Is there a Bayes rule for when some of the random variables are discrete and continuous?

Question

Let $X$ be a continuous random variable, and $Y$ be a discrete random variable

Is there a "Bayes rule" for:

$P[Y | X = x]$

where $P$ is the probability mass function of $Y$?

Any reference help!

Answer 1

The proper statement of Bayes law for this case would be that:

Given any continuous variate $X$ with range $R_X$ and discrete variate $Y$ and some joint distribution of $X$ and $Y$ (they need not be independent) then for any set $B \subset R_X$ such that $P (X \in B) > 0$ and any value $y$, $$ P( Y = y | X\in B )= \frac{P( (Y = y) \wedge (X\in B) )}{P(X\in B)} $$ You can also go the other way, provided that $y$ is in the range of $Y$ with non-zero probability: $$ P(X\in B | Y = y )= \frac{P( (Y = y) \wedge (X\in B) )}{P(Y = y)} $$

Answer 2

There's also another identity in the case when $\ X\ $ has a density function $\ \phi\ $ with respect to some measure $\ \mu\ $ on its codomain. When this is the case $\ Y, X\ $ will also have a joint mass-density function $\ \psi\ $ satisfying the equation $\ P\left((Y=y)\, \land \left(X\in B\right)\right)=\int_B\ \psi\left(y,x\right)d\mu\left(x\right)\ $, and we then have $$ P\left(Y=y\,\vert\, X=x\,\right)= \frac{\psi\left(y,x\right)}{\phi\left(x\right)}\ $$ for all $\ y\ $ and $\mu$-almost all $\ x\ $.