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I would like to solve a problem by the Bayesian approach.
I have measurement samples $\texttt{y}_{1}$ and $\texttt{y}_{2}$, they are Gaussian independent with (scalar) means $\theta_{i}$; $i=1,2$ and the same variance $\sigma^{2}$. I want to test whether $\theta_{1}>\theta_{2}$ (hyp. $H_{1}$) or instead $\theta_{1}\leq\theta_{2}$ (hyp. $H_{2}$). Assuming I know the joint a posteriori density of the two random variables $\theta_{i}$; $i=1,2$, what is the expression that I should use to compute the two conditional probabilities $P(H_{i}|\texttt{y}=y)$, $i=1,2$?
EDIT: with $P(H_{i}|\texttt{y}=y)$, $i=1,2$ obviously I mean $P(\theta_{1}>\theta_{2}|y_{1},y_{2})$ and $P(\theta_{1}\leq\theta_{2}|y_{1},y_{2})$.
Thanks!

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    $\begingroup$ $P(\theta_1>\theta_2\mid y_1,y_2)$ and $P(\theta_1\leq \theta_2\mid y_1,y_2)$? $\endgroup$ Jun 13 at 3:51
  • $\begingroup$ Yes! I can't write them formally. $\endgroup$
    – Empty
    Jun 13 at 8:36

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