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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 likelihoodism and aic based 47 for each of the proposed foundations the interpretation of likelihood is different the four interpretations are described in the subsections below frequentist interpretation edit this section is empty you can help by adding to it march 2019 bayesian interpretation edit in bayesian inference although one can speak about the likelihood of any proposition or random variable given another random variable for example the likelihood of a parameter value or of a statistical model see marginal likelihood given specified data or other evidence 48 49 50 51 the likelihood function remains the same entity with the additional interpretations of i a conditional density of the data given the parameter since the parameter is then a random variable and ii a measure or amount of information brought by the data about the parameter value or even the model 48 49 50 51 52 due to the introduction of a probability structure on the parameter space or on the collection of models it is possible that a parameter value or a statistical model have a large likelihood value for given data and yet have a low probability or vice versa 50 52 this is often the case in medical contexts 53 following bayes rule the likelihood when seen as a conditional density can be multiplied by the prior probability density of the parameter and then normalized to give a posterior probability density 48 49 50 51 52 more generally the likelihood of an unknown quantity x textstyle x given another unknown quantity y textstyle y is proportional to the probability of y textstyle y given x textstyle x 48 49 50 51 52 likelihoodist interpretation edit this section includes a list of general references but lacks sufficient corresponding inline citations please help improve this section by introducing more precise citations april 2019 learn how and when to remove this message in frequentist statistics the likelihood function is itself a statistic that summarizes a single sample from a population whose calculated value depends on a choice of several parameters θ 1 θ p where p is the count of parameters in some already selected statistical model the value of the likelihood serves as a figure of merit for the choice used for the parameters and the parameter set with maximum likelihood is the best choice given the data available the specific calculation of the likelihood is the probability that the observed sample would be assigned assuming that the model chosen and the values of the several parameters θ give an accurate approximation of the frequency distribution of the population that the observed sample was drawn from heuristically it makes sense that a good choice of parameters is those which render the sample actually observed the maximum possible post hoc probability of having happened wilks theorem quantifies the heuristic rule by showing that the difference in the logarithm of the likelihood generated by the estimate s parameter values and the logarithm of the likelihood generated by population s true but unknown parameter values is asymptotically χ 2 distributed each independent sample s maximum likelihood estimate is a separate estimate of the true parameter set describing the population sampled successive estimates from many independent samples will cluster together with the population s true set of parameter values hidden somewhere in their midst the difference in the logarithms of the maximum likelihood and adjacent parameter sets likelihoods may be used to draw a confidence region on a plot whose co ordinates are the parameters θ 1 θ p the region surrounds the maximum likelihood estimate and all points parameter sets within that region differ at most in log likelihood by some fixed value the χ 2 distribution given by wilks theorem converts the region s log likelihood differences into the confidence that the population s true parameter set lies inside the art of choosing the fixed log likelihood difference is to make the confidence acceptably high while keeping the region acceptably small narrow range of estimates as more data are observed instead of being used to make independent estimates they can be combined with the previous samples to make a single combined sample and that large sample may be used for a new maximum likelihood estimate as the size of the combined sample increases the size of the likelihood region with the same confidence shrinks eventually either the size of the confidence region is very nearly a single point or the entire population has been sampled in both cases the estimated parameter set is essentially the same as the population parameter set aic based interpretation edit this section needs expansion you can help by adding missing information march 2019 under the aic paradigm likelihood is interpreted within the context of information theory 54 55 56 see also edit bayes factor conditional entropy conditional probability empirical likelihood likelihood principle likelihood ratio test likelihoodist statistics maximum likelihood estimation principle of maximum entropy pseudolikelihood score statistics notes edit see exponential family interpretation references edit zellner arnold 1971 an introduction to bayesian inference in econometrics new york wiley pp 13 14 isbn 0 471 98165 6 billingsley patrick 1995 probability and measure third ed john wiley sons pp 422 423 shao jun 2003 mathematical statistics 2nd ed springer 4 4 1 gouriéroux christian monfort alain 1995 statistics an...
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