  We introduce a Bayesian approach to discovering patterns in structurally
complex processes. The proposed method of Bayesian Structural Inference (BSI)
relies on a set of candidate unifilar HMM (uHMM) topologies for inference of
process structure from a data series. We employ a recently developed exact
enumeration of topological epsilon-machines. (A sequel then removes the
topological restriction.) This subset of the uHMM topologies has the added
benefit that inferred models are guaranteed to be epsilon-machines,
irrespective of estimated transition probabilities. Properties of
epsilon-machines and uHMMs allow for the derivation of analytic expressions for
estimating transition probabilities, inferring start states, and comparing the
posterior probability of candidate model topologies, despite process internal
structure being only indirectly present in data. We demonstrate BSI's
effectiveness in estimating a process's randomness, as reflected by the Shannon
entropy rate, and its structure, as quantified by the statistical complexity.
We also compare using the posterior distribution over candidate models and the
single, maximum a posteriori model for point estimation and show that the
former more accurately reflects uncertainty in estimated values. We apply BSI
to in-class examples of finite- and infinite-order Markov processes, as well to
an out-of-class, infinite-state hidden process.
