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In Bayesian statistics a standard way to perform a Lindley significance test for the hypothesis $\theta=\theta_0$, where $\theta_0$ is the suggested value for $\theta$ at the $\alpha$ level of significane would be to construct an interval such that $Pr(c_l<\theta<c_h |Y)=1-\alpha$. Then if the value $\theta_0$ falls into the interval $[c_l,c_h]$ we accept, we otherwise we reject. However, as soon as we are in the multivariate case I struggle with finding the correct representation of the credible regions. Assume the standard linear regression model $=X\beta+\epsilon$ with conjugate conditional uninformed prior for $\beta$ and $\sigma^2$ and gaussian likelihood. This results in a multivariate student-t marginal posterior distribution for $\beta$: $$\beta|Y \sim t_N (\hat{\beta},\overline{\Sigma},T)$$ with parameters for example defined in this question. Lets assume I want to investigate the hypothesis that $\beta=\beta_0$ in this case. I understand that I can perform a Bayesian F-test by computing: $$\zeta=\frac{1}{N-1}(\hat{\beta}-\beta_0)'\hat{\Sigma}^{-1}(\hat{\beta}-\beta_0)$$ Now I can check whether $\zeta$ is smaller than the $1-\alpha$ quantile of the $F(N,T)$ distribution. If this is the case the null hypothesis is not rejected.

Is this way correct? Why can I do not compute credible regions for each of the elements of $\beta$ individually and reject the $H_0$ if one of the elements $\beta_{0,i}$ falls outside the region? To me this sounds more straightforward, but I do not find the argument that proofs me wrong. Looking forward to each and every comment.

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Why can I do not compute credible regions for each of the elements of β individually and reject the H0 if one of the elements β0,i falls outside the region

because the marginal posteriors associated to each component of $\beta$ have no reason to be independent. So you cannot simply assess each dimension independently. See this simple plot where the 5% rejection area for a bivariate zero mean normal distribution is given in red and where the 5% rejection area for each dimension independenlty are given using horizontal and vertical lines:

enter image description here

as you can see the overlap between the delineated regions is only partial.

Then, concerning the strategy you proposed for the multivariate case, maybe I missed some points but I am a bit lost. But as you mentioned it at the beginning of your question you can simply check whether $\beta_0$ lies into the region of highest probability covering $1-\alpha$% of the multivariate density. This can be done using simulation with a pretty good accuracy if $dim(\beta)$ is not to large.

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