There are normal distributions with known means and standard deviations. The first distribution is a Bayesian prior distribution with known mean1 and known SD1. The second distribution is a Bayesian posterior distribution with known mean3 and known SD3. I'm interested in inferring what the mean (mean2) and standard deviation (SD2) of the likelihood distribution is. I'm using the Gaussian PDF.
N(mean3, SD3) = C x N(mean2, SD2) x N(mean1, SD1)
What is mean2 and SD2 for any given mean3, SD3, mean1, and SD1?
This is one of the basic steps in the Expectation Propagation algorithm. See section 3.2 of A family of algorithms for approximate Bayesian inference, where the formulas you want are given by (3.32,3.33). Keep in mind that the variance of the likelihood might turn out to be negative, so SD2 would be imaginary. This is why it is better to write the algorithm in terms of variances rather than standard deviations.