Editing Likelihood function (section)

===In general===

In [[Probability theory#Measure-theoretic probability theory|measure-theoretic probability theory]], the [[Probability density function|density function]] is defined as the [[Radon–Nikodym theorem|Radon–Nikodym derivative]] of the probability distribution relative to a common dominating measure.<ref>{{citation |first=Patrick |last=Billingsley | author-link= Patrick Billingsley|title=Probability and Measure |publisher= [[John Wiley & Sons]] |edition=Third |year=1995 |pages=422–423 |mode=cs1 }}</ref> The likelihood function is this density interpreted as a function of the parameter, rather than the random variable.<ref name="Shao03">{{citation| first= Jun| last= Shao| year= 2003 | title= Mathematical Statistics | edition= 2nd | publisher= Springer | at= §4.4.1 |mode=cs1 }}</ref> Thus, we can construct a likelihood function for any distribution, whether discrete, continuous, a mixture, or otherwise. (Likelihoods are comparable, e.g. for parameter estimation, only if they are Radon–Nikodym derivatives with respect to the same dominating measure.)

The above discussion of the likelihood for discrete random variables uses the [[counting measure]], under which the probability density at any outcome equals the probability of that outcome.