Denoising the Future: Top-p Distributions for Moving Through Time

TL;DR

This work tackles the computational burden of inference in dynamic probabilistic models, particularly Hidden Markov Models, by pruning to the top-$p$ most probable future events and renormalizing to form a top-$p$ HMM. The authors derive a per-step error bound of $1-p$ and a time-evolving bound of $\delta(P^k,Q^k) \le \frac{1-p}{\gamma}$, where $\gamma$ is the minimal mixing rate, and demonstrate substantial sparsity and runtime gains across several HMMs, with a total-variation error below 0.09 for $p=0.9$. Empirical results on Bell HMM, Uniform HMM, and LM HMM show sparsity above 0.9 and speedups of at least an order of magnitude, validating the approach while highlighting potential biases in minority transitions. The method offers a practical pathway to denoised, efficient inference suitable for edge computing and other resource-constrained settings, with a clear error bound to guide usage.

Abstract

Inference in dynamic probabilistic models is a complex task involving expensive operations. In particular, for Hidden Markov Models, the whole state space has to be enumerated for advancing in time. Even states with negligible probabilities are considered, resulting in computational inefficiency and increased noise due to the propagation of unlikely probability mass. We propose to denoise the future and speed up inference by using only the top-p states, i.e., the most probable states with accumulated probability p. We show that the error introduced by using only the top-p states is bound by p and the so-called minimal mixing rate of the underlying model. Moreover, in our empirical evaluation, we show that we can expect speedups of at least an order of magnitude, while the error in terms of total variation distance is below 0.09.

Theorems & Definitions (1)

• definition thmcounterdefinition: Hidden Markov Model hmm