Computationally tractable fitting of graphical models : the cost and benefits of decomposable Bayesian and penalized likelihood approaches : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Albany, New Zealand
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2012
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Massey University
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Abstract
Gaussian graphical models are a useful tool for eliciting information about relationships
in data with a multivariate normal distribution. In the rst part of this thesis
we demonstrate that partial correlation graphs facilitate di erent and better insight
into high-dimensional data than sample correlations. This raises the question of
which method one should use to model and estimate the parameters. In the second,
and major part, we take a more theoretical focus examining the costs and bene ts of
two popular approaches to model selection and parameter estimation (penalized likelihood
and decomposable Bayesian) when the true graph is non-decomposable.
We rst consider the e ect a restriction to decomposable models has on the estimation
of both the inverse covariance matrix and the covariance matrix. Using the
variance as a measure of variability we compare non-decomposable and decomposable
models. Here we nd that, if the true model is non-decomposable, the variance
of estimates is demonstrably larger when a decomposable model is used. Although
the cost in terms of accuracy is fairly small when estimating the inverse covariance
matrix, this is not the case when estimation of the covariance matrix is the goal. In
this case using a decomposable model caused up to 200-fold increases in the variance
of estimates.
Finally we compare the latest decomposable Bayesian method (the feature-inclusion
stochastic search) with penalized likelihood methods (graphical lasso and adaptive
graphical lasso) on measures of model selection and prediction performance. Here
we nd that graphical lasso is clearly outclassed on all measures by both adaptive
graphical lasso and feature-inclusion stochastic search. The sample size and the
ultimate goal of the estimation will determine whether adaptive graphical lasso or
feature-inclusion stochastic search is better.
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Graphical modeling (Statistics), Bayesian statistical decision theory