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x _ theta frac 1 h int _ x_ j x_ j h f x mid theta dx the first fundamental theorem of calculus provides that lim h 0 1 h x j x j h f x θ d x f x j θ displaystyle lim _ h to 0 frac 1 h int _ x_ j x_ j h f x mid theta dx f x_ j mid theta then a r g m a x θ l θ x j a r g m a x θ lim h 0 l θ x x j x j h a r g m a x θ lim h 0 1 h x j x j h f x θ d x a r g m a x θ f x j θ displaystyle begin aligned mathop operatorname arg max _ theta mathcal l theta mid x_ j mathop operatorname arg max _ theta left lim _ h to 0 mathcal l theta mid x in x_ j x_ j h right 4pt mathop operatorname arg max _ theta left lim _ h to 0 frac 1 h int _ x_ j x_ j h f x mid theta dx right 4pt mathop operatorname arg max _ theta f x_ j mid theta end aligned therefore a r g m a x θ l θ x j a r g m a x θ f x j θ displaystyle mathop operatorname arg max _ theta mathcal l theta mid x_ j mathop operatorname arg max _ theta f x_ j mid theta and so maximizing the probability density at x j textstyle x_ j amounts to maximizing the likelihood of the specific observation x j textstyle x_ j in general edit in measure theoretic probability theory the density function is defined as the radon nikodym derivative of the probability distribution relative to a common dominating measure 2 the likelihood function is this density interpreted as a function of the parameter rather than the random variable 3 thus we can construct a likelihood function for any distribution whether discrete continuous a mixture or otherwise likelihoods are comparable e g for parameter estimation only if they are radon nikodym derivatives with respect to the same dominating measure the above discussion of the likelihood for discrete random variables uses the counting measure under which the probability density at any outcome equals the probability of that outcome likelihoods for mixed continuous discrete distributions edit the above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components suppose that the distribution consists of a number of discrete probability masses p k θ textstyle p_ k theta and a density f x θ textstyle f x mid theta where the sum of all the p textstyle p s added to the integral of f textstyle f is always one assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component the likelihood function for an observation from the continuous component can be dealt with in the manner shown above for an observation from the discrete component the likelihood function for an observation from the discrete component is simply l θ x p k θ displaystyle mathcal l theta mid x p_ k theta where k textstyle k is the index of the discrete probability mass corresponding to observation x textstyle x because maximizing the probability mass or probability at x textstyle x amounts to maximizing the likelihood of the specific observation the fact that the likelihood function can be defined in a way that includes contributions that are not commensurate the density and the probability mass arises from the way in which the likelihood function is defined up to a constant of proportionality where this constant can change with the observation x textstyle x but not with the parameter θ textstyle theta regularity conditions edit in the context of parameter estimation the likelihood function is usually assumed to obey certain conditions known as regularity conditions these conditions are assumed in various proofs involving likelihood functions and need to be verified in each particular application for maximum likelihood estimation the existence of a global maximum of the likelihood function is of the utmost importance by the extreme value theorem it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist 4 while the continuity assumption is usually met the compactness assumption about the parameter space is often not as the bounds of the true parameter values might be unknown in that case concavity of the likelihood function plays a key role more specifically if the likelihood function is twice continuously differentiable on the k dimensional parameter space θ textstyle theta assumed to be an open connected subset of r k textstyle mathbb r k there exists a unique maximum θ θ textstyle hat theta in theta if the matrix of second partials h θ 2 l θ i θ j i j 1 1 n i n j displaystyle mathbf h theta equiv left frac partial 2 l partial theta _ i partial theta _ j right _ i j 1 1 n_ mathrm i n_ mathrm j is negative definite for every θ θ textstyle theta in theta at which the gradient l l θ i i 1 n i textstyle nabla l equiv left frac partial l partial theta _ i right _ i 1 n_ mathrm i vanishes and if the likelihood function approaches a constant on the boundary of the parameter space θ textstyle partial theta i e lim θ θ l θ 0 displaystyle lim _ theta to partial theta l theta 0 which may include the points at infinity if θ textstyle theta is unbounded mäkeläinen and co authors prove this result using morse theory while informally appealing to a mountain pass property 5 mascarenhas restates their proof using the mountain pass theorem 6 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 7 in particular for almost all x textstyle x and for all θ θ textstyle theta in theta log f θ r 2 log f θ r θ s 3 log f θ r θ s θ t displaystyle frac partial log f partial theta _ r quad frac partial 2 log f partial theta _ r partial theta _ s quad frac partial 3 log f partial theta _ r partial theta _ s partial theta _ t exist for all r s t 1 2 k textstyle r s t 1 2 ldots k in order to ensure the existence of a taylor expansion second for almost all x textstyle x and for every θ θ textstyle theta in theta it must be that f θ r f r x 2 f θ r θ s f r s x 3 f θ r θ s θ t h r s t x displaystyle left frac partial f partial theta _ r right f_ r x quad left frac partial 2 f partial theta _ r partial theta _ s right f_ rs x quad left frac partial 3 f partial theta _ r partial theta _ s partial theta _ t right h_ rst x where h textstyle h is such that h r s t z d z m textstyle int _ infty infty h_ rst z dz leq m infty this boundedness of the derivatives is needed to allow for differentiation under the integral sign and lastly it is assumed that the information matrix i θ log f θ r log f θ s f d z displaystyle mathbf i theta int _ infty infty frac partial log f partial theta _ r frac partial log f partial theta _ s f dz is positive definite and i θ textstyle left mathbf i theta right is finite this ensures that the score has a finite variance 8 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 9 10 and therefore to justify a laplace approximation of the posterior in large samples 11 likelihood ratio and relative likelihood edit see also pseudo r squared likelihood ratio edit this section is about the likelihood ratio in general for the use of likelihood ratios in interpreting diagnostic tests see likelihood ratios in diagnostic testing for the statistical test to compare goodness of fit see likelihood ratio test a likelihood ratio is the ratio of any two specified likelihoods frequently written as λ θ 1 θ 2 x l θ 1 x l θ 2 x displaystyle lambda theta _ 1 theta _ 2 mid x frac mathcal l theta _ 1 mid x mathcal l theta _ 2 mid x the likelihood ratio is central to likelihoodist statistics the law of likelihood states that the 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 12 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 is used in bayes rule stated in terms of odds bayes rule states that the posterior odds of two alternatives a 1 displaystyle a_ 1 and a 2 displaystyle a_ 2 given an event b displaystyle b is the prior odds times the likelihood ratio as an equation o a 1 a 2 b o a 1 a 2 λ a 1 a 2 b displaystyle o a_ 1 a_ 2 mid b o a_ 1 a_ 2 cdot lambda a_ 1 a_ 2 mid b the likelihood ratio is not directly used in aic based statistics instead what is used is the relative likelihood of models see below in evidence based medicine likelihood ratios are used in diagnostic testing to assess the value of performing a diagnostic test relative likelihood function edit see also relative likelihood 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 θ is θ textstyle hat theta relative plausibilities of other θ values may be found by comparing the likelihoods of those other values with the likelihood of θ textstyle hat theta the relative likelihood of θ is defined to be 13 14 15 16 17 r θ l θ x l θ x displaystyle r theta frac mathcal l theta mid x mathcal l hat theta mid x thus the relative likelihood is the likelihood ratio discussed above with the fixed denominator l θ textstyle mathcal l hat theta this corresponds to standardizing the likelihood to have a maximum of 1 likelihood region edit a likelihood region is the set of all values of θ whose relative likelihood is greater than or equal to a given threshold in terms of percentages a p likelihood region for θ is defined to be 13 15 18 θ r θ p 100 displaystyle left theta r theta geq frac p 100 right if θ is a single real parameter a p likelihood region will usually comprise an interval of real values if the region does comprise an interval then it is called a likelihood interval 13 15 19 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 θ is a single real parameter then under certain conditions a 14 65 likelihood interval about 1 7 likelihood for θ will be the same as a 95 confidence interval 19 20 coverage probability 13 18 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 e 2 likelihood interval is the same as the 0 954 confidence interval assuming difference in df s to be 1 18 19 likelihoods that eliminate nuisance parameters edit in many cases the likelihood is a function of more than one parameter but interest focuses on the estimation of only one or at most a few of them with the others being considered as nuisance parameters several alternative approaches have been developed to eliminate such nuisance parameters so that a likelihood can be written as a function of only the parameter or parameters of interest the main approaches are profile conditional and marginal likelihoods 20 21 these approaches are also useful when a high dimensional likelihood surface needs to be reduced to one or two parameters of interest in order to allow a graph profile likelihood edit it is possible to reduce the dimensions by concentrating the likelihood function for a subset of parameters by expressing the nuisance parameters as functions of the parameters of interest and replacing them in the likelihood function 22 23 in general for a likelihood function depending on the parameter vector θ textstyle mathbf theta that can be partitioned into θ θ 1 θ 2 textstyle mathbf theta left mathbf theta _ 1 mathbf theta _ 2 right and where a correspondence θ 2 θ 2 θ 1 textstyle mathbf hat theta _ 2 mathbf hat theta _ 2 left mathbf theta _ 1 right can be determined explicitly concentration reduces computational burden of the original maximization problem 24 for instance in a linear regression with normally distributed errors y x β u textstyle mathbf y mathbf x beta u the coefficient vector could be partitioned into β β 1 β 2 textstyle beta left beta _ 1 beta _ 2 right and consequently the design matrix x x 1 x 2 textstyle mathbf x left mathbf x _ 1 mathbf x _ 2 right maximizing with respect to β 2 textstyle beta _ 2 yields an optimal value function β 2 β 1 x 2 t x 2 1 x 2 t y x 1 β 1 textstyle beta _ 2 beta _ 1 left mathbf x _ 2 mathsf t mathbf x _ 2 right 1 mathbf x _ 2 mathsf t left mathbf y mathbf x _ 1 beta _ 1 right using this result the maximum likelihood estimator for β 1 textstyle beta _ 1 can then be derived as β 1 x 1 t i p 2 x 1 1 x 1 t i p 2 y displaystyle hat beta _ 1 left mathbf x _ 1 mathsf t left mathbf i mathbf p _ 2 right mathbf x _ 1 right 1 mathbf x _ 1 mathsf t left mathbf i mathbf p _ 2 right mathbf y where p 2 x 2 x 2 t x 2 1 x 2 t textstyle mathbf p _ 2 mathbf x _ 2 left mathbf x _ 2 mathsf t mathbf x _ 2 right 1 mathbf x _ 2 mathsf t is the projection matrix of x 2 textstyle mathbf x _ 2 this result is known as the frisch waugh lovell theorem since graphically the procedure of concentration is equivalent to slicing the likelihood surface along the ridge of values of the nuisance parameter β 2 textstyle beta _ 2 that maximizes the likelihood function creating an isometric profile of the likelihood function for a given β 1 textstyle beta _ 1 the result of this procedure is also known as profile likelihood 25 26 in addition to being graphed the profile likelihood can also be used to compute confidence intervals that often have better small sample properties than those based on asymptotic standard errors calculated from the full likelihood 27 28 conditional likelihood edit sometimes it is possible to fi...
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