Editing Likelihood function (section)

====Likelihoodist interpretation====
{{more footnotes needed|date=April 2019}}
In frequentist statistics, the likelihood function is itself a [[statistic]] that summarizes a single sample from a population, whose calculated value depends on a choice of several parameters ''θ''<sub>1</sub> ... ''θ''<sub>p</sub>, where ''p'' is the count of parameters in some already-selected [[statistical model]]. The value of the likelihood serves as a figure of merit for the choice used for the parameters, and the parameter set with maximum likelihood is the best choice, given the data available.

The specific calculation of the likelihood is the probability that the observed sample would be assigned, assuming that the model chosen and the values of the several parameters '''''θ''''' give an accurate approximation of the [[frequency distribution]] of the population that the observed sample was drawn from. Heuristically, it makes sense that a good choice of parameters is those which render the sample actually observed the maximum possible ''post-hoc'' probability of having happened. [[Wilks' theorem]] quantifies the heuristic rule by showing that the difference in the logarithm of the likelihood generated by the estimate's parameter values and the logarithm of the likelihood generated by population's "true" (but unknown) parameter values is asymptotically [[chi-squared distribution|χ<sup>2</sup> distributed]].

Each independent sample's maximum likelihood estimate is a separate estimate of the "true" parameter set describing the population sampled. Successive estimates from many independent samples will cluster together with the population's "true" set of parameter values hidden somewhere in their midst. The difference in the logarithms of the maximum likelihood and adjacent parameter sets' likelihoods may be used to draw a [[confidence region]] on a plot whose co-ordinates are the parameters ''θ''<sub>1</sub> ... ''θ''<sub>p</sub>. The region surrounds the maximum-likelihood estimate, and all points (parameter sets) within that region differ at most in log-likelihood by some fixed value. The [[chi-squared distribution|χ<sup>2</sup> distribution]] given by [[Wilks' theorem]] converts the region's log-likelihood differences into the "confidence" that the population's "true" parameter set lies inside. The art of choosing the fixed log-likelihood difference is to make the confidence acceptably high while keeping the region acceptably small (narrow range of estimates).

As more data are observed, instead of being used to make independent estimates, they can be combined with the previous samples to make a single combined sample, and that large sample may be used for a new maximum likelihood estimate. As the size of the combined sample increases, the size of the likelihood region with the same confidence shrinks. Eventually, either the size of the confidence region is very nearly a single point, or the entire population has been sampled; in both cases, the estimated parameter set is essentially the same as the population parameter set.