Bayes Factor with Lindley Paradox and Tow Standard Methods in Model

Abstract

For any statistical analysis, Model selection is necessary and required. In many cases of selection, Bayes factor is one of the important basic elements. For the unilateral hypothesis testing problem, we extend the harmony of frequency and Bayesian evidence to the generalized p-value of unilateral hypothesis testing problem, and study the harmony of generalized P-value and posterior probability of original hypothesis. For the problem of single point hypothesis testing, the posterior probability of the Bayes evidence under the traditional Bayes testing method, that is, the Bayes factor or the single point original hypothesis is established, is analyzed , a phenomenon known as the Lindley paradox, which is at odds with the classical frequency evidence of p-value. At this point, many statisticians have been worked for this from both frequentist and Bayesian perspective. In this paper , I am going to focus on Bayesian approach to model selection , s tarting from Bayes factors and go ing within Lindley Paradox, which also briefly talks about partial and fractional Bayes factor. Trying to use a simple way to consider this paradox is the thing what I want to do in the paper. On the other hand, a detailed derivation of BIC and AIC is given in S ection 4. The guiding principle of selecting the optimal model is to investigate from two aspects: one is to maximize the likelihood function, the other is to minimize the number of unknown parameters in the model. The larger the likelihood function value, the better the model fitting, but we can not simply measure the model fitting accuracy, which leads to more and more unknown parameters in the model, and the model that becomes more and more complex would have caused an overmatch. Therefore, a good model should be the combination of the fitting accuracy and the number of unknown parameters to optimize the configuration.