Markov Chain Monte Carlo sampling for conditional tests: A link between permutation tests and algebraic statistics

We consider conditional tests for non-negative discrete exponential families. We develop two Markov Chain Monte Carlo (MCMC) algorithms which allow us to sample from the conditional space and to perform approximated tests. The first algorithm is based on the MCMC sampling described by Sturmfels. The...

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Bibliographic Details
Published in:arXiv.org 2017-07
Main Authors: Fontana, Roberto, Crucinio, Francesca Romana
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Approximation
Bayesian analysis
Computer simulation
Distribution functions
Markov analysis
Markov chains
Monte Carlo simulation
Orbits
Partitions
Permutations
Sampling
Statistical tests
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Summary:We consider conditional tests for non-negative discrete exponential families. We develop two Markov Chain Monte Carlo (MCMC) algorithms which allow us to sample from the conditional space and to perform approximated tests. The first algorithm is based on the MCMC sampling described by Sturmfels. The second MCMC sampling consists in a more efficient algorithm which exploits the optimal partition of the conditional space into orbits of permutations. We thus establish a link between standard permutation and algebraic-statistics-based sampling. Through a simulation study we compare the exact cumulative distribution function (cdf) with the approximated cdfs which are obtained with the two MCMC samplings and the standard permutation sampling. We conclude that the MCMC sampling which exploits the partition of the conditional space into orbits of permutations gives an estimated cdf, under \(H_0\), which is more reliable and converges to the exact cdf with the least steps. This sampling technique can also be used to build an approximation of the exact cdf when its exact computation is computationally infeasible.
ISSN:2331-8422