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$A$ is an event. (underlying phenomenon) $y$ is a continuous random variable s.t. $f_{Y \mid A}$ and $f_{Y|A^c}$ are known (observation) seek $\b{P}(A| Y = y)$.
Example
Binary signal $S $ transmitted,
$\b{P}({S = 1}) = p, \b{P}({S = -1}) = 1 - p$.
received signal is $S+N$, where $N $ is standard normal, $S, N $ are independent.
We seek $\b{P}(S = 1 | Y = y)$ where $Y = S + N $.
$A = \set{ S = 1 }$
$\b{P}(S = 1 | Y = y) = \frac{ \b{P}(S = 1) f_{Y | S = 1}(y)}{ \b{P}(S = 1)f_{Y \mid S =1}(y) + \b{P}(S = -1) f_{ Y | S = -1 }(y)}$
Conditioned on $S = 1, y$ is $N + 1 $,
$f_{Y | S =1}(y) = \frac{1}{\sqrt{2\pi}{(1)}}e^{-\frac{(x-{1})^2}{2{(1)}^2}} $
$f_{Y | S =-1}(y) = \frac{1}{\sqrt{2\pi}{(1)}}e^{-\frac{(x-{(-1)})^2}{2{(1)}^2}}$
Theorem
$$\b{P}(A | Y = y) = \frac{ \b{P}(A)f_{Y \mid A}(y)}{ \b{P}(A)f_{Y \mid A}(y) + \b{P}(A^c)f_{Y \mid A^c}(y)} = \frac{ \b{P}(A)f_{Y \mid A}(y)}{ f_Y{y}}$$