Determining the Likelihood function

1. Nov 7, 2008

daviddoria

I was under the impression that the likelihood function was simply the probability density function but viewing the parameter theta as the variable instead of the observations x. Ie
p(x|theta) = L(theta|x)

However, the likelihood function is no longer a probability function

See Example 1 here:
http://en.wikipedia.org/wiki/Likelihood_function

Can anyone explain this a little bit?

Thanks,
Dave

2. Nov 7, 2008

Right. The observations are fixed in the context of a likelihood function.

Yeah, it doesn't typically integrate to 1 (and sometimes doesn't integrate at all). Note that in most pdf's, the data and parameters play very different roles, so we shouldn't expect the function to have the same integral over each of them. Also, in maximum likelihood estimation, the parameter is assumed to be a fixed, unknown constant (and NOT a random variable), so there is no distribution assigned to it.

I could probably give a better answer if you could explain why you think the likelihood function should be a probability function.

3. Nov 8, 2008

daviddoria

maybe I used a bad example

I considered a Gaussian random variable with variance 1,

so f(x|u=3) is a Gaussian distribution centered at 3.

Then L(u|x=5) is a Gaussian distribution centered at 5.

Is that correct? That must just be a "special" case and hence a bad example?

Dave

4. Nov 10, 2008