I have a discrete variable, let's call it X, and a continuous variable called Z.
I know the posterior f(Z|X=xi) has Gaussian Distribution with expected value ui and variance sigma2i (and I know these for all possible xi).
Also I know that, according to the Bayes' theorem:
f(Z|X=xi) = c * f(Z) * L(X=xi|Z), where the constant is there to normalize.
The problem is, I have access to Z and want to find the xi that maximizes the likelihood L(X=xi|Z).
L(X=xi|Z) = f(Z|X=xi) / (c * f(Z))
but that constant could be different for every xi... How should I go about it?
Basically I want to use this to discretize values.
For example, I measure the speed of a ball in m/s. I have 3 possible discrete speeds (slow, medium, fast). And I assume that for each discrete speed, the measured speed will have a gaussian distribution with a given mean and variance.
What I want is, given the measured speed, find the most likely discrete speed. The "intuitive" is just to take the one with the highest posterior, but I'm not sure it's correct.