Let $a,b,c$ be independent normally distributed variables with mean $0$ and standard deviation $\sigma_a,\sigma_b,\sigma_c$, respectively. What is the posterior distribution of the variables $a,b,c$ and that of the sum $a+b+c$, after observing signals $$ Y_a = b+c+\alpha \epsilon_a, Y_b = c+a+\beta \epsilon_b, Y_c = a+b+ \gamma \epsilon_c, $$ where $\epsilon_a,\epsilon_b,\epsilon_c$ are i.i.d. random variables with normalized normal distribution $N(0,1)$.
Motivating Story: A company has three employees. Each day two of the three employees are at work and observations are made about how much is produced on each day. How can the manager update his belief about the productivity of each employee, and what can he conclude about the productivity the company would have if all three workers were required work each day? Assume that the productivity is additive (workers work independently).
Note that $\tilde a = \tfrac 1 2 (Y_b + Y_c - Y_a)$ is an unbiased estimator of $a$, however this estimator does not need to be very efficient. For example if either of the parameters $\alpha,\beta,$ or $\gamma$ is large relative to $\sigma_a$, then the estimator $\tilde a$ would have larger variance then the prior distribution of $a$.