In statistics, a

**likelihood function**is a conditional probability function considered a function of its

*second*argument with its first agument held fixed, thus:

*B*is the equivalence class of functions

*L*(

*b*) is immaterial; all that matters are ratios of the form

*L*(

*b*

_{2})/

*L*(

*b*

_{1}), since these are invariant with respect to the constant of proportionality.

** Likelihood** as a solitary term is a shorthand for

**.**

*likelihood function*
In a sense, likelihood works backwards from probability: given *B*, we use the conditional probability P(*A* | *B*) to reason about *A*, and, given *A*, we use the likelihood function P(*A* | *B*) to reason about *B*.
This mode of reasoning is formalized in Bayes' theorem;
note the appearance of a likelihood function for *B* given *A* in the numerator:

In the colloquial language, "likelihood" is one of several informal synomyms for "probability", but throughout this article we use only the technical definition.

## Likelihood function of a parametrized model

where θ is the parameter (in the case of discrete distributions, the probability density functions are probability "mass" functions) the**likelihood function**is

*x*is the observed outcome of an experiment. In other words, when

*f*(

*x*| θ) is viewed as a function of

*x*with θ fixed, it is a probability density function, and when viewed as a function of θ with

*x*fixed, it is a likelihood function.

*Note:* This is *not* the same as the probability that those parameters are the right ones, given the observed sample. Attempting to interpret the likelihood of a hypothesis given observed evidence as the probability of the hypothesis is a common error, with potentially disastrous real-world consequences in medicine, engineering or jurisprudence. See prosecutor's fallacy for an example of this.

## Example

In symbols, we can say the above as

Another way of saying this is to reverse it and say that "the likelihood of*p*= 0.5 given the observation 'HH' is 0.25", i.e.,

_{H}

- .

*probability*of

*p*= 0.5 given the observation is 0.25.

_{H}
To take an extreme case, on this basis we can say "the likelihood of *p _{H}* = 1 given the observation 'HH' is 1". But it is clearly not the case that the

*probability*of

*p*= 1 given the observation is 1: the event 'HH' can occur for any

_{H}*p*> 0 (and often does, in reality, for

_{H}*p*roughly 0.5).

_{H}The likelihood function does not in general follow all the axioms of probability: for example, the integral of a likelihood function is not in general 1.

This is because integration of the likelihood density function *L* is performed over all possible values of the model parameters (in this case, *p*_{H}), while integration of a probability density function *f* is performed over the random variables (which in this case take on the four pairs of values 'TT', 'TH', 'HT' and 'HH').

In this example, the integral of the likelihood density over the interval [0, 1] in *p _{H}* is 1/3, demonstrating again that the likelihood density function cannot be interpreted as a probability density function for

*p*.

_{H}
On the other hand, given any particular value of p_{H}, e.g. p_{H}=0.5, the integral of the probability density function over the domain of the random variables **is** 1.

See also:

- Bayesian inference
- Likelihood principle
- Likelihood-ratio test
- Maximum likelihood method
- The principle of maximum entropy is also related to the maximum likelihood method
- Minimum entropy method