Can the variance of my posterior distribution reflect the amount of missing data from my sample? I have treatment and outcome data (both binary, 100 samples) from two exchangeable populations which I'd like to contrast. The outcome data is partially missing, which was anticipated. Furthermore, the missing data mechanism can be considered to be completely at random. The thing is, I was wondering how would one go about incorporating somehow the missing data information (ammount) into the posterior distribution variance? that is, I'd expect my inference to be more precise have I observed missing outcome from 2 samples instead of 10, for instance.
I'm fairly new to Bayesian analysis, I hope the question makes sense. References would be much appreciated, as well as software/packages recommendation. Thanks in advance.

When your target variable is missing completely at random, you can simply delete the observations without biasing your results (Gelman pg. 530) Consequently, when $k$ observations are missing the values of the target variable, you will have $n-k$ observations left.  Clearly, this will be incorporated in the posterior in a natural fashion - those observations simply won't be there to update the posterior distribution.
Consider a simplified version of your problem: you have generated 100 Gaussian variates from a distribution with unknown mean and known variance $ = 1$.  However, 20 of these observations are "missing".  Your prior is improper: $\mu \sim U(-\infty, \infty)$.  The resultant posterior will be that $\mu \sim N(\bar{x}, 1/80)$.  If you had all 100 observations available to you, the posterior would be that $\mu \sim N(\bar{x}, 1/100)$.  Clearly, in the missing data case, you have a larger posterior variance ($1/80$) than in the full data case ($1/100$).
Consequently, in the lucky situation where the target variable is missing at random, you don't need to do any extra work to have the missingness incorporated in your posterior - it happens naturally.
