Bayesian hypothesis testing of two parameters

Question

This question is about hypothesis testing in the Bayesian framework which I am new to.

Suppose I have two independent Poisson models with parameters $\lambda_1$ and $\lambda_2$ such that $X \sim Pois(\lambda_1)$ and $Y \sim Pois(\lambda_2)$. $n$ observations are drawn from each of the distributions $X$ and $Y$. If I provide priors $\pi (\lambda_1)$ and $\pi (\lambda_2)$, I can get the posteriors $\pi (\lambda_1 |x)$ and $\pi (\lambda_2 |y)$.

My question is how do I test the hypothesis that $\lambda_1 = \lambda_2$ against the alternative hypothesis $\lambda_1 < \lambda_2$ using a Bayesian approach?

What I did was to find the 95% credible interval for $\lambda_1$ which I denoted as $C_1$. And then I set my null hypothesis to be $H_0:\lambda_2 \in C_1$ vs $H_1:\lambda_2>max (C_1)$ and test to minimize Bayes risk. Is this an appropriate way of testing the hypothesis? If not, what would be a better way?

Thanks.

Answer 1

Point hypothesis are not common(reasonable) in a Bayesian framework since their probability, under a continuous model, is zero.

Something that seems to fit in your context is the calculation of the posterior stress-strength coefficient

$$\theta=P(\lambda_1\lt \lambda_2) = \int_{0}^{\infty}\int_{0}^{l_2}\pi_{\lambda_1\vert{\bf x}}(l_1)\pi_{\lambda_2\vert{\bf x}}(l_2)dl_1 dl_2.$$

The posterior densities involved in this expression can be approximated using a kernel density estimator and the posterior simulations.

However, note that this is only quantifying the uncertainty about $H:\lambda_1\lt \lambda_2$ and that $P(\lambda_1=\lambda_2\vert \text{Observations})=0$. Then, you may want to think what is the question of interest that you want to answer.