Discovering the hidden structure of complex dynamic systems(3)

we propose a new approximation that circumvents this bottleneck. Roughly speaking, this approximation estimates the posterior probability of the event for which we want statistics by a product of small factors. As we show, this estimate is quite good, and can be computed e ciently as a by-product of the inference algorithm. Our nal contribution addresses a fundamental problem in learning models for dynamic systems. In order to learn models that are statistically robust and computationally tractable, we must often introduce hidden variables into our structure. These variables serve many roles: enabling the Markov property, capturing hidden in uences of the observables, etc. It is possible, in theory, to discover hidden variables simply by

introducing them into the model and hoping that the search algorithm learns their meaning automatically (as in Friedman et al. 1998]). We propose a technique that allows for a more targeted search for hidden variables. Our approach is based on the observation that, in DBNs, ignoring a hidden variable typically results in a violation of the Markov property. For example, in a DBN for tracking trafc, eliminating the velocity variable would result in a non-Markovian dependence of the current location on the location at the two previous time slices. Our algorithm exploits this property by explicitly searching for violations of the Markov property. For example, a signi cant correlation between a variable A(+1) and some other earlier variable B (? ) (given the Parents of A(+1) ) indicates that A must have additional inuences not present in our model. In the general case there will be an additional hidden process that in uences both A and B, and possibly other variables as well. We thus add an extra variable in our model to account for this hidden process. We believe that our algorithmic ideas combine to provide a viable solution to the problem of learning complex dynamic systems from data. We provide some empirical results on real and synthetic data that, while very preliminary, are quite promising in their ability to scale up to larger systems.t t d t

2 Preliminaries: Learning DBNs2.1 Dynamic Bayesian NetworksA dynamic Bayesian network (DBN) is a model of the evolution of a stochastic system over time. We choose to model the system evolution using a sequence of discrete time points at regular intervals. At each point in time, the instantaneous state of a process is specied in terms of a set of attributes X1;:::; X . We use X ( ) to denote the random variable corresponding to the attribute X at time t. A DBN represents a distribution over (possibly in nite) trajectories of the system. In order to allow such a distribution to be represented, we usually make two assumptions. The Markov assumption states that the future is independent of the past given the present. More precisely, for all t, we have the conditional independence I (X(+1); fX(0);::: X(?1) g j X( ) ). This assumption allows us to reduce t

he representation problem to that of specifying P (X(0) ) and P (X(+1) j X( ) ) for all t. The second assumption is that the process is stationary (or time-invariant ), i.e., that P (X(+1) j X( )) is the same for all t. Given these assumptions, we can specify the proban t i i t t t t t t t

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