Testing point null hypotheses is a very common activity in various applied situations. However, the existing Bayesian testing procedure may give evidence which does not agree with the classical frequentist p-value in many point null testing situations. A typical example for this is the well known Lindley's paradox (Lindley in Biometrika 44:187-192, 1957). In this paper we propose an alternative testing procedure in the Bayesian framework. It is shown that for many classical testing examples, the Bayesian evidence derived by our new testing procedure is not contradictory to its frequentist counterpart any more. In fact, the new Bayesian evidence under the noninformative prior is usually coincident with the frequentist observed significance level.

The Behrens-Fisher problem concerns the inferences for the difference between the means of two normal populations without making any assumption about the variances. Although the problem has been extensively studied in the literature, researchers cannot agree on its solution at present. In this paper, we propose a new method for dealing with the Behrens-Fisher problem in the Bayesian framework. The Bayesian evidence for testing the equality of two normal means and a credible interval at a specified level for the difference between the means are derived. Simulation studies are carried out to evaluate the performance of the provided Bayesian evidence.

The problem of reconciling the frequentist and Bayesian evidence in testing statistical hypotheses has been extensively studied in the literature. Most of the existing work considers cases without the nuisance parameters which is not the frequently encountered situation since the presence of the nuisance parameters is very common in practice. In this paper, we consider the reconcilability of the Bayesian evidence against the null hypothesis H 0 in terms of the posterior probability of H 0 being true and the frequentist evidence against H 0 in terms of the P value in testing normal means where the nuisance parameters are present. The reconcilability of evidence can be obtained both for testing a normal mean and for the Behrens-Fisher problem.

The problem of comparing the frequentist evidence and the Bayesian evidence in the one-sided testing problems has been widely treated and many researches revealed that these two methods can reach an agreement approximately. However, most of the previous work dealt mainly with situations without nuisance parameters. Since the presence of nuisance parameters is very common in practice, whether these two kinds of evidence still reach an agreement is a problem worthy of study. In this article, we establish in a systematic way under the exponential distributions the agreement of the Bayesian evidence and the generalized frequentist evidence (the generalized P -value) for a variety of one-sided testing problems where the nuisance parameters are involved.

With the advent of the "Internet +" era, many teaching methods such as flip classrooms, mixed teaching, and MOOC have provided many innovative teaching methods for the majority of educators. The classroom structure has also undergone major changes in the "Internet +" era. The teaching structure has entered a mixed stage, and the teacher-student relationship tends to be equal. Teachers should change from being taught in the classroom standing in front of podium to being integrated into the "guidance" of students, using the "diagnosis" of big data, and "helping" after hiding in the net cloud. This article takes "Machinery Manufacturing Technology Foundation" as the application object, analyzes some teaching viewpoints of the "Internet +" era" Machinery Manufacturing Technology Foundation" teaching reform, and provides reference for college engineering teachers to adapt to the opportunities and challenges of the "Internet +" era.