Cost-aware Generalized $α$-investing for Multiple Hypothesis Testing

We consider the problem of sequential multiple hypothesis testing with nontrivial data collection cost. This problem appears, for example, when conducting biological experiments to identify differentially expressed genes in a disease process. This work builds on the generalized $\alpha$-investing framework that enables control of the false discovery rate in a sequential testing setting. We make a theoretical analysis of the long term asymptotic behavior of $\alpha$-wealth which motivates a consideration of sample size in the $\alpha$-investing decision rule. Using the game theoretic principle of indifference, we construct a decision rule that optimizes the expected return (ERO) of $\alpha$-wealth and provides an optimal sample size for the test. We show empirical results that a cost-aware ERO decision rule correctly rejects more false null hypotheses than other methods. We extend cost-aware ERO investing to finite-horizon testing which enables the decision rule to hedge against the risk of unproductive tests. Finally, empirical tests on a real data set from a biological experiment show that cost-aware ERO produces actionable decisions as to which tests to conduct and if so at what sample size.

Maximum a-Posteriori Estimation for the Gaussian Mixture Model via Mixed Integer Nonlinear Programming

We present a global optimization approach for solving the classical maximum a-posteriori (MAP) estimation problem for the Gaussian mixture model. Our approach formulates the MAP estimation problem as a mixed-integer nonlinear optimization problem (MINLP). Our method provides a certificate of global optimality, can accommodate side constraints, and is extendable to other finite mixture models. We propose an approximation to the MINLP hat transforms it into a mixed integer quadratic program (MIQP) which preserves global optimality within desired accuracy and improves computational aspects. Numerical experiments compare our method to standard estimation approaches and show that our method finds the globally optimal MAP for some standard data sets, providing a benchmark for comparing estimation methods.