# Bayesian hypothesis testing and its applications

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This dissertation consists of three distinct but related research projects. The first two projects focus on the Bayesian hypothesis test for correlations and partial correlations. The third project deals with the Bayesian test for equality of means in high dimensions. In the first project, we derive Bayes factor-based testing procedures for the presence of a correlation and a partial correlation. The proposed Bayesian tests are obtained by restricting the class of the alternative hypotheses to maximize the probability of rejecting the null hypothesis when the Bayes factor is larger than a specified threshold. It turns out that they depend simply on the frequentist t-statistics with the associated critical values and can thus be easily calculated by using a spreadsheet in Excel and, in fact, by just adding one more step after one has performed the frequentist correlation tests. In addition, they can yield an identical decision with the frequentist paradigm, provided that the evidence threshold of the Bayesian tests is determined by the significance level of the frequentist paradigm. We illustrate the performance of the proposed procedures through simulated and real-data examples. In the second project, we follow the work of Held and Ott [22] and propose a sample-size adjusted minimum Bayes factor (minBF) for testing the presence of a correlation and a partial correlation. The proposed minBF is related to the two-sided p-value from the frequentist test and can be easily calculated using either a pocket calculator or spreadsheets, so long as the researcher is familiar with the frequentist paradigm. It turns out that the minBF increases with an increasing sample size, which implies that the maximal evidence of the two-sided p-value decreases with increasing sample size. Simulation studies and two real-data applications are provided for illustrative purposes. In the third project, we study the problem of testing the equality of two mean vectors in high dimensional settings. We consider a permutation test from a Bayesian point of view by hierarchical clustering, which might make more efficient use of covariance structure at high dimensions. Moreover, our proposed Bayes factor is invariant under linear transformations of the marginal distributions and has a closed-form expression. We showed by simulation that our proposed Bayesian procedure has higher power than competing tests in the literature.