I've read a number of different explanations trying to understand the likelihood function, and I understand the purpose of it, but some statements sound contradictory.
Consider observed data X, model parameters M, likelihood function L(M|X), and probability P(X|M).
I keep seeing it written that L(M|X) = P(X|M). At the same time, they say that the likelihood is not the same thing as the probability and L is not a probability density function (pdf).
What does it mean to say they are not the same thing but equal? How could L be equal to a pdf, but not be a pdf?
Reference to one of the places I've read this: http://www.stat.cmu.edu/~larry/=stat705/Lecture6.pdf