Suppose I have two independent random deviates $A$ and $B$ sampled from Gaussian (Normal) distributions with means $\mu_a$ and $\mu_b$ and standard deviations $\sigma_a$ and $\sigma_b$. I can't observe $A$ or $B$ directly, but see only their difference $C = A - B$.
Given that I observe $C=c$, what's $Pr\{A = a | C=c\}$ ?
Seems like a job for Bayes' rule, and its easy to write down
$$Pr\{A=a | c\} = {Pr\{C | A\} Pr\{A\} \over Pr\{C\}}$$
From the assumptions above $Pr\{A\}$ ~ Normal($\mu_a, \sigma_a^2$), and $$Pr\{C | A\}$$ $$= Pr\{C=A-B | A\}$$ $$= Pr\{B = A-C | A\}$$, which is also ~ Normal($\mu_b, \sigma_b^2$) (we've conditioned on $A$, so we just want the probability that $B$ equals some value)
...however this leads to a dense thicket of algebra I can't seem to climb out of. Any handy tricks or references I should examine? Based on simulations the solution is Gaussian, but it's some complex function of $\mu_a, \sigma_a, \mu_b, and \sigma_b$, which I can't seem to derive. Thanks!