[Paper Review] The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains

Randal Douc, François Roueff+1 more|arXiv (Cornell University)|Sep 30, 2015

Bayesian Methods and Mixture Models19 references3 citations

TL;DR

This paper establishes that, under mild conditions, the asymptotic normalized log-likelihood for partially observed Markov chains is maximized precisely over the equivalence class of the true parameter, ensuring equivalence-class consistency of the maximum likelihood estimator (MLE). The key insight relies on the uniqueness of the invariant distribution of the complete-data Markov chain, regardless of its convergence rate, and applies to hidden Markov models (HMMs) and observation-driven time series models.

ABSTRACT

This paper deals with a parametrized family of partially observed bivariate Markov chains. We establish that, under very mild assumptions, the limit of the normalized log-likelihood function is maximized when the parameters belong to the equivalence class of the true parameter, which is a key feature for obtaining the consistency of the maximum likelihood estimators (MLEs) in well-specified models. This result is obtained in the general framework of partially dominated models. We examine two specific cases of interest, namely, hidden Markov models (HMMs) and observation-driven time series models. In contrast with previous approaches, the identifiability is addressed by relying on the uniqueness of the invariant distribution of the Markov chain associated to the complete data, regardless its rate of convergence to the equilibrium.

Motivation & Objective

• To establish the consistency of the maximum likelihood estimator (MLE) in partially observed Markov models under minimal assumptions.
• To address the identifiability problem by focusing on the equivalence class of the true parameter rather than point identification.
• To show that the maximizing set of the asymptotic normalized log-likelihood corresponds exactly to parameters yielding the same observation distribution as the true parameter.
• To extend consistency results to general partially dominated models, particularly HMMs and observation-driven time series models.
• To demonstrate that uniqueness of the invariant distribution of the complete-data Markov chain is sufficient for equivalence-class consistency, irrespective of the convergence rate to equilibrium.

Proposed method

• Formalizes the asymptotic normalized log-likelihood in the context of partially observed Markov chains with latent and observed components.
• Uses the framework of partially dominated models to analyze the limit of the normalized log-likelihood function.
• Relies on the uniqueness of the invariant distribution of the complete-data Markov chain as the central technical condition.
• Applies results from Markov chain ergodicity theory, including polynomial and geometric ergodicity, to ensure existence and uniqueness of the stationary distribution.
• Employs coupling and moment conditions (e.g., existence of moments of order β > 0) to verify ergodicity and integrability of the invariant measure.
• Uses contradiction arguments involving return time moments to prove non-geometric ergodicity when exponential moments fail, supporting the main result.

Experimental results

Research questions

• RQ1 Under what conditions is the maximum likelihood estimator consistent in partially observed Markov models with latent states?
• RQ2 Can the maximizing set of the asymptotic normalized log-likelihood be characterized without requiring geometric ergodicity?
• RQ3 Is the uniqueness of the invariant distribution of the complete-data Markov chain sufficient for equivalence-class consistency of the MLE?
• RQ4 How does this framework apply to hidden Markov models (HMMs) and observation-driven time series models?
• RQ5 Can consistency be established even when the latent chain is only polynomially ergodic and not geometrically ergodic?

Key findings

• The asymptotic normalized log-likelihood is maximized precisely over the equivalence class of the true parameter, ensuring equivalence-class consistency of the MLE.
• Uniqueness of the invariant distribution of the complete-data Markov chain is sufficient for this consistency, regardless of the rate of convergence to equilibrium.
• For the HMM example with state dynamics $ X_k = (X_{k-1} + U_k - m)^+ $, the model is polynomially ergodic and admits a unique stationary distribution under mild moment conditions.
• The MLE is shown to be consistent in distribution even when geometric ergodicity fails, as demonstrated by the non-finite moment condition $ ho > 1 $ for return times.
• The proof technique avoids exponential separation conditions and instead relies on moment bounds and Doob's maximal inequality to establish non-geometric ergodicity.
• The result applies broadly to observation-driven models such as GARCH(1,1), where the conditional likelihood is tractable but consistency is non-trivial to prove.