I think after all the reading I've done I still don't fully understand MAP estimation. I came across a problem that's leaving me dumbfounded.
Suppose $A$ ~ $N(0,\sigma^2_1) $ and $\epsilon$ ~ $N(0,\sigma^2_2)$
Let's say $a$ is drawn from the distribution of $A$ and that we observe $b$, which is conditioned on $a$, coming from a random variable $B$:
$b = a + \epsilon$
The questions asks me to find a MAP estimate of $a$ given $b$.
This seems so simple and yet I'm confused - I thought the point of MAP estimation was to use Bayes rule to determine the posterior by multiplying the likelihood and the prior - but I don't see what the prior on $a$ is here. Am I missing something basic?