Is there a reason why you cannot compute these quantities from the posteriors? From your MCMC trace you can compute expected values, variance, or any summaries for any parameter, combination, or function of them. If that is not a quantity computed during estimation just compute a new vector with the and then compute its variance.

I think you could compute each term of the expression separately, since it follows from the properties of random variables, but it’s probably easier to just compute ab + \sigma for all samples and find the summaries from that.

Glad it helped. Sometimes nice analytical expressions are derived from probability theory or statistics, but the frequentist approach still require you to find the summaries that are not always readily available. One of the coolest thing of Bayesian statistics has to be that you are always working with the posterior distributions, so it’s easy to visualize and think about the quantities of interest.