I start with some prior, $\Theta$, with PMF $f_{\theta}(\theta)$. $\theta$ is the probability of an event, $X$, yielding a success. Then, I use Monte-Carlo simulation to sample a $\theta$ from $\Theta$, and then I simulate whether $X$ would yield a success. I then create the posterior $f_{\Theta | X}(\theta | x_1)$. I repeat this process: sample a $\theta$ from the posterior, use it simulate whether $X$ yields a success, and incorporate that into a new posterior, $f_{\Theta | X} (\theta | x_1, x_2)$. As the number of repetitions of this process goes to infinity, will this converge to the same distribution? I presume not.
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