# Bayesian statistics

• Aug 21st 2009, 11:03 AM
pedrosorio
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?
• Aug 21st 2009, 01:26 PM
CaptainBlack
Quote:

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
• Aug 21st 2009, 01:41 PM
pedrosorio
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.
• Aug 22nd 2009, 10:15 AM
pedrosorio
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)?