The infinite Viterbi alignment and decay-convexity

TL;DR

This paper introduces decay-convexity conditions for hidden Markov models on $\

Contribution

It develops a novel approach to establish the existence and bounds of the infinite Viterbi alignment in high-dimensional settings.

Findings

Quantitative bounds on the convergence to the infinite Viterbi alignment.

Approximate estimation via parallelization is scalable and accurate.

Application to neural population activity modeling.

Abstract

The infinite Viterbi alignment is the limiting maximum a-posteriori estimate of the unobserved path in a hidden Markov model as the length of the time horizon grows. For models on state-space $\mathbb{R}^{d}$ satisfying a new ``decay-convexity'' condition, we develop an approach to existence of the infinite Viterbi alignment in an infinite dimensional Hilbert space. Quantitative bounds on the distance to the infinite Viterbi alignment, which are the first of their kind, are derived and used to illustrate how approximate estimation via parallelization can be accurate and scaleable to high-dimensional problems because the rate of convergence to the infinite Viterbi alignment does not necessarily depend on $d$. The results are applied to approximate estimation via parallelization and a model of neural population activity.