[Paper Review] Markov chain Monte Carlo test of toric homogeneous Markov chains

Akimichi Takemura, Hisayuki Hara|arXiv (Cornell University)|Apr 20, 2010

Markov Chains and Monte Carlo Methods13 references7 citations

TL;DR

This paper develops Markov bases for conditional goodness-of-fit tests of toric homogeneous Markov chains (THMC), a curved exponential family model that serves as the envelope of the standard homogeneous Markov chain. It provides a complete Markov basis for two-state chains of arbitrary length and for three-time-point chains with any finite state space, enabling exact conditional inference via MCMC, with a numerical example showing rejection of homogeneity due to poor asymptotic approximation.

ABSTRACT

Markov chain models are used in various fields, such behavioral sciences or econometrics. Although the goodness of fit of the model is usually assessed by large sample approximation, it is desirable to use conditional tests if the sample size is not large. We study Markov bases for performing conditional tests of the toric homogeneous Markov chain model, which is the envelope exponential family for the usual homogeneous Markov chain model. We give a complete description of a Markov basis for the following cases: i) two-state, arbitrary length, ii) arbitrary finite state space and length of three. The general case remains to be a conjecture. We also present a numerical example of conditional tests based on our Markov basis.

Motivation & Objective

• To develop exact conditional tests for toric homogeneous Markov chain (THMC) models, which are the envelope exponential family of standard homogeneous Markov chains.
• To address the limitation of large-sample chi-squared approximations in small-sample settings for Markov chain model testing.
• To derive a complete Markov basis for THMC models in cases where the usual homogeneous Markov chain model is a curved exponential family.
• To enable exact inference via Markov chain Monte Carlo (MCMC) using the derived Markov bases.
• To investigate structural properties of Markov bases for general S and T, proposing a conjecture on extended simple loops.

Proposed method

• Constructs the THMC model as an envelope exponential family with free parameters γi and βij, allowing normalization only over the total probability.
• Uses the Markov basis methodology (Diaconis and Sturmfels) to derive moves that preserve sufficient statistics under the null hypothesis of homogeneity.
• Applies the distance reduction argument of Takemura and Aoki to prove completeness of the Markov basis for S=2 and T≥3.
• Employs a computer algebra system (4ti2) for computational verification in small cases, such as S=3, T=5.
• Defines and analyzes 'extended simple loops' in edge-sign pattern graphs to conjecture a general structure for Markov bases.
• Performs MCMC sampling using the derived Markov basis to compute exact p-values for test statistics like Pearson’s chi-squared.

Experimental results

Research questions

• RQ1 What is the complete Markov basis for the toric homogeneous Markov chain model when the state space has two states and arbitrary length?
• RQ2 What is the complete Markov basis for the THMC model when the chain has three time points and an arbitrary finite state space?
• RQ3 How can exact conditional tests be performed for THMC models using MCMC when asymptotic approximations are unreliable?
• RQ4 What structural properties characterize the Markov basis for general S and T, and can they be described via extended simple loops in edge-sign pattern graphs?
• RQ5 Is there a uniform bound on the number of extended simple loops needed to generate all moves in a Markov basis for a given S?

Key findings

• A complete Markov basis is derived for two-state toric homogeneous Markov chains of any length T≥3, with moves based on simple loops.
• A complete Markov basis is derived for three-time-point chains with any finite state space S, using moves involving at most three extended simple loops.
• The MCMC-based exact test on the marijuana use data yields an exact p-value of 0.0184, slightly lower than the asymptotic p-value of 0.0212, indicating rejection of the THMC model.
• The asymptotic chi-squared approximation is found to be poor for the data due to low cell frequencies, highlighting the need for exact methods.
• The authors conjecture that for each S, there exists a bound kS such that all moves in the Markov basis are sums of at most kS extended simple loops, with kS=S as a strong candidate.
• For S=2, the Markov basis consists of moves that are sums of at most two extended simple loops, and for S=3, T=4, moves involve up to three such loops.