Let X=( ) be a random vector and
a statistical model parametrized by , the parameter vector in the parameter space . The likelihood function is a map given by
In other words, the likelikhood function is functionally the same in form as a probability density function. However, the emphasis is changed from the to the . The pdf is a function of the 's while holding the parameters 's constant, is a function of the parameters 's, while holding the 's constant.
When there is no confusion, is abbreviated to be .
The parameter vector such that for all is called a maximum likelihood estimate, or MLE, of .