Bayes multiple decision functions

Abstract

This paper deals with the problem of simultaneously making many (M) binary decisions based on one realization of a random data matrix X. M is typically large and X will usually have M rows associated with each of the M decisions to make, but for each row the data may be low dimensional. Such problems arise in many practical areas such as the biological and medical sciences, where the available dataset is from microarrays or other high-throughput technology and with the goal being to decide which among of many genes are relevant with respect to some phenotype of interest; in the engineering and reliability sciences; in astronomy; in education; and in business. A Bayesian decision-theoretic approach to this problem is implemented with the overall loss function being a cost-weighted linear combination of Type I and Type II loss functions. The class of loss functions considered allows for use of the false discovery rate (FDR), false nondiscovery rate (FNR), and missed discovery rate (MDR) in assessing the quality of decision. Through this Bayesian paradigm, the Bayes multiple decision function (BMDF) is derived and an efficient algorithm to obtain the optimal Bayes action is described. In contrast to many works in the literature where the rows of the matrix X are assumed to be stochastically independent, we allow a dependent data structure with the associations obtained through a class of frailty-induced Archimedean copulas. In particular, non-Gaussian dependent data structure, which is typical with failure-time data, can be entertained. The numerical implementation of the determination of the Bayes optimal action is facilitated through sequential Monte Carlo techniques. The theory developed could also be extended to the problem of multiple hypotheses testing, multiple classification and prediction, and high-dimensional variable selection. The proposed procedure is illustrated for the simple versus simple hypotheses setting and for the composite hypotheses setting through simulation studies. The procedure is also applied to a subset of a microarray data set from a colon cancer study.

Classes of multiple decision functions strongly controlling FWER and FDR

“…Certainly, another approach to eliminating the need to specify a fixed P 1 to obtain the ROC functions is to use a Bayesian approach. This programme was partly done in [24], though more work is still needed to clarify and fully understand this approach.…”

“…Recent papers dealing with multiple decision functions (MDFs) with certain optimality properties are [12,5,13,6,14–19]. On the other hand, papers proposing MDFs with a Bayes or empirical Bayes flavor, are [20,21,1, 22]; and more recently, [23,24]. …”

The effects of pediatric epilepsy on a language connectome

“…Empirically, however, simulations in various cases, including the statistical model considered in Section Local Network Features, showed very similar results to BH, except for some rare cases were it showed slightly better results. Several real data applications in gene expression microarrays analysis support these observations [Muller et al, 2004;Sarkar et al, 2008;Wu and Pena, 2013]. The BH method has become widely adopted because of its ease of computation and implementation.…”