The infinite Viterbi alignment and decay-convexity
This work addresses a foundational challenge in statistical inference for high-dimensional hidden Markov models, with applications in fields like neuroscience, though it appears incremental in extending existing theory.
The paper tackles the problem of estimating the infinite Viterbi alignment in hidden Markov models on high-dimensional state spaces by introducing a decay-convexity condition, resulting in quantitative bounds on convergence that enable scalable approximate estimation via parallelization without dependence on dimension.
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.