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An Information-Theoretic Empirical Bayes Method and its Application to a Systems Biology Model
(2017)
This thesis covers the development and application of an empirical Bayes method to
the problem of parameter estimation in systems biology. The goal was to provide a general
and practical solution to the Bayesian inverse problem in the case of high dimensional
parameter spaces making use of present cohort-data. We show that the maximum penalized likelihood estimator (MPLE) with information penalty is based on natural, information-theoretic considerations
and admits the desirable property of transformation invariance. Due to its concavity, the objective function is computationally feasible and its mesh-free Monte-Carlo approximation enables its application to high-dimensional problems eluding the curse of dimensionality.
We furthermore show how to apply the developed methods to a real world problem by
the means of Markov chain Monte-Carlo sampling (MCMC), affirming its proficiency in a practical scenario.
When dealing with Bayesian inference the choice of the prior often remains a debatable question. Empirical Bayes methods offer a data-driven solution to this problem by estimating the prior itself from an ensemble of data. In the nonparametric case, the maximum likelihood estimate is known to overfit the data, an issue that is commonly tackled by regularization. However, the majority of regularizations are ad hoc choices which lack invariance under reparametrization of the model and result in inconsistent estimates for equivalent models. We introduce a nonparametric, transformation-invariant estimator for the prior distribution. Being defined in terms of the missing information similar to the reference prior, it can be seen as an extension of the latter to the data-driven setting. This implies a natural interpretation as a trade-off between choosing the least informative prior and incorporating the information provided by the data, a symbiosis between the objective and empirical Bayes methodologies.