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We present Bayesian models and computational methods for the problem of matching predictions from molecular studies with known biological pathway databases- the problem of pathway annotation of summary results of an experiment or observational study. In areas such as cancer genomics, linking quantified, experimentally defined gene expression signatures with known biological pathway gene sets is essential to improving the understanding of the complexity of molecular pathways related to outcome. Our probabilistic pathway annotation (PROPA) analysis involves new models for formal assessment and rankings of pathways putatively linked to an experimental or observational phenotype, integrates qualitative biological information into the analysis, and generates coherent inferences on uncertainties about gene pathway membership that can inform the revision of pathway databases. Our analysis relies on simulation-based computation in high-dimensional models, and introduces a novel extension of variational methods for computation of model evidence, or marginal likelihood functions, that are central to the comparison of multiple biological pathways. Examples highlight the methodology using both simulated and real data, and we develop detailed cases studies in breast cancer genomics involving hormonal pathways and pathway activities underlying cellular responses to lactic acidosis in breast cancer. The second study demonstrates the application of the method in decomposing the complexity of gene expression-based predictions about interacting biological pathway activation from both experimental (in vitro) and observational (in vivo) human cancer data.

The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data and expert opinion. Initially, the posterior distribution was approximated by samplingimportance-resampling, which is simple to implement, easy to interpret, transparent to users and gave acceptable results for most countries. For some countries, however, this is not computationally efficient because the posterior distribution tends to be concentrated around nonlinear ridges and can also be multimodal. We propose instead Incremental Mixture Importance Sampling (IMIS), which iteratively builds up a better importance sampling function. This retains the simplicity and transparency of sampling importance resampling, but is much more efficient computationally. It also leads to a simple estimator of the integrated likelihood that is the basis for Bayesian model comparison and model averaging. In simulation experiments and on real data it outperformed both sampling importance resampling and three publicly available generic Markov chain Monte Carlo algorithms for this

University of Glasgow for providing all the resources used during the implementation procedure and the researchers in the Inference Group for their help and support. Special thanks to: My supervisor, Dr. Vladislav Vyshemirsky, who was always able to find some time to explain and correct things; and without whom this work would have never been accomplished. My second supervisor, Prof. Mark Girolami, whose scientific guidance and advice were of great importance. In biochemical models defined by systems of ordinary differential equations, there is always a level of uncertainty regarding the appropriate parameter values. Bayesian methods of parameter inference and evidence-based model comparison are considered to be sound methods to handle such uncertainty; not assigning fixed values to the parameters, but using probability distributions, formally taking prior knowledge into account. BioBayes is a software

Minimum description length (MDL) model selection, in its modern NML formulation, involves a model complexity term which is equivalent to minimax/maximin regret. When the data are discrete-valued, the complexity term is a logarithm of a sum of maximized likelihoods over all possible data-sets. Because the sum has an exponential number of terms, its evaluation is in many cases intractable. In the continuous case, the sum is replaced by an integral for which a closed form is available in only a few cases. We present an approach based on Monte Carlo sampling, which works for all model classes, and gives strongly consistent estimators of the minimax regret. The estimates convergence almost surely to the correct value with increasing number of iterations. For the important class of Markov models, one of the presented estimators is particularly efficient: in empirical experiments, accuracy that is sufficient for model selection is usually achieved already on the first iteration, even for long sequences.

The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Bayesian analysis, the evaluation of marginal likelihoods remains difficult. We consider the behavior of the wellknown harmonic mean estimator (Newton and Raftery, 1994) of the marginal likelihood, which converges almost-surely but may have infinite variance and so may not obey a central limit theorem. We give examples illustrating the convergence in distribution of the harmonic mean estimator to a one-sided stable law with characteristic exponent 1 < α < 2. While the harmonic mean estimator does converge almost surely, we show that it does so at rate n −ǫ where ǫ = 1 − α −1 is often as small as 0.10 or 0.01. In such a case, the reduction of Monte Carlo sampling error by a factor of two requires increasing the Monte Carlo sample size by a factor of 2 1/ǫ, or in excess of 2.5 ·10 30 when ǫ = 0.01. We explore the possibility of estimating the parameters of the limiting stable distribution to provide accelerated convergence.

The task of calculating marginal likelihoods arises in a wide array of statistical inference problems, including the evaluation of Bayes factors for model selection and hypothesis testing. Although Markov chain Monte Carlo methods have simplified many posterior calculations needed for practical Bayesian analysis, the evaluation of marginal likelihoods remains difficult. We consider the behavior of the wellknown harmonic mean estimator (Newton and Raftery, 1994) of the marginal likelihood, which converges almost-surely but may have infinite variance and so may not obey a central limit theorem.

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