Discovering the hidden structure of complex dynamic systems(4)

Dynamic Bayesian networks provide a compact and natural representation for complex dynamic systems. However, in many cases, there is no expert available from whom a model can be elicited. Learning provides an alternative approach for constructing models of

bilistic model of a DBN using two Bayesian network (BN) fragments. The rst is a prior network B0 that speci es a distribution over initial states X(0) . The second is a transition network B!, which represents the transition probability from states X( ) to states X(+1). The transition network is a BN fragment over 0 the nodes fX1;:::; X; X1;:::; X 0 g. A node X rep( ) 0 represents X (+1) . The nodes X resents X and X in the network are forced to be roots (i.e., have no parents), and are not associated with conditional probability distributions. We denote the parents of X 0 in the graph by Pa(X 0 ). Each node X 0 is associated with a conditional probability distribution (CPD), which speci es P (X 0 j Pa(X 0 )). The transition probability from one stateQ to another x0| P (x0 j x)| is then x de ned to be P (x0 j u ), where u is the value in x; x0 of the variables in Pa(X 0). A DBN de nes a distribution over in nite trajectories of states. In practice, we reason only about a nite time interval 0;:::; T . To do this reasoning, we can notionally\unroll" the DBN structure into a long BN over X(0);:::; X( ) . In time slice 0, the parents of X (0) and its CPD are those speci ed in the prior network B0; in slice t+1, the parents of X (+1) and its CPD are those speci ed for X 0 in B! . Thus, given a DBN B= (B0; B! ), the joint distribution over X(0);:::; X( ) is P (x(0);:::; x( ) )t t n t n t i i i i i i i i i i i i i i i T i t i i T B T

it is thus an estimate of how well a given candidate ts the empirical data. The log-likelihood depends on the su cient statistics that summarize the frequencies of the relevant events in the data. For any event y over X; X0, we de ne

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