can say Q ɛ {A, Â}. Since the node orders of S and Q are same, we apply lemma to deduce S ɛ {Q, Q}, and thus, S ɛ {A, Â, A, Â}. To begin with, S was any chromosome from the universal set U. Thus we can say, U = {A, Â, A, Â}. We have proved that starting with our initial population it is theoretically possible to reach every chromosome of the solution space, a necessary (but not sufficient) condition for a good genetic algorithm. In order to test the importance of enriching the population, we compare the results of the networks learned when a randomly generated population (of size p) is used vs. when the random population is enriched and the enriched population (of size p) is used. To compare the results on a fair basis, in each iteration of the latter learning process, we generate only p (instead of p) daughters. It is observed that with an enriched population better networks are learned. To present the results for this test we use a 6 node network. For the network, we are able to learn the optimal structure using an enriched population, whereas the best network learned in the case of a random population has a Hamming Distance value of. We also compare the behavior of our algorithm when samesized random and enriched populations are used. For small population sizes, the networks learned using enriched population are better, but for larger sizes the random population is already rich enough, that enriching it does not improve the learning process any further. 7 Mutations Mutation is a function that randomly changes one or more genes of a chromosome. It prevents the learning process from getting stuck at any local optima. Acyclic nature of our chromosomes is preserved easily in our mutation function. We mutate both the node order and the connectivity matrix of our chromosomes.

13 We toggle the bits of the connectivity matrix based on a mutation probability. Our mutation operation for node order involves swapping two nodes randomly based on the mutation probability. 8 Results Given a Bayesian network (network structure + probability tables), the Logic Sampling method is used to generate a data set [7]. The Acyclic Bayesian Net Generator is used to learn the Bayesian network structure. For the learning process, we generate an enriched population of size p as described in section 6. In each iteration of the genetic algorithm, we produce p daughter chromosomes, and discard the worst p chromosomes (chromosomes with the lowest score) among the union of daughters and parents. The logarithm of Bayesian Dirichlet is used as the scoring function [, ]. The best network learned is compared with the optimal network. We test our algorithm on Bayesian networks of different sizes, including the ASIA (an 8 node network) [], the ALARM (a 7 node network) [], a network with 6 nodes and a node subset of ALARM. Sometimes we learn a network that has a better score than that of the original network. The term optimal network denotes the original network, unless a better network is learned over the span of all the tests performed. In our tests, we are able to learn the optimal structure for all networks, except for one case, the ALARM network. In the case of the ALARM network, we are able to learn a nearly optimal solution. (The network we learn has a score value of -, whereas the score of the original ALARM is -.) We test our algorithm by varying population size of the genetic algorithms. As the population size increases, the probability that the optimal structure is learned approaches a value of. 9 Conclusions We have presented the Acyclic Bayesian Net Generator, a new technique to learn the Bayesian network structure using genetic algorithms. Acyclic nature is inherent to the encoding scheme and is easily preserved during the learning process. In this technique, the genetic algorithm operations such as crossover and mutation are closed operations. In our method, throwing out cycle forming arcs, instead of reversing them, causes the algorithm to converge slower and produces a poorer fit Bayesian network. We hypothesize that other methods suffer when arcs are thrown out. In the Acyclic Bayesian Net Generator, we do not assume any predefined variable ordering for the structure learning. Our algorithm learns the optimal node ordering and the optimal network simultaneously. This is better than a two step approach in which the first step learns an optimal node ordering and the second step learns the actual structure.

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