Oftentimes, a “likelihood function” is generalized from that definition to refer to any non-normalized function which is still treated like a probability distri

Oftentimes, a “likelihood function” is generalized from that definition to refer to any non-normalized function which is still treated like a probability distribution. Any function with a finite integral over the real line can be turned into a probability distribution easily, and there are other functions that can be used in places where you would normally use a probability function, even when they don’t integrate to a finite value over the real line. (For example, improper priors in Bayesian analysis.) “Function that can in some contexts be treated like a probability distribution” is just too long.