Thread: Bayesian statistics

1. Bayesian statistics

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?

2. Originally Posted by pedrosorio
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?
What exactly are you trying to do?

CB

3. 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.

4. I know that c is a normalizing constant, that is:

1/c = Integral wrt Z of f(z) * L(X=xi|z)

I think integral equals P(X=xi)?