is used in Bayes' rule. Stated in terms of odds, Bayes' rule states that the posterior odds of two alternatives, and , given an event , is the prior odds, times the likelihood ratio. As an equation: The likelihood ratio is not directly used in AIC-based statistics. Instead, what is used is the relative likelihood of models (see below). Relative likelihood function Since the actual value of the likelihood function depends on the sample, it is often convenient to work with a standardized measure. Suppose that the maximum likelihood estimate for the parameter is . Relative plausibilities of other values may be found by comparing the likelihoods of those other values with the likelihood of . The relative likelihood of is defined to be Thus, the relative likelihood is the likelihood ratio (discussed above) with the fixed denominator . This corresponds to standardizing the likelihood to have a maximum of 1. Likelihood region A likelihood region is the set of all values of whose relative likelihood is greater than or equal to a given threshold. In terms of percentages, a % likelihood region for is defined to be If is a single real parameter, a % likelihood region will usually comprise an interval of real values. If the region does comprise an interval, then it is called a likelihood interval. Likelihood intervals, and more generally likelihood regions, are used for interval estimation within likelihoodist statistics: they are similar to confidence intervals in frequentist statistics and credible intervals in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of coverage probability (frequentism) or posterior probability (Bayesianism). Given a model, likelihood intervals can be compared to confidence intervals. If is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for will be the same as a 95% confidence interval (19/20 coverage probability). In a slightly different formulation suited to the use of log-likelihoods (see Wilks' theorem), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a chi-squared distribution with degrees-of-freedom (df) equal to the difference in df's between the two models (therefore, the −2 likelihood interval is the same as the 0.954 confidence interval; assuming difference in df's to be 1). Likelihoods that eliminate nuisance parameters In many cases, the likelihood is a function of more than one parameter but