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bf 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 to a change of coordinates so in these coordinates the log likelihood of an exponential family is given by the simple formula ℓ η x η t x a η displaystyle ell boldsymbol eta mid x langle boldsymbol eta mathbf t x rangle a boldsymbol eta in words the log likelihood of an exponential family is inner product of the natural parameter η displaystyle boldsymbol eta and the sufficient statistic t x displaystyle mathbf t x minus the normalization factor log partition function a η displaystyle a boldsymbol eta thus for example the maximum likelihood estimate can be computed by taking derivatives of the sufficient statistic t and the log partition function a example the gamma distribution edit the gamma distribution is an exponential family with two parameters α textstyle alpha and β textstyle beta the likelihood function is l α β x β α γ α x α 1 e β x displaystyle mathcal l alpha beta mid x frac beta alpha gamma alpha x alpha 1 e beta x finding the maximum likelihood estimate of β textstyle beta for a single observed value x textstyle x looks rather daunting its logarithm is much simpler to work with log l α β x α log β log γ α α 1 log x β x displaystyle log mathcal l alpha beta mid x alpha log beta log gamma alpha alpha 1 log x beta x to maximize the log likelihood we first take the partial derivative with respect to β textstyle beta log l α β x β α β x displaystyle frac partial log mathcal l alpha beta mid x partial beta frac alpha beta x if there are a number of independent observations x 1 x n textstyle x_ 1 ldots x_ n then the joint log likelihood will be the sum of individual log likelihoods and the derivative of this sum will be a sum of derivatives of each individual log likelihood log l α β x 1 x n β log l α β x 1 β log l α β x n β n α β i 1 n x i displaystyle begin aligned frac partial log mathcal l alpha beta mid x_ 1 ldots x_ n partial beta frac partial log mathcal l alpha beta mid x_ 1 partial beta cdots frac partial log mathcal l alpha beta mid x_ n partial beta frac n alpha beta sum _ i 1 n x_ i end aligned to complete the maximization procedure for the joint log likelihood the equation is set to zero and solved for β textstyle beta β α x displaystyle widehat beta frac alpha bar x here β textstyle widehat beta denotes the maximum likelihood estimate and x 1 n i 1 n x i textstyle textstyle bar x frac 1 n sum _ i 1 n x_ i is the sample mean of the observations background and interpretation edit historical remarks edit see also history of statistics and history of probability the term likelihood has been in use in english since at least late middle english 39 its formal use to refer to a specific function in mathematical statistics was proposed by ronald fisher 40 in two research papers published in 1921 41 and 1922 42 the 1921 paper introduced what is today called a likelihood interval the 1922 paper introduced the term method of maximum likelihood quoting fisher i n 1922 i proposed the term likelihood in view of the fact that with respect to the parameter it is not a probability and does not obey the laws of probability while at the same time it bears to the problem of rational choice among the possible values of the parameter a relation similar to that which probability bears to the problem of predicting events in games of chance whereas however in relation to psychological judgment likelihood has some resemblance to probability the two concepts are wholly distinct 43 the concept of likelihood should not be confused with probability as mentioned by sir ronald fisher i stress this because in spite of the emphasis that i have always laid upon the difference between probability and likelihood there is still a tendency to treat likelihood as though it were a sort of probability the first result is thus that there are two different measures of rational belief appropriate to different cases knowing the population we can express our incomplete knowledge of or expectation of the sample in terms of probability knowing the sample we can express our incomplete knowledge of the population in terms of likelihood 44 fisher s invention of statistical likelihood was in reaction against an earlier form of reasoning called inverse probability 45 his use of the term likelihood fixed the meaning of the term within mathematical statistics a w f edwards 1972 established the axiomatic basis for use of the log likelihood ratio as a measure of relative support for one hypothesis against another the support function is then the natural logarithm of the likelihood function both terms are used in phylogenetics but were not adopted in a general treatment of the topic of statistical evidence 46 interpretations under different foundations edit among statisticians there is no consensus about what the foundation of statistics should be there are four main paradigms that have been proposed for the foundation frequentism bayesianism likelih...