We investigate the efficiency of a marginal likelihood estimator where the product of the marginal posterior distributions is used as an importance sampling function. The approach is generally applicable to multi-block parameter vector settings, does not require additional Markov Chain Monte Carlo (MCMC) sampling and is not dependent on the type of MCMC scheme used to sample from the posterior. The proposed approach is applied to normal regression models, finite normal mixtures and longitudinal Poisson models, and leads to accurate marginal likelihood estimates.

Acute myeloid leukemia (AML) is a severe, mostly fatal hematopoietic malignancy. We were interested in whether transcriptomic-based machine learning could predict AML status without requiring expert input. Using 12,029 samples from 105 different studies, we present a large-scale study of machine learning-based prediction of AML in which we address key questions relating to the combination of machine learning and transcriptomics and their practical use. We find data-driven, high-dimensional approaches-in which multivariate signatures are learned directly from genome-wide data with no prior knowledge-to be accurate and robust. Importantly, these approaches are highly scalable with low marginal cost, essentially matching human expert annotation in a near-automated workflow. Our results support the notion that transcriptomics combined with machine learning could be used as part of an integrated-omics approach wherein risk prediction, differential diagnosis, and subclassification of AML are achieved by genomics while diagnosis could be assisted by transcriptomic-based machine learning.

The power-expected-posterior (PEP) prior provides an objective, automatic, consistent and parsimonious model selection procedure. At the same time it resolves the conceptual and computational problems due to the use of imaginary data. Namely, (i) it dispenses with the need to select and average across all possible minimal imaginary samples, and (ii) it diminishes the effect that the imaginary data have upon the posterior distribution. These attributes allow for large sample approximations, when needed, in order to reduce the computational burden under more complex models. In this work we generalize the applicability of the PEP methodology, focusing on the framework of generalized linear models (GLMs), by introducing two new PEP definitions which are in effect applicable to any general model setting. Hyper-prior extensions for the power parameter that regulates the contribution of the imaginary data are introduced. We further study the validity of the predictive matching and of the model selection consistency, providing analytical proofs for the former and empirical evidence supporting the latter. For estimation of posterior model and inclusion probabilities we introduce a tuning-free Gibbs-based variable selection sampler. Several simulation scenarios and one real life example are considered in order to evaluate the performance of the proposed methods compared to other commonly used approaches based on mixtures of g-priors. Results indicate that the GLM-PEP priors are more effective in the identification of sparse and parsimonious model formulations.

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Applications of high-dimensional regression often involve multiple sources or types of covariates. We propose methodology for this setting, emphasizing the "wide data" regime with large total dimensionality p and sample size n!p. We focus on a flexible ridgetype prior with shrinkage levels that are specific to each data type or source and that are set automatically by empirical Bayes. All estimation, including setting of shrinkage levels, is formulated mainly in terms of inner product matrices of size nˆn. This renders computation efficient in the wide data regime and allows scaling to problems with millions of features. Furthermore, the proposed procedures are free of user-set tuning parameters. We show how sparsity can be achieved by post-processing of the Bayesian output via constrained minimization of a certain Kullback-Leibler divergence. This yields sparse solutions with adaptive, source-specific shrinkage, including a closed-form variant that scales to very large p. We present empirical results from a simulation study based on real data and a case study in Alzheimer's disease involving millions of features and multiple data sources.