Reductions of Hidden Information Sources

In all but special circumstances, measurements of time-dependent processes reflect internal structures and correlations only indirectly. Building predictive models of such hidden information sources requires discovering, in some way, the internal states and mechanisms. Unfortunately, there are often many possible models that are observationally equivalent. Here we show that the situation is not as arbitrary as one would think. We show that generators of hidden stochastic processes can be reduced to a minimal form and compare this reduced representation to that provided by computational mechanics—the -machine. On the way to developing deeper, measure-theoretic foundations for the latter, we introduce a new two-step reduction process. The first step (internal-event reduction) produces the smallest observationally equivalent σ-algebra and the second (internal-state reduction) removes σ-algebra components that are redundant for optimal prediction. For several classes of stochastic dynamical systems these reductions produce representations that are equivalent to -machines.