  This paper studies the problem of ergodicity of transition probability
matrices in Markovian models, such as hidden Markov models (HMMs), and how it
makes very difficult the task of learning to represent long-term context for
sequential data. This phenomenon hurts the forward propagation of long-term
context information, as well as learning a hidden state representation to
represent long-term context, which depends on propagating credit information
backwards in time. Using results from Markov chain theory, we show that this
problem of diffusion of context and credit is reduced when the transition
probabilities approach 0 or 1, i.e., the transition probability matrices are
sparse and the model essentially deterministic. The results found in this paper
apply to learning approaches based on continuous optimization, such as gradient
descent and the Baum-Welch algorithm.
