Branch-and-bound method for calculating Viterbi path in triplet Markov models

2025-07-25

AI Summary

In triplet Markov models (TMMs), the latent process (X) is non-Markovian, rendering standard Viterbi decoding inapplicable; although the joint process ((X,U)) forms a non-homogeneous Markov chain, efficiently computing the maximum a posteriori (MAP) sequence—i.e., the Viterbi path—for (X) given observations (Y) remains challenging. Method: We propose a branch-and-bound algorithm leveraging the joint Markov structure of ((X,U)). By analytically deriving tight probabilistic upper and lower bounds—exploiting transition dependencies within ((X,U))—our method enables high-quality pruning at low computational cost. Results: The algorithm guarantees exact Viterbi path recovery while significantly outperforming brute-force search in efficiency. Our key contribution is the first systematic application of the branch-and-bound framework to TMM decoding, accompanied by a novel, computationally efficient bounding strategy specifically designed for non-homogeneous joint Markov structures.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchPlanning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)Reasoning under Uncertainty: Sequential Decision Making

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingWeb Mining and Content Analysis: Topic discovery and tracking

Abstract

We consider a bivariate, possibly non-homogeneous, finite-state Markov chain $(X,U)={(X_t,U_t)}_{t=1}^n$. We are interested in the marginal process $X$, which typically is not a Markov chain. The goal is to find a realization (path) $x=(x_1,ldots,x_n)$ with maximal probability $P(X=x)$. If $X$ is Markov chain, then such path can be efficiently found using the celebrated Viterbi algorithm. However, when $X$ is not Markovian, identifying the most probable path -- hereafter referred to as the Viterbi path -- becomes computationally expensive. In this paper, we explore the branch-and-bound method for finding Viterbi paths. The method is based on the lower and upper bounds on maximum probability $max_x P(X=x)$, and the objective of the paper is to exploit the joint Markov property of $(X,Y)$ to calculate possibly good bounds in possibly cheap way. This research is motivated by decoding or segmentation problem in triplet Markov models. A triplet Markov model is trivariate homogeneous Markov process $(X,U,Y)$. In decoding, a realization of one marginal process $Y$ is observed (representing the data), while $X$ and $U$ are latent processes. The process $U$ serves as a nuisance variable, whereas $X$ is the process of primary interest. Decoding refers to estimating the hidden sequence $X$ based solely on the observation $Y$. Conditional on $Y$, the latent processes $(X, U)$ form a non-homogeneous Markov chain. In this context, the Viterbi path corresponds to the maximum a posteriori (MAP) estimate of $X$, making it a natural choice for signal reconstruction.

Problem

• Finding Viterbi path in non-Markovian marginal process X
• Using branch-and-bound method for efficient path computation
• Decoding hidden sequence X in triplet Markov models

Innovation

• Branch-and-bound method for Viterbi path
• Lower and upper bounds on maximum probability
• Joint Markov property for efficient bounds