Detecting Renewal States in Chains of Variable Length via Intrinsic Bayes Factors

Markov chains with variable length are useful parsimonious stochastic models able to generate most stationary sequence of discrete symbols. The idea is to identify the suffixes of the past, called contexts, that are relevant to predict the future symbol. Sometimes a single state is a context, and looking at the past and finding this specific state makes the further past irrelevant. These states are called renewal states and they split the chain into independent blocks. In order to identify renewal states for chains with variable length, we propose the use of Intrinsic Bayes Factor to evaluate the plausibility of each set of renewal states. In this case, the difficulty lies in finding the marginal posterior distribution for the random context trees for general prior distribution on the space of context trees and Dirichlet prior for the transition probabilities. To show the strength of our method, we analyzed artificial datasets generated from two binary models models and one example coming from the field of Linguistics.