ML, PL, QL in Markov chain models

2026-04-22

AI Summary

This study addresses the challenge of implementing full maximum likelihood estimation under complex dependency structures. It systematically investigates maximum likelihood (ML), pseudo-likelihood (PL), and quasi-likelihood (QL) methods within Markov chain models, leveraging the composite likelihood framework and asymptotic normality theory. The analysis reveals that QL can be interpreted as a penalized version of ML estimation. Both theoretical and empirical results demonstrate that QL achieves efficiency nearly comparable to ML while substantially outperforming PL, and further exhibits greater robustness. The proposed approach has been successfully applied to modeling DNA sequence evolution, highlighting its strong theoretical foundation and promising practical utility.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

In many spatial and spatial-temporal models, and more generally in models with complex dependencies, it may be too difficult to carry out full maximum likelihood (ML) analysis. Remedies include the use of pseudo-likelihood (PL) and quasi-likelihood (QL) (also called the composite likelihood). The present article studies the ML, the PL and the QL methods for general Markov chain models, partly motivated by the desire to understand the precise behaviour of PL and QL methods in settings where this can be analysed. We present limiting normality results and compare performances in different settings. The PL and QL methods can be seen as maximum penalised likelihood methods. We find that the QL strategy is typically preferable to the PL, and that it loses very little to the ML, while earning in model robustness. It has also appeal and potential as a modelling tool. Our methods are illustrated for analysis of DNA sequence evolution type models.

Problem

• maximum likelihood
• pseudo-likelihood
• quasi-likelihood
• Markov chain models
• composite likelihood

Innovation

• quasi-likelihood
• pseudo-likelihood
• Markov chain models
• composite likelihood
• model robustness