# Compute conditional probabilities in Bayesian hypothesis testing

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!

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