Analytic Properties of Trackable Weak Models

We present new results on inferring the hidden states in trackable weak models. A weak model is a directed graph where each node has a set of colors which may be emitted when that node is visited. A hypothesis is a node sequence consistent with a given color sequence. A weak model is trackable if the worst case number of hypotheses grows polynomially in the sequence length. We show that the number of hypotheses in strongly-connected trackable models is bounded by a constant. We also consider the problem of reconstructing which branch was taken at a node with same-colored out-neighbors, and show that it is always eventually possible to identify which branch was taken if the model is strongly connected and trackable. We illustrate these properties by employing standard tools for analyzing Markov chains. In addition, we present new results for the entropy rates of weak models. These theorems indicate that the combination of trackability and strong connectivity simplifies the task of reconstructing which nodes were visited. This work has implications for problems which can be described in terms of an agent traversing a colored graph, such as the reconstruction of hidden states in a hidden Markov model (HMM).