Likelihood functions

I have vaguely been learning about likelihood functions and was wondering if there is a method for converting them to probability density functions, specifically for the binomial distribution (for example to calculate confidence intervals). I cannot seem to find any suggestion that this is possible or impossible. Any help?

Since likelihood functions are derived from probability distributions, you can simply reverse the process and recover the probability density from a likelihood; but this would be rather pointless and is not, I think, what you mean. In general, the answer to your question is no; there are plenty of likelihood functions that integrate to infinity, and therefore cannot be normalized to be a probability density. (Take the uniform distribution on $\left(0, \theta\right)$, for example).

The properties of likelihood functions lead you off in another direction entirely. Although likelihood functions are not themselves probability distributions, the logarithm of a likelihood ratio is. So if you take the likelihood evaluated at one value of the parameter, divide this by the value of the likelihood at another parameter, and take the logarithm, you do get a random variable. You can then use this fact to either set up a statistical test, or to derive (likelihood based) confidence intervals. This turns out to be a very good thing to do, as by the Neyman-Pearson lemma this statistical test is the most powerful test you can construct (i.e. it is the test that makes the best use of the data you have available).

In fact the log likelihood ratio is asymptotically distributed as a $\chi^2$ random variable, making these tests very easy to carry out in practice. The likelihood ratio test is ubiquitous in higher statistics!

While that makes a lot of sense, is it technically possible to simply scale a likelihood function (that doesn't integrate to infinity) and have a probability distribution?