As MCMC (Markov Chain Monte Carlo) theory was set up, Bayes Theory could be employed in the application, because by using MCMC, it is possible for us to summarize the posterior distribution of the unknown parameter. In the process of computing the posterior, choosing the right prior is important. In recent years, the intrinsic prior draws more and more attentions of the statisticians because the use of intrinsic priors for model selection has proven to provide sensible prior for a wide variety of model selection problem, especially in the nested model scenario. For the univariate normal test, it is relatively easier to get the intrinsic prior and to check the property of it, such as proper or improper, or the type of the distribution. However, for the multivariate normal test, it is not an easy job. Then, how to get the intrinsic prior under di erent hypotheses for multivariate normal tests; Is it proper or improper? And is there any relationship between the intrinsic prior and the di erence dimension of the unknown parameters of the two models. The goal of my thesis is to study these problems. The thesis is organized as follows: in Chapter II, we recall basic theory which we will need for our discussion; in Chapter III, we carry out our arguments and prove main theorems; in Chapter IV, we summarize the results and give the conclusions; in the appendix, we list some basic probability and statistics concepts and theorems in favor of readers; at the end, an index of frequently used concepts are included.