Bayesian hypothesis testing

Question

Let $x_1,\ldots,x_4$ be a sample taken from the uniform dstribution with the density $$ f_{\theta}(x)=\theta^{-1} \cdot 1_{(0,\theta)}(x). $$ Assume that $\theta$ is a random variable with the density $$ \pi (\theta)=\frac{4}{3}\theta^4 e^{-2 \theta}\cdot 1_{(0,\infty)}(\theta). $$ We reject $H_0: \theta\leq 3$ (with $H_1: \theta > 3$) for all such $(x_1,x_2,x_3,x_4)$ for which the posteriori probability of the set $\{\theta: \theta>3\}$ is greater than $\frac{1}{2}$. Calculate the significance level of a test.

Could you help me please with this exercise? It is taken from the actuary exam organized in my country - for me this is the hardest exercise ever. I even don't know what significance level of a test is in Bayesian contecst.

My work so far: I am able to calculate a posteriori probability: it has the shifted exponential distribution with the density $$ l(\theta|x_1,\ldots,x_4)=2e^{-2(\theta-\max\{x_i\})}1_{(\max\{x_i\},\infty)}(\theta). $$

Answer 1

To repeat my comments, since they appear to have been helpful:

Presumably you have to start by calculating the $k$ such that you will reject $H_0$ if $\max\{x_i\}\gt k$.

I might guess at calculating the significance as the probability that $\max\{X_i\}\gt k$ conditioned on $\theta \le 3$ using the prior distribution, i.e. the probability of erroneously rejecting $H_0$ when it is true.