is negative definite for every at which the gradient vanishes, and if i.e. the likelihood function approaches a constant on the boundary of the parameter space, which may include the points at infinity if is unbounded. Mäkeläinen et al. prove this result using Morse theory while informally appealing to a mountain pass property. Mascarenhas restates their proof using the mountain pass theorem. In the proofs of consistency and asymptotic normality of the maximum likelihood estimator, additional assumptions are made about the probability densities that form the basis of a particular likelihood function. These conditions were first established by Chanda. In particular, for almost all , and for all exist for all in order to ensure the existence of a Taylor expansion. Second, for almost all and for every it must be that where is such that This boundedness of the derivatives is needed to allow for differentiation under the integral sign. And lastly, it is assumed that the information matrix, is positive definite and is finite. This ensures that the score has a finite variance. The above conditions are sufficient, but not necessary. That is, a model that does not meet these regularity conditions may or may not have a maximum likelihood estimator of the properties mentioned above. Further, in case of non-independently or non-identically distributed observations additional properties may need to be assumed. In Bayesian statistics, almost identical regularity conditions are imposed on the likelihood function in order to proof asymptotic normality of the posterior probability, and therefore to justify a Laplace approximation of the posterior in large samples. Likelihood ratio and relative likelihood Likelihood ratio A likelihood ratio is the ratio of any two specified likelihoods, frequently written as: The likelihood ratio is central to likelihoodist statistics: the law of likelihood states that degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio. In frequentist inference, the likelihood ratio is the basis for a test statistic, the so-called likelihood-ratio test. By the Neyman–Pearson lemma, this is the most powerful test for comparing two simple hypotheses at a given significance level. Numerous other tests can be viewed as likelihood-ratio tests or approximations thereof. The asymptotic distribution of the log-likelihood ratio, considered as a test statistic, is given by Wilks' theorem. The likelihood ratio is also of central importance in Bayesian inference, where it is known as the Bayes factor, and