# Conditional Likelihood Function

A Conditional Likelihood Function is a likelihood function that is a conditional probability.

**Counter-Example(s):****See:**Conditional Likelihood Estimation, Conditional Expectation Function, Sufficient Statistic, Hypergeometric Distribution, Fisher's Exact Test.

## References

### 2014

- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/likelihood_function#Conditional_likelihood Retrieved:2014-12-10.
- Sometimes it is possible to find a sufficient statistic for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters.
One example occurs in 2×2 tables, where conditioning on all four marginal totals leads to a conditional likelihood based on the non-central hypergeometric distribution. This form of conditioning is also the basis for Fisher's exact test.

- Sometimes it is possible to find a sufficient statistic for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters.

### 2009

- (Gentle, 2009) ⇒ James E. Gentle. (2009). “Computational Statistics." Springer. ISBN:978-0-387-98143-7
- QUOTE: The likelihood function arises from a probability density, but it is not a probability density function. It does not in any way relate to a “probability” associated with the parameters or the model.
Although non-statisticians will often refer to the “likelihood of an observation”, in statistics, we use the term “likelihood” to refer to a model or a distribution given observations.

In a multiparameter case, we may be interested in only some of the parameters. There are two ways of approaching this, use of a profile likelihood or of a conditional likelihood.

Let [math]θ= (θ_1, θ_2)[/math]. If θ_2 is fixed, the likelihood L(θ_1 ;θ_2, y) is called a profile likelihood or concentrated likelihood of θ_1 for given θ_2 and y.

If the PDFs can be factored so that one factor includes [math]θ_2[/math] and some function of the sample, [math]S(y)[/math], and the other factor, given [math]S(y)[/math], is free of [math]θ_2[/math], then this factorization can be carried into the likelihood. Such a likelihood is called a conditional likelihood of [math]θ_1[/math] given [math]S(y)[/math].

- QUOTE: The likelihood function arises from a probability density, but it is not a probability density function. It does not in any way relate to a “probability” associated with the parameters or the model.