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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 find a sufficient statistic for the nuisance parameters and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters 29 one example occurs in 2 2 tables where conditioning on all four marginal totals leads to a conditional likelihood based on the non central hypergeometric distribution this form of conditioning is also the basis for fisher s exact test marginal likelihood edit main article marginal likelihood sometimes we can remove the nuisance parameters by considering a likelihood based on only part of the information in the data for example by using the set of ranks rather than the numerical values another example occurs in linear mixed models where considering a likelihood for the residuals only after fitting the fixed effects leads to residual maximum likelihood estimation of the variance components partial likelihood edit a partial likelihood is an adaption of the full likelihood such that only a part of the parameters the parameters of interest occur in it 30 it is a key component of the proportional hazards model using a restriction on the hazard function the likelihood does not contain the shape of the hazard over time products of likelihoods edit the likelihood given two or more independent events is the product of the likelihoods of each of the individual events λ a x 1 x 2 λ a x 1 λ a x 2 displaystyle lambda a mid x_ 1 land x_ 2 lambda a mid x_ 1 cdot lambda a mid x_ 2 this follows from the definition of independence in probability the probabilities of two independent events happening given a model is the product of the probabilities this is particularly important when the events are from independent and identically distributed random variables such as independent observations or sampling with replacement in such a situation the likelihood function factors into a product of individual likelihood functions the empty product has value 1 which corresponds to the likelihood given no event being 1 before any data the likelihood is always 1 this is similar to a uniform prior in bayesian statistics but in likelihoodist statistics this is not an improper prior because likelihoods are not integrated log likelihood edit see also log probability log likelihood function is the logarithm of the likelihood function often denoted by a lowercase l or ℓ displaystyle ell to contrast with the uppercase l or l textstyle mathcal l for the likelihood because logarithms are strictly increasing functions maximizing the likelihood is equivalent to maximizing the log likelihood but for practical purposes it is more convenient to work with the log likelihood function in maximum likelihood estimation in particular since most common probability distributions notably the exponential family are only logarithmically concave 31 32 and concavity of the objective function plays a key role in the maximization given the independence of each event the overall log likelihood of intersection equals the sum of the log likelihoods of the individual events this is analogous to the fact that the overall log probability is the sum of the log probability of the individual events in addition to the mathematical convenience from this the adding process of log likelihood has an intuitive interpretation as often expressed as support from the data when the parameters are estimated using the log likelihood for the maximum likelihood estimation each data point is used by being added to the total log likelihood as the data can be viewed as an evidence that support the estimated parameters this process can be interpreted as support from independent evidence adds and the log likelihood is the weight of evidence interpreting negative log probability as information content or surprisal the support log likelihood of a model given an event is the negative of the surprisal of the event given the model a model is supported by an event to the extent that the event is unsurprising given the model a logarithm of a likelihood ratio is equal to the difference of the log likelihoods log l a l b log l a log l b ℓ a ℓ b displaystyle log frac mathcal l a mathcal l b log mathcal l a log mathcal l b ell a ell b just as the likelihood given no event being 1 the log likelihood given no event is 0 which corresponds to the value of the empty sum without any data there is no support for any models graph edit the graph of the log likelihood is called the support curve in the univariate case 33 in the multivariate case the concept generalizes into a support surface over the parameter space it has a relation to but is distinct from the support of a distribution the term was coined by a w f edwards 33 in the context of statistical hypothesis testing i e whether or not the data support one hypothesis or parameter value being tested more than any other the log likelihood function being plotted is used in the computation of the score the gradient of the log likelihood and fisher information the curvature of the log likelihood thus the graph has a direct interpretation in the context of maximum likelihood estimation and likelihood ratio tests likelihood equations edit if the log likelihood function is smooth its gradient with respect to the parameter known as the score and written s n θ θ ℓ n θ textstyle s_ n theta equiv nabla _ theta ell _ n theta exists and allows for the application of differential calculus the basic way to maximize a differentiable function is to find the stationary points the points where the derivative is zero since the derivative of a sum is just the sum of the derivatives but the derivative of a product requires the product rule it is easier to compute the stationary points of the log likelihood of independent events than for the likelihood of independent events the equations defined by the stationary point of the score function serve as estimating equations for the maximum likelihood estimator s n θ 0 displaystyle s_ n theta mathbf 0 in that sense the maximum likelihood estimator is implicitly defined by the value at 0 textstyle mathbf 0 of the inverse function s n 1 e d θ textstyle s_ n 1 mathbb e d to theta where e d textstyle mathbb e d is the d dimensional euclidean space and θ textstyle theta is the parameter space using the inverse function theorem it can be shown that s n 1 textstyle s_ n 1 is well defined in an open neighborhood about 0 textstyle mathbf 0 with probability going to one and θ n s n 1 0 textstyle hat theta _ n s_ n 1 mathbf 0 is a consistent estimate of θ textstyle theta as a consequence there exists a sequence θ n textstyle left hat theta _ n right such that s n θ n 0 textstyle s_ n hat theta _ n mathbf 0 asymptotically almost surely and θ n p θ 0 textstyle hat theta _ n xrightarrow text p theta _ 0 34 a similar result can be established using rolle s theorem 35 36 the second derivative evaluated at θ textstyle hat theta known as fisher information determines the curvature of the likelihood surface 37 and thus indicates the precision of the estimate 38 exponential families edit further information exponential family the log likelihood is also particularly useful for exponential families of distributions which include many of the common parametric probability distributions the probability distribution function and thus likelihood function for exponential families contain products of factors involving exponentiation the logarithm of such a function is a sum of products again easier to differentiate than the original function an exponential family is one whose probability density function is of the form for some functions writing textstyle langle rangle for the inner product p x θ h x exp η θ t x a θ displaystyle p x mid boldsymbol theta h x exp big langle boldsymbol eta boldsymbol theta mathbf t x rangle a boldsymbol theta big each of these terms has an interpretation a but simply switching from probability to likelihood and taking logarithms yields the sum ℓ θ x η θ t x a θ log h x displaystyle ell boldsymbol theta mid x langle boldsymbol eta boldsymbol theta mathbf t x rangle a boldsymbol theta log h x the η θ textstyle boldsymbol eta boldsymbol theta and h x textstyle h x each correspond...
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