Tractable Inference for Complex Stochastic Processes (1301.7362v1)

Abstract: The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory. In the case of a stochastic system, these tasks typically involve the use of a belief state- a probability distribution over the state of the process at a given point in time. Unfortunately, the state spaces of complex processes are very large, making an explicit representation of a belief state intractable. Even in dynamic Bayesian networks (DBNs), where the process itself can be represented compactly, the representation of the belief state is intractable. We investigate the idea of maintaining a compact approximation to the true belief state, and analyze the conditions under which the errors due to the approximations taken over the lifetime of the process do not accumulate to make our answers completely irrelevant. We show that the error in a belief state contracts exponentially as the process evolves. Thus, even with multiple approximations, the error in our process remains bounded indefinitely. We show how the additional structure of a DBN can be used to design our approximation scheme, improving its performance significantly. We demonstrate the applicability of our ideas in the context of a monitoring task, showing that orders of magnitude faster inference can be achieved with only a small degradation in accuracy.

• The paper introduces a tractable belief state approximation technique for DBNs, reducing error accumulation in dynamic stochastic systems.
• It leverages exponential contraction of the Kullback-Leibler divergence to bound errors over successive updates.
• Empirical tests on WATER and BAT networks demonstrate significant speed improvements with minimal accuracy loss.

Tractable Inference for Complex Stochastic Processes

The paper "Tractable Inference for Complex Stochastic Processes" by Xavier Boyen and Daphne Koller addresses the challenges in reasoning about complex stochastic systems, particularly within the context of dynamic Bayesian networks (DBNs). The authors explore the problem of maintaining a belief state—a probability distribution over a system's states—that evolves over time in a computationally feasible manner. They propose strategies for approximating belief states to make the inference process tractable without letting errors accumulate to unacceptable levels.

The paper begins by highlighting the necessity of belief states in monitoring and controlling dynamic systems, especially within stochastic realms where systems are partially observable. Traditional models such as Hidden Markov Models and Kalman Filters are insufficient for representing complex networks due to their large state spaces. Even DBNs, which introduce compact representations through state variable decomposition and conditional independence, fall short when processing belief states, as dependencies tend to proliferate over time.

To address these challenges, the authors propose a scheme that involves maintaining compact approximations of belief states. This approach is grounded in the insight that errors in belief states contract exponentially over consecutive updates. The paper introduces a theoretical framework demonstrating that stochastic processes inherently reduce the divergence between approximate and true belief states, thereby bounding the errors indefinitely. This contraction result is novel in its focus on the relative entropy (Kullback-Leibler divergence) and leverages the structure of DBNs to tailor approximations further, enhancing inference efficiency.

The authors extend their focus to structured processes composed of weakly interacting subprocesses, crafting an approximation that decomposes belief states into products of independent states. They prove that with this structure-aware methodology, the contraction rate improves, thus reinforcing the reliability and efficiency of inference. This insight paves the way for applications in areas that deal with large-scale, stochastic environments.

Empirical evaluations were conducted using networks such as the WATER network, associated with water purification systems, and the BAT network, related to freeway traffic monitoring. The results indicate orders of magnitude speed improvements with minimal accuracy degradation. Variations in clustering, approximation schemes, and the connectivity of subprocesses were tested, showing the sensitivity of error bounds to the structural properties of the network, with structured decompositions yielding the best performance.

The implications of this research are substantial. Practically, it provides scopes where DBNs can be efficiently employed in monitoring tasks despite high-dimensional state spaces. Theoretically, it deepens understanding of error dynamics in approximate inference, offering a rigorous basis for tackling challenges in temporal probabilistic reasoning.

In future developments, this work suggests exploring other representations for belief states, particularly those allowing conditional independence or context-sensitive dependencies. There is also potential for extending these results to continuous processes or those involving backward inference, enhancing applications in learning and decision-making.

This paper stands as a significant step toward managing complexity in stochastic systems through structures that leverage inherent properties for computational efficiency, without compromising significantly on accuracy. As AI continues to engage with dynamic, uncertain environments, these insights will be critical in crafting robust, practical solutions.