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.
Highlights
The advent of computer-automated high-throughput data-gathering technology, epitomized by the microarray, has led to the generation of so-called “large M, small n” data sets, which are those characterized by a large number, M, of variables, which are observed or measured on a relatively small number, n, of subjects or units
CONCLUDING REMARKS Bayes multiple decision function (BMDF) defines a class of multiple decision procedures that achieve optimality in the Bayesian framework associated with a general class of loss functions
The results in Theorem 1 describe the form of the BMDF and provide an efficient algorithm of finding it in multiple testing settings
Summary
The advent of computer-automated high-throughput data-gathering technology, epitomized by the microarray, has led to the generation of so-called “large M , small n” data sets, which are those characterized by a large number, M , of variables (hereon called genes for historical reasons), which are observed or measured on a relatively small number, n, of subjects or units. In addition to this decision-theoretic framework, we implement a Bayesian approach to decision-making by putting a prior probability on the unknown state of reality θ. Another major contribution is an efficient algorithm for computationally finding the Bayes multiple decision action, an algorithm that has computational order of at most O(M 2 log M ).
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