On Large Lag Smoothing for Hidden Markov Models

TL;DR

This paper introduces a multilevel Monte Carlo method with optimal transport for efficient large lag smoothing in hidden Markov models, significantly reducing computational cost compared to traditional methods.

Contribution

It proposes a novel MLMC approach using Knothe-Rosenblatt rearrangement for large lag smoothing in HMMs, improving efficiency and error bounds.

Findings

Achieves mean square error of O(ε^2) with cost O(ε^{-2})

Reduces computational complexity from O(nε^{-2}) to O(ε^{-2})

Demonstrates effectiveness through numerical examples

Abstract

In this article we consider the smoothing problem for hidden Markov models (HMM). Given a hidden Markov chain $\{X_n\}_{n\geq 0}$ and observations $\{Y_n\}_{n\geq 0}$, our objective is to compute $\mathbb{E}[\varphi(X_0,\dots,X_k)|y_{0},\dots,y_n]$ for some real-valued, integrable functional $\varphi$ and $k$ fixed, $k \ll n$ and for some realisation $(y_0,\dots,y_n)$ of $(Y_0,\dots,Y_n)$. We introduce a novel application of the multilevel Monte Carlo (MLMC) method with a coupling based on the Knothe-Rosenblatt rearrangement. We prove that this method can approximate the afore-mentioned quantity with a mean square error (MSE) of $\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$ with a cost of $\mathcal{O}(\epsilon^{-2})$. This is in contrast to the same direct Monte Carlo method, which requires a cost of $\mathcal{O}(n\epsilon^{-2})$ for the same MSE. The approach we suggest is, in general, not possible to implement, so the optimal transport methodology of \cite{span} is used, which directly approximates our strategy. We show that our theoretical improvements are achieved, even under approximation, in several numerical examples.