Probability inference of an action from a continuous outcome

Question

Assume person A takes an action, it could be either $a_1$ or $a_2$ with $a_1>a_2$, we cannot observe A's action but a signal $x$, with $x=a_i+\epsilon$. $\epsilon$ follows a normal distribution $N(\mu,\sigma)$. My question: how to infer the probability that person A takes action $a_1$ conditional on x, i.e. $P(a_1|x)$. Thanks!

Answer 1

Let the distribution over the $a_i$s be denoted as $P(a_i)$. Then, by Baye's theorem, $$ P(a_1 \vert x) = \frac{f(x\vert a_1)P(a_1)}{f(x)} = \frac{f(x\vert a_1)P(a_1)}{\sum\limits_{i=1,2} f(x\vert a_i)P(a_i)} = \frac{\displaystyle \exp({-\frac{1}{2\sigma}(x-\mu-a_1)^2})P(a_1)}{\displaystyle \sum\limits_{i=1,2} \displaystyle \exp({-\frac{1}{2\sigma}(x-\mu-a_i)^2})P(a_i)}\,. $$