Bayesian Tail Probability Estimation and Model Selection

Bayesian statistics is a prevalent and important field in statistics that assigns Bayesian probabilities, which represent a state of knowledge, to unknown quantities. We study Bayesian statistics with its applications through two projects in this report.In the first project, we investigate the reasons that the Bayesian estimator of the tail probability is always higher than the frequentist estimator. Sufficient conditions for this phenomenon are established by looking at Taylor series approximations about the tail and by using Jensen's Inequality, both of which point to the convexity of the distribution function.The second project is about redefining the Bayesian information criterion (BIC) in the model selection procedure using the effective sample size, which has a better theoretical foundation in the circumstance that mixed-effects models involve. Numerical experiment results are also given by comparing the performance of our new BIC with other widely used BICs.