Thursday, August 16, 2012

The Likelihood Principle

The so-called "Likelihood Principle" forms the foundation of both classical (frequentist) statistics, as well as Bayesian statistics. So, as an econometrician, whether you rely on Maximum Likelihood estimation and the associated asymptotic tests, or if you prefer to adopt a Bayesian approach to inference, this principle is of fundamental importance to you.

What is this principle? Suppose that x is the value of a (possibly vector-valued) random variable, X, whose density depends on a vector of parameters, θ. Then, the Likelihood Principle states that:

"All the information about θ obtainable from an experiment is contained in the likelihood function for θ given x. Two likelihood functions for θ (from the same or different experiments) contain the same information about θ if they are proportional to one another."  (Berger and Wolpert, 1988, p.19).