Likelihood is the probability of a set of observations provided the value of some parameter or the set of parameters. For instance, the likelihood of the random sample of n observations

 

with probability distribution, f(x,θ) which can be  given by
 
This function is the basis of the maximum likelihood estimation. In number of applications the likelihood includes number of parameters, only a few of which are of interest to investigator. The remaining nuisance parameters are essential in order that the model makes the sense physically, but their values are largely irrelevant of the investigation and the conclusions to be made. Since there are troubles in dealing with likelihoods which depend on a large number of incidental parameters (for instance, maximizing the likelihood will be more tough) some form of modified likelihood is sought which comprises as few of the uninteresting parameters as possible. The number of possibilities is available. For instance, the marginal likelihood, removes the nuisance parameters by integrating them out of the likelihood. The profile likelihood with respect to parameters of interest, is the original likelihood, partly maximized with respect to the nuisance parameters.