Definition: Given observed data \(X=x\), the likelihood function for \(\theta\) is defined as
\[L(\theta|x) = f(x|\theta).\]Here, \(f(x|\theta)\) can be a density function, point mass function, or a mixture of the two.
In practice, it is typically more sensible to work with the log-likelihood instead, which is simply the (natural) log of the likelihood:
\[\ell(\theta|x) = \log L(\theta|x).\]