## 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).

From a frequentist viewpoint, the (random) likelihood function is a minimal sufficient statistic for θ. Once you have the likelihood function, you have all of the relevant information about this parameter.

A terrific free resource that you may wish to check out is the electronic version of the second edition of Berger and Wopert's The Likelihood Principle, which is available courtesy of Project Euclid. It's available here, chapter-by-chapter. If you want a single pdf file of the full monograph, I've put it together here.

Reference

Berger, J. O. and R. L. Wolpert (1988). The Likelihood Principle (2nd ed.). Lecture Notes - Monograph Series, Vol. 6, Institute of Mathematical Statistics, Hayward, CA.