Hybrid Methods for Running MCMC over $I\times J\times K$ Contingency Tables

We consider an $I \times J\times K$ table with cell counts $X_{ijk} \geq 0$ for $i = 1, \ldots , I$, $j = 1, \ldots , J$ and $k = 1, \ldots , K$ under the no-three-way interaction model. In this paper, we propose a Markov Chain Monte Carlo (MCMC) scheme connecting the set of all contingency tables by all basic moves of $2 \times 2 \times 2$ minors with allowing $X_{ijk} \geq -1$ combined with simulated annealing. In addition, we propose a hybrid scheme of MCMC with basic moves by allowing $-1$ in cell counts combined with simulated annealing and Hit and Run (HAR) algorithm proposed by Andersen and Diaconis in order to improve a mixing time. We also compare this hybrid scheme with a hybrid method of sequential importance sampling (SIS) and MCMC introduced by Kahle, Yoshida, and Garcia-Puente. We apply these hybrid schemes to simulated and empirical data on Naval officer and enlisted population.