Exact Inference Using a Regression/Attraction Method
of Markov Chain Monte Carlo
Philip J. O'Neil
AnVil Informatics, Inc.
In recent years much work has been focused on developing exact
tests for the analysis of discrete data using log
linear or logistic regression models. A parametric model is
tested for a data set by conditioning on the value of a sufficient
statistic and determining the probability of obtaining
another data set as extreme or more extreme relative to the general
model, where extremeness is determined by the value of a
test statistic.
Exact determination of these probabilities can be infeasible
for high dimensional problems and asymptotic approximations to
them are often inaccurate when there are small data entries
and/or there are many nuisance parameters. In these cases
Monte Carlo methods can be used to estimate exact probabilities
by randomly generating data sets (tables) that match the sufficient
statistic of the original table. However, naive Monte
Carlo methods produce tables that are usually far from matching
the sufficient statistic. The Markov chain Monte Carlo method
presented in this talk (the regression/attraction approach) uses
attraction to concentrate the distribution around the set of
tables that match the sufficient statistic, and uses regression
to take advantage of information in tables that ``almost'' match.
It is also more general than others in that it does not require
the sufficient statistic to be linear, and it can be adapted to
problems involving continuous data.
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