# Using Monte Carlo approximation for posterior expectation

We have two groups of people, members of each group have a chance of getting a disease $$\theta_i$$. The groups have $$n_i$$ members and $$Y_i$$ diseased members. Group 1: $$n_1=100, Y_1=21$$, Group 2: $$n_2=200,Y_2=45$$. We assume that $$\theta_i$$ has a $$Beta(1,1)$$ distribution. If I did this correctly, the posterior distribution for $$\theta_i$$ is given as $$\theta_i^{x_i}(1-\theta_i)^{x_i} {n_i\choose Y_i}$$. What I really want, though, is the posterior expectation of the odds ratio: $$\frac{\theta_1(1-\theta_2)}{\theta_2(1-\theta_1)}$$. It is suggested that I use R to employ Monte Carlo approximation, by generating samples from both posterior distributions. I am not sure how to do this. It's probably quite basic, but I haven't been properly introduced to R.

• The question has nothing to do with R in the sense that you can solve it with any programming language: the posterior distribution on $\theta_1$ is standard, the posterior distribution on $\theta_2$ is standard, hence you can simulate samples from both and deduce a sample of$$\frac{\theta_1(1-\theta_2)}{\theta_2(1-\theta_1)}$$ – Xi'an Oct 3 '18 at 18:09