# Likelihood function – Wikipedia

Function related to statistics and probability hypothesis
The likelihood function ( frequently just called the likelihood ) describes the joint probability of the observe data as a function of the parameters of the choose statistical model. [ 1 ] For each specific argument value θ { \displaystyle \theta } in the parameter outer space, the likelihood function p ( X | θ ) { \displaystyle p ( X|\theta ) } therefore assigns a probabilistic prediction to the observed data X { \displaystyle adam } . Since it is basically the product of sampling densities, the likelihood generally encapsulates both the data-generating process american samoa well as the missing-data mechanism that produced the respect sample. To emphasize that the likelihood is a function ( although not a probability concentration officiate ) of the parameters, [ a ] while the sample distribution is taken as given, it is frequently written as L ( θ ∣ X ) { \displaystyle { \mathcal { L } } ( \theta \mid X ) } . According to the likelihood principle, all of the information a given sample provides about θ { \displaystyle \theta } is expressed in the likelihood function. [ 2 ] In utmost likelihood estimate, the value which maximizes the probability of observing the given sample, i.e. θ ^ = argmax θ ∈ Θ ⁡ L ( θ ∣ X ) { \displaystyle { \hat { \theta } } =\operatorname { argmax } _ { \theta \in \Theta } { \mathcal { L } } ( \theta \mid X ) } , serves as a point estimate for the parameter of the distribution from which the sample was drawn. meanwhile in bayesian statistics, the likelihood function serves as the conduit through which sample information influences p ( θ ∣ X ) { \displaystyle p ( \theta \mid X ) } , the buttocks probability of the parameter, via Bayes ‘ rule. [ 3 ]

## definition

The likelihood routine is normally defined differently for discrete and continuous probability distributions. A general definition is besides possible, as discussed below.

### Discrete probability distribution

Let X { \displaystyle x } be a discrete random varying with probability mass serve phosphorus { \displaystyle phosphorus } depending on a parameter θ { \displaystyle \theta }. then the function

L ( θ ∣ x ) = p θ ( x ) = P θ ( X = x ), { \displaystyle { \mathcal { L } } ( \theta \mid x ) =p_ { \theta } ( ten ) =P_ { \theta } ( X=x ), }

considered as a function of θ { \displaystyle \theta }, is the likelihood function, given the consequence x { \displaystyle adam } of the random variable star X { \displaystyle ten }. sometimes the probability of “ the measure x { \displaystyle x } of X { \displaystyle adam } for the parameter value θ { \displaystyle \theta } ” is written as P ( X = x | θ ) or P ( X = x ; θ ). The likelihood is peer to the probability that a especial consequence x { \displaystyle x } is observe when the true value of the parameter is θ { \displaystyle \theta }, it is equal to the probability concentration over x { \displaystyle adam }, it is not a probability density over the parameter θ { \displaystyle \theta }. The likelihood, L ( θ ∣ x ) { \displaystyle { \mathcal { L } } ( \theta \mid adam ) } , should not be confused with phosphorus ( θ ∣ x ) { \displaystyle phosphorus ( \theta \mid ten ) } , which is the buttocks probability of θ { \displaystyle \theta } given the data x { \displaystyle ten }. Given no event ( no data ), the probability and thus likelihood is 1 ; [ citation needed ] any non-trivial consequence will have a lower likelihood .

#### model

phosphorus H 2 { \displaystyle p_ { \text { H } } ^ { 2 } } figure 1. The likelihood routine ( ) for the probability of a coin landing heads-up ( without anterior cognition of the coin ‘s comeliness ), given that we have observed HH . phosphorus H 2 ( 1 − phosphorus H ) { \displaystyle p_ { \text { H } } ^ { 2 } ( 1-p_ { \text { H } } ) } name 2. The likelihood officiate ( ) for the probability of a coin landing heads-up ( without prior cognition of the coin ‘s comeliness ), given that we have observed HHT. Consider a dim-witted statistical exemplar of a coin somersault : a single argument p H { \displaystyle p_ { \text { H } } } that expresses the “ fairness ” of the coin. The parameter is the probability that a mint lands heads up ( “ H ” ) when tossed. p H { \displaystyle p_ { \text { H } } } can take on any prize within the range 0.0 to 1.0. For a perfectly carnival mint, phosphorus H = 0.5 { \displaystyle p_ { \text { H } } =0.5 } . Imagine flipping a fair coin doubly, and observing the follow data : two heads in two tosses ( “ HH ” ). Assuming that each consecutive coin flip is i.i.d., then the probability of observing HH is

P ( HH ∣ p H = 0.5 ) = 0.5 2 = 0.25. { \displaystyle P ( { \text { HH } } \mid p_ { \text { H } } =0.5 ) =0.5^ { 2 } =0.25. }

Hence, given the note data HH, the likelihood that the model parameter p H { \displaystyle p_ { \text { H } } } equals 0.5 is 0.25. mathematically, this is written as

L ( phosphorus H = 0.5 ∣ HH ) = 0.25. { \displaystyle { \mathcal { L } } ( p_ { \text { H } } =0.5\mid { \text { HH } } ) =0.25. }

This is not the same as saying that the probability that p H = 0.5 { \displaystyle p_ { \text { H } } =0.5 }, given the observation HH, is 0.25. ( For that, we could apply Bayes ‘ theorem, which implies that the posterior probability is proportional to the likelihood times the anterior probability. ) Suppose that the coin is not a bonny coin, but rather it has p H = 0.3 { \displaystyle p_ { \text { H } } =0.3 } . then the probability of getting two heads is

P ( HH ∣ phosphorus H = 0.3 ) = 0.3 2 = 0.09. { \displaystyle P ( { \text { HH } } \mid p_ { \text { H } } =0.3 ) =0.3^ { 2 } =0.09. }

hence

L ( p H = 0.3 ∣ HH ) = 0.09. { \displaystyle { \mathcal { L } } ( p_ { \text { H } } =0.3\mid { \text { HH } } ) =0.09. }

More generally, for each value of p H { \displaystyle p_ { \text { H } } }, we can calculate the comparable likelihood. The consequence of such calculations is displayed in Figure 1. In Figure 1, the integral of the likelihood over the interval [ 0, 1 ] is 1/3. That illustrates an crucial aspect of likelihoods : likelihoods do not have to integrate ( or sum ) to 1, unlike probabilities .

### continuous probability distribution

Let X { \displaystyle ten } be a random variable following an absolutely continuous probability distribution with the concentration function fluorine { \displaystyle f } ( a affair of x { \displaystyle x } ) which depends on a argument θ { \displaystyle \theta }. then the function

L ( θ ∣ x ) = farad θ ( x ), { \displaystyle { \mathcal { L } } ( \theta \mid x ) =f_ { \theta } ( ten ), \, }

considered as a function of θ { \displaystyle \theta }, is the likelihood function ( of θ { \displaystyle \theta }, given the consequence x { \displaystyle adam } of X { \displaystyle X } ). Sometimes the concentration function for “ the respect x { \displaystyle ten } of X { \displaystyle adam } given the parameter value θ { \displaystyle \theta } ” is written as farad ( x ∣ θ ) { \displaystyle farad ( x\mid \theta ) } . The likelihood function, L ( θ ∣ x ) { \displaystyle { \mathcal { L } } ( \theta \mid x ) }, should not be confused with fluorine ( θ ∣ x ) { \displaystyle degree fahrenheit ( \theta \mid x ) } ; the likelihood is equal to the probability density of the observe consequence, x { \displaystyle x }, when the true value of the parameter is θ { \displaystyle \theta }, and therefore it is equal to a probability concentration over the result x { \displaystyle ten }, i.e. the likelihood function is not a concentration over the argument θ { \displaystyle \theta }. Put plainly, L ( θ ∣ x ) { \displaystyle { \mathcal { L } } ( \theta \mid ten ) } is to hypothesis test ( finding the probability of varying outcomes given a set of parameters defined in the null hypothesis ) as farad ( θ ∣ x ) { \displaystyle degree fahrenheit ( \theta \mid x ) } is to inference ( finding the probable parameters given a particular consequence ) .

### In general

In measure-theoretic probability theory, the concentration function is defined as the Radon–Nikodym derived function of the probability distribution relative to a park dominate measure. [ 4 ] The likelihood serve is that density interpreted as a function of the parameter ( possibly a vector ), rather than the possible outcomes. [ 5 ] This provides a likelihood function for any statistical mannequin with all distributions, whether discrete, absolutely continuous, a mixture or something else. ( Likelihoods will be comparable, e.g. for parameter appraisal, alone if they are Radon–Nikodym derivatives with deference to the lapp predominate measure. ) The discussion above of likelihood with discrete probabilities is a extra case of this using the count measure, which makes the probability density at any consequence adequate to the probability of that unmarried consequence .

### Likelihood affair of a parameterized model

Among many applications, we consider here one of broad theoretical and practical importance. Given a parameterized syndicate of probability concentration functions ( or probability mass functions in the sheath of discrete distributions )

ten ↦ fluorine ( x ∣ θ ), { \displaystyle x\mapsto f ( x\mid \theta ), \ ! }

where θ { \displaystyle \theta } is the parameter, the likelihood function is

θ ↦ farad ( adam ∣ θ ), { \displaystyle \theta \mapsto f ( x\mid \theta ), \ ! }

written

L ( θ ∣ x ) = degree fahrenheit ( adam ∣ θ ), { \displaystyle { \mathcal { L } } ( \theta \mid x ) =f ( x\mid \theta ), \ ! }

where x { \displaystyle x } is the watch consequence of an experiment. In early words, when farad ( x ∣ θ ) { \displaystyle degree fahrenheit ( x\mid \theta ) } is viewed as a function of x { \displaystyle ten } with θ { \displaystyle \theta } fixed, it is a probability concentration function, and when viewed as a officiate of θ { \displaystyle \theta } with x { \displaystyle x } fixed, it is a likelihood routine. This is not the same as the probability that those parameters are the right ones, given the note sample. Attempting to interpret the likelihood of a hypothesis given observe evidence as the probability of the hypothesis is a common error, with potentially black consequences. See prosecutor ‘s fallacy for an case of this. From a geometric point of view, if we consider fluorine ( x ∣ θ ) { \displaystyle fluorine ( x\mid \theta ) } as a function of two variables then the family of probability distributions can be viewed as a family of curves parallel to the x { \displaystyle ten } -axis, while the syndicate of likelihood functions is the orthogonal curves parallel to the θ { \displaystyle \theta } -axis .

#### Likelihoods for continuous distributions

The use of the probability concentration in specifying the likelihood function above is apologize as follows. Given an observation x joule { \displaystyle x_ { joule } } , the likelihood for the interval [ x j, x joule + hydrogen ] { \displaystyle [ x_ { joule }, x_ { j } +h ] } , where planck’s constant > 0 { \displaystyle planck’s constant > 0 } . observe that

argmax θ ⁡ L ( θ ∣ x ∈ [ x j, x joule + hydrogen ] ) = argmax θ ⁡ 1 planck’s constant L ( θ ∣ x ∈ [ x j, x joule + planck’s constant ] ) { \displaystyle \operatorname { argmax } _ { \theta } { \mathcal { L } } ( \theta \mid x\in [ x_ { j }, x_ { joule } +h ] ) =\operatorname { argmax } _ { \theta } { \frac { 1 } { hydrogen } } { \mathcal { L } } ( \theta \mid x\in [ x_ { joule }, x_ { j } +h ] ) }

since planck’s constant { \displaystyle heat content } is cocksure and constant. Because

argmax θ ⁡ 1 henry L ( θ ∣ x ∈ [ x joule, x joule + heat content ] ) = argmax θ ⁡ 1 heat content Pr ( x joule ≤ x ≤ x j + planck’s constant ∣ θ ) = argmax θ ⁡ 1 henry ∫ x joule x joule + henry f ( x ∣ θ ) vitamin d x, { \displaystyle \operatorname { argmax } _ { \theta } { \frac { 1 } { hydrogen } } { \mathcal { L } } ( \theta \mid x\in [ x_ { joule }, x_ { joule } +h ] ) =\operatorname { argmax } _ { \theta } { \frac { 1 } { hydrogen } } \Pr ( x_ { joule } \leq x\leq x_ { joule } +h\mid \theta ) =\operatorname { argmax } _ { \theta } { \frac { 1 } { heat content } } \int _ { x_ { j } } ^ { x_ { j } +h } degree fahrenheit ( x\mid \theta ) \, dx, }

where f ( ten ∣ θ ) { \displaystyle fluorine ( x\mid \theta ) } is the probability concentration officiate, it follows that

argmax θ ⁡ L ( θ ∣ x ∈ [ x joule, x j + henry ] ) = argmax θ ⁡ 1 heat content ∫ x j x joule + hydrogen fluorine ( ten ∣ θ ) d ten { \displaystyle \operatorname { argmax } _ { \theta } { \mathcal { L } } ( \theta \mid x\in [ x_ { j }, x_ { joule } +h ] ) =\operatorname { argmax } _ { \theta } { \frac { 1 } { henry } } \int _ { x_ { j } } ^ { x_ { joule } +h } fluorine ( x\mid \theta ) \, dx }

The first fundamental theorem of calculus provides that

lim planck’s constant → 0 + 1 planck’s constant ∫ x joule x j + planck’s constant f ( ten ∣ θ ) five hundred x = degree fahrenheit ( ten joule ∣ θ ). { \displaystyle { \begin { aligned } & \lim _ { h\to 0^ { + } } { \frac { 1 } { h } } \int _ { x_ { joule } } ^ { x_ { j } +h } fluorine ( x\mid \theta ) \, dx=f ( x_ { joule } \mid \theta ) .\end { aligned } } }

then

argmax θ ⁡ L ( θ ∣ x joule ) = argmax θ ⁡ [ lim planck’s constant → 0 + L ( θ ∣ x ∈ [ x joule, x j + heat content ] ) ] = argmax θ ⁡ [ lim henry → 0 + 1 h ∫ x j x j + h degree fahrenheit ( x ∣ θ ) d x ] = argmax θ ⁡ fluorine ( x j ∣ θ ). { \displaystyle { \begin { aligned } & \operatorname { argmax } _ { \theta } { \mathcal { L } } ( \theta \mid x_ { j } ) =\operatorname { argmax } _ { \theta } \left [ \lim _ { h\to 0^ { + } } { \mathcal { L } } ( \theta \mid x\in [ x_ { j }, x_ { j } +h ] ) \right ] \\ [ 4pt ] = { } & \operatorname { argmax } _ { \theta } \left [ \lim _ { h\to 0^ { + } } { \frac { 1 } { planck’s constant } } \int _ { x_ { j } } ^ { x_ { joule } +h } f ( x\mid \theta ) \, dx\right ] =\operatorname { argmax } _ { \theta } degree fahrenheit ( x_ { joule } \mid \theta ) .\end { aligned } } }

consequently ,

argmax θ ⁡ L ( θ ∣ x j ) = argmax θ ⁡ degree fahrenheit ( ten joule ∣ θ ), { \displaystyle \operatorname { argmax } _ { \theta } { \mathcal { L } } ( \theta \mid x_ { joule } ) =\operatorname { argmax } _ { \theta } fluorine ( x_ { j } \mid \theta ), \ ! }

and so maximizing the probability concentration at x joule { \displaystyle x_ { j } } amounts to maximizing the likelihood of the specific observation x j { \displaystyle x_ { j } } .

### Likelihoods for mix continuous–discrete distributions

The above can be extended in a bare way to allow circumstance of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses p k θ { \displaystyle p_ { thousand } \theta } and a concentration farad ( ten ∣ θ ) { \displaystyle farad ( x\mid \theta ) }, where the summarize of all the p { \displaystyle phosphorus } ‘s add to the built-in of farad { \displaystyle fluorine } is constantly 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 part, 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 serve for an observation from the discrete component is merely

L ( θ ∣ x ) = phosphorus thousand ( θ ), { \displaystyle { \mathcal { L } } ( \theta \mid x ) =p_ { kelvin } ( \theta ), \ ! }

where k { \displaystyle potassium } is the index of the discrete probability mass corresponding to observation x { \displaystyle ten }, because maximizing the probability mass ( or probability ) at x { \displaystyle x } amounts to maximizing the likelihood of the specific observation. The fact that the likelihood function can be defined in a direction 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 { \displaystyle ten }, but not with the parameter θ { \displaystyle \theta } .

### regularity conditions

In the context of argument estimate, the likelihood function is normally assumed to obey sealed conditions, known as regularity conditions. These conditions are assumed in diverse proof involving likelihood functions, and need to be verified in each particular application. For maximum likelihood estimate, the being of a ball-shaped maximal of the likelihood serve is of the extreme importance. By the extreme value theorem, it suffices that the likelihood function is continuous on a compress parameter space for the utmost likelihood calculator to exist. [ 6 ] While the continuity assumption is normally met, the concentration assumption about the parameter space is often not, as the bounds of the true parameter values are obscure. In that case, concavity of the likelihood function plays a key function. More specifically, if the likelihood function is twice continuously differentiable on the k-dimensional argument quad Θ { \displaystyle \, \Theta \, } assumed to be an open connected subset of R k, { \displaystyle \, \mathbb { R } ^ { kilobyte } \ ;, } there exists a singular maximum θ ^ ∈ Θ { \displaystyle { \hat { \theta } } \in \Theta } if the matrix of irregular partials

H ( θ ) ≡ [ ∂ 2 L ∂ θ iodine ∂ θ j ] one, joule = 1, 1 north one, nitrogen joule { \displaystyle \mathbf { H } ( \theta ) \equiv \left [ \, { \frac { \partial ^ { 2 } L } { \, \partial \theta _ { iodine } \, \partial \theta _ { joule } \, } } \, \right ] _ { one, j=1,1 } ^ { n_ { \mathrm { one } }, n_ { \mathrm { joule } } } \ ; }negative definite for every θ ∈ Θ { \displaystyle \, \theta \in \Theta \, } ∇ L ≡ [ ∂ L ∂ θ one ] one = 1 nitrogen one { \displaystyle \ ; \nabla L\equiv \left [ \, { \frac { \partial L } { \, \partial \theta _ { one } \, } } \, \right ] _ { i=1 } ^ { n_ { \mathrm { one } } } \ ; }

and if

lim θ → ∂ Θ L ( θ ) = 0, { \displaystyle \lim _ { \theta \to \partial \Theta } L ( \theta ) =0\ ;, }

i.e. the likelihood affair approaches a changeless on the limit of the parameter space, ∂ Θ, { \displaystyle \ ; \partial \Theta \ ;, } which may include the points at eternity if Θ { \displaystyle \, \Theta \, } is boundless. Mäkeläinen et al. prove this result using Morse theory while colloquially appealing to a mountain pass property. [ 7 ] Mascarenhas restates their proof using the batch passing theorem. [ 8 ] In the proof of consistency and asymptotic normality of the utmost likelihood calculator, extra assumptions are made about the probability densities that form the basis of a particular likelihood serve. These conditions were first established by Chanda. [ 9 ] In finical, for about all x { \displaystyle ten }, and for all θ ∈ Θ, { \displaystyle \, \theta \in \Theta \, , }

∂ log ⁡ f ∂ θ roentgen, ∂ 2 log ⁡ f ∂ θ gas constant ∂ θ sulfur, ∂ 3 log ⁡ fluorine ∂ θ roentgen ∂ θ s ∂ θ triiodothyronine { \displaystyle { \frac { \partial \log fluorine } { \partial \theta _ { r } } } \, ,\quad { \frac { \partial ^ { 2 } \log degree fahrenheit } { \partial \theta _ { gas constant } \partial \theta _ { sulfur } } } \, ,\quad { \frac { \partial ^ { 3 } \log f } { \partial \theta _ { gas constant } \, \partial \theta _ { second } \, \partial \theta _ { thyroxine } } } \, }

exist for all gas constant, south, t = 1, 2, …, k { \displaystyle \, radius, mho, t=1,2, \ldots, k\, } in club to ensure the universe of a Taylor expansion. Second, for about all ten { \displaystyle adam } and for every θ ∈ Θ { \displaystyle \, \theta \in \Theta \, } it must be that

| ∂ f ∂ θ radius | < F gas constant ( x ), | ∂ 2 degree fahrenheit ∂ θ roentgen ∂ θ s | < F roentgen south ( x ), | ∂ 3 degree fahrenheit ∂ θ roentgen ∂ θ s ∂ θ t | < H radius randomness metric ton ( x ) { \displaystyle \left| { \frac { \partial farad } { \partial \theta _ { gas constant } } } \right|

where H { \displaystyle H } is such that ∫ − ∞ ∞ H radius sulfur t ( z ) d z ≤ M < ∞. { \displaystyle \, \int _ { -\infty } ^ { \infty } H_ { rst } ( z ) \mathrm { d } z\leq M < \infty \ ;. } This finiteness of the derivatives is needed to allow for differentiation under the integral polarity. And last, it is assumed that the information matrix ,

I ( θ ) = ∫ − ∞ ∞ ∂ log ⁡ fluorine ∂ θ radius ∂ logarithm ⁡ fluorine ∂ θ s f five hundred z { \displaystyle \mathbf { I } ( \theta ) =\int _ { -\infty } ^ { \infty } { \frac { \partial \log degree fahrenheit } { \partial \theta _ { radius } } } \ { \frac { \partial \log f } { \partial \theta _ { sulfur } } } \ f\ \mathrm { five hundred } z }

is positive definite and | I ( θ ) | { \displaystyle \, \left|\mathbf { I } ( \theta ) \right|\, } is finite. This ensures that the score has a finite variance. [ 10 ] 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 utmost likelihood calculator of the properties mentioned above. Further, in sheath of non-independently or non-identically distribute observations extra properties may need to be assumed. In bayesian statistics, about identical regularity conditions are imposed on the likelihood affair in order to justify the Laplace approximation of the posterior probability. [ 11 ]

## Likelihood proportion and relative likelihood

### Likelihood proportion

A likelihood ratio is the proportion 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 degree to which data ( considered as testify ) 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 alleged likelihood-ratio test. By the Neyman–Pearson lemma, this is the most knock-down 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 proportion, considered as a screen statistic, is given by Wilks ‘ theorem. The likelihood ratio is besides of central importance in bayesian inference, where it is known as the Bayes agent, 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 proportion. As an equality :

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. rather, what is used is the relative likelihood of models ( see below ) .

### relative likelihood serve

Since the actual value of the likelihood routine depends on the sample, it is frequently convenient to work with a standardize measure. Suppose that the maximum likelihood estimate for the parameter θ is θ ^ { \displaystyle { \hat { \theta } } } . proportional plausibilities of other θ values may be found by comparing the likelihoods of those other values with the likelihood of θ ^ { \displaystyle { \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 adam ) } { { \mathcal { L } } ( { \hat { \theta } } \mid x ) } }. }

frankincense, the proportional likelihood is the likelihood proportion ( discussed above ) with the fix denominator L ( θ ^ ) { \displaystyle { \mathcal { L } } ( { \hat { \theta } } ) } . This corresponds to standardizing the likelihood to have a utmost of 1 .

#### Likelihood region

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 one real argument, a p % likelihood region will normally comprise an interval of real values. If the region does comprise an time interval, then it is called a likelihood interval. [ 13 ] [ 15 ] [ 19 ] Likelihood intervals, and more broadly likelihood regions, are used for time interval appraisal within likelihoodist statistics : they are like to confidence intervals in frequentist statistics and credible intervals in bayesian statistics. likelihood intervals are interpreted immediately in terms of proportional likelihood, not in terms of coverage probability ( frequentism ) or back tooth probability ( Bayesianism ). Given a exemplar, likelihood intervals can be compared to confidence intervals. If θ is a single very 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 unlike formulation suited to the use of log-likelihoods ( see Wilks ‘ theorem ), the test statistic is twice the deviation in log-likelihoods and the probability distribution of the trial statistic is approximately a chi-squared distribution with degrees-of-freedom ( df ) equal to the dispute in df ‘s between the two models ( consequently, the e−2 likelihood interval is the same as the 0.954 confidence time interval ; assuming deviation in df ‘s to be 1 ). [ 18 ] [ 19 ]

## Likelihoods that eliminate pain parameters

In many cases, the likelihood is a serve of more than one parameter but pastime focuses on the estimate of only one, or at most a few of them, with the others being considered as nuisance parameters. several alternate approaches have been developed to eliminate such nuisance parameters, so that a likelihood can be written as a function of only the argument ( or parameters ) of pastime : the main approaches are profile, conditional, and fringy likelihoods. [ 20 ] [ 21 ] These approaches are besides 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

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 { \displaystyle \mathbf { \theta } } that can be partitioned into θ = ( θ 1 : θ 2 ) { \displaystyle \mathbf { \theta } =\left ( \mathbf { \theta } _ { 1 } : \mathbf { \theta } _ { 2 } \right ) } , and where a parallelism θ ^ 2 = θ ^ 2 ( θ 1 ) { \displaystyle \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 trouble. [ 24 ] For example, in a analogue regression with normally distributed errors, y = X β + u { \displaystyle \mathbf { yttrium } =\mathbf { X } \beta +u } , the coefficient vector could be partitioned into β = [ β 1 : β 2 ] { \displaystyle \beta =\left [ \beta _ { 1 } : \beta _ { 2 } \right ] } ( and consequently the design matrix X = [ X 1 : x 2 ] { \displaystyle \mathbf { X } =\left [ \mathbf { X } _ { 1 } : \mathbf { X } _ { 2 } \right ] } ). Maximizing with respect to β 2 { \displaystyle \beta _ { 2 } } yields an optimum respect function β 2 ( β 1 ) = ( X 2 T X 2 ) − 1 X 2 T ( yttrium − X 1 β 1 ) { \displaystyle \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 utmost likelihood calculator for β 1 { \displaystyle \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 { ten } _ { 1 } \right ) ^ { -1 } \mathbf { X } _ { 1 } ^ { \mathsf { T } } \left ( \mathbf { I } -\mathbf { P } _ { 2 } \right ) \mathbf { yttrium } }

where P 2 = X 2 ( X 2 T X 2 ) − 1 X 2 T { \displaystyle \mathbf { P } _ { 2 } =\mathbf { X } _ { 2 } \left ( \mathbf { X } _ { 2 } ^ { \mathsf { T } } \mathbf { X } _ { 2 } \right ) ^ { -1 } \mathbf { X } _ { 2 } ^ { \mathsf { T } } } is the expulsion matrix of X 2 { \displaystyle \mathbf { X } _ { 2 } } . This leave is known as the Frisch–Waugh–Lovell theorem. Since diagrammatically the operation of concentration is equivalent to slicing the likelihood airfoil along the ridge of values of the nuisance argument β 2 { \displaystyle \beta _ { 2 } } that maximizes the likelihood function, creating an isometric line profile of the likelihood function for a given β 1 { \displaystyle \beta _ { 1 } }, the leave of this routine is besides known as profile likelihood. [ 25 ] [ 26 ] In summation to being graphed, the profile likelihood can besides be used to compute confidence intervals that often have better small-sample properties than those based on asymptotic standard errors calculated from the wide likelihood. [ 27 ] [ 28 ]

### conditional likelihood

sometimes it is potential to find a sufficient statistic for the pain parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters. [ 29 ] One model occurs in 2×2 tables, where conditioning on all four borderline totals leads to a conditional likelihood based on the non-central hypergeometric distribution. This form of condition is besides the basis for Fisher ‘s accurate quiz .

### borderline likelihood

sometimes we can remove the nuisance parameters by considering a likelihood based on only separate of the data in the data, for example by using the set of ranks quite than the numeric values. Another example occurs in linear interracial models, where considering a likelihood for the residuals only after fitting the fix effects leads to residual maximum likelihood estimate of the variance components .

### partial derivative likelihood

A partial derivative likelihood is an adaptation of the full likelihood such that lone a function of the parameters ( the parameters of interest ) occur in it. [ 30 ] It is a identify component of the proportional hazards model : using a restriction on the hazard function, the likelihood does not contain the shape of the guess over time .

## Products of likelihoods

The likelihood, given two or more independent events, is the product of the likelihoods of each of the individual events :

Λ ( A ∣ ten 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 merchandise of the probabilities. This is particularly significant when the events are from mugwump and identically circulate random variables, such as independent observations or sampling with substitute. 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 constantly 1. This is similar to a uniform anterior in bayesian statistics, but in likelihoodist statistics this is not an improper anterior because likelihoods are not integrated .
Log-likelihood function is a logarithmic transformation of the likelihood routine, frequently denoted by a lowercase l or ℓ { \displaystyle \ell } , to contrast with the capital L or L { \displaystyle { \mathcal { L } } } for the likelihood. Because logarithm are strictly increasing functions, maximizing the likelihood is equivalent to maximizing the log-likelihood. But for hardheaded purposes it is more convenient to work with the log-likelihood function in maximum likelihood estimate, in particular since most common probability distributions —notably the exponential family —are lone logarithmically concave, [ 31 ] [ 32 ] and concave shape of the objective affair plays a key function 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 accession to the numerical convenience from this, the adding march of log-likelihood has an intuitive rendition, as much expressed as “ support ” from the data. When the parameters are estimated using the log-likelihood for the maximal likelihood estimate, 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 “ accompaniment from independent evidence adds”, and the log-likelihood is the “ weight of evidence ”. Interpreting negative log-probability as information contented or surprise, the support ( log-likelihood ) of a model, given an event, is the minus of the surprise of the event, given the model : a model is supported by an event to the extent that the consequence is unsurprising, given the model. A logarithm of a likelihood ratio is equal to the difference of the log-likelihoods :

logarithm ⁡ L ( A ) L ( B ) = log ⁡ L ( A ) − log ⁡ L ( B ) = ℓ ( A ) − ℓ ( B ). { \displaystyle \log { \frac { L ( A ) } { L ( B ) } } =\log L ( A ) -\log L ( B ) =\ell ( A ) -\ell ( B ). }

merely 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 .

### Likelihood equations

If the log-likelihood function is politic, its gradient with respect to the argument, known as the score and written randomness n ( θ ) ≡ ∇ θ ℓ north ( θ ) { \displaystyle s_ { north } ( \theta ) \equiv \nabla _ { \theta } \ell _ { normality } ( \theta ) } , exists and allows for the application of differential tartar. The basic manner 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 summarize 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 autonomous events. The equations defined by the stationary point of the score officiate serve as estimating equations for the maximum likelihood calculator .

sulfur newton ( θ ) = 0 { \displaystyle s_ { newton } ( \theta ) =\mathbf { 0 } }

In that sense, the maximum likelihood calculator is implicitly defined by the measure at 0 { \displaystyle \mathbf { 0 } } of the inverse function s nitrogen − 1 : e vitamin d → Θ { \displaystyle s_ { normality } ^ { -1 } : \mathbb { E } ^ { vitamin d } \to \Theta } , where E d { \displaystyle \mathbb { E } ^ { five hundred } } is the d-dimensional Euclidean space, and Θ { \displaystyle \Theta } is the parameter space. Using the inverse officiate theorem, it can be shown that second normality − 1 { \displaystyle s_ { n } ^ { -1 } } is chiseled in an clear neighborhood about 0 { \displaystyle \mathbf { 0 } } with probability going to one, and θ ^ n = south n − 1 ( 0 ) { \displaystyle { \hat { \theta } } _ { normality } =s_ { newton } ^ { -1 } ( \mathbf { 0 } ) } is a reproducible estimate of θ { \displaystyle \theta }. As a consequence there exists a sequence { θ ^ north } { \displaystyle \left\ { { \hat { \theta } } _ { nitrogen } \right\ } } such that s normality ( θ ^ nitrogen ) = 0 { \displaystyle s_ { nitrogen } ( { \hat { \theta } } _ { newton } ) =\mathbf { 0 } } asymptotically about surely, and θ ^ n → p θ 0 { \displaystyle { \hat { \theta } } _ { north } { \xrightarrow { \text { phosphorus } } } \theta _ { 0 } } . [ 33 ] A similar resultant role can be established using Rolle ‘s theorem. [ 34 ] [ 35 ] The second derivative instrument evaluated at θ ^ { \displaystyle { \hat { \theta } } }, known as Fisher data, determines the curvature of the likelihood surface, [ 36 ] and thus indicates the preciseness of the estimate. [ 37 ]

### exponential families

The log-likelihood is besides particularly utilitarian 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 routine is a summarize of products, again easier to differentiate than the original function. An exponential class is one whose probability density officiate is of the shape ( for some functions, writing ⟨ −, − ⟩ { \displaystyle \langle -, -\rangle } for the inner intersection ) :

p ( x ∣ θ ) = h ( x ) exp ⁡ ( ⟨ η ( θ ), T ( x ) ⟩ − A ( θ ) ). { \displaystyle p ( x\mid { \boldsymbol { \theta } } ) =h ( ten ) \exp { \Big ( } \langle { \boldsymbol { \eta } } ( { \boldsymbol { \theta } } ), \mathbf { T } ( ten ) \rangle -A ( { \boldsymbol { \theta } } ) { \Big ) }. }

Each of these terms has an rendition, [ boron ] but merely switching from probability to likelihood and taking logarithm 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 } ( ten ) \rangle -A ( { \boldsymbol { \theta } } ) +\log h ( x ). }

The η ( θ ) { \displaystyle { \boldsymbol { \eta } } ( { \boldsymbol { \theta } } ) } and henry ( x ) { \displaystyle hydrogen ( x ) } each correspond to a change of coordinates, therefore in these coordinates, the log-likelihood of an exponential family is given by the simpleton 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 inside product of the natural parameter η { \displaystyle { \boldsymbol { \eta } } } and the sufficient statistic T ( x ) { \displaystyle \mathbf { T } ( adam ) } , minus the standardization factor ( log-partition serve ) A ( η ) { \displaystyle A ( { \boldsymbol { \eta } } ) } . Thus for example the utmost likelihood estimate can be computed by taking derivatives of the sufficient statistic T and the log-partition function A .

#### example : the da gamma distribution

The gamma distribution is an exponential family with two parameters, α { \displaystyle \alpha } and β { \displaystyle \beta } . The likelihood function is

L ( α, β ∣ x ) = β α Γ ( α ) x α − 1 east − β x. { \displaystyle { \mathcal { L } } ( \alpha, \beta \mid x ) = { \frac { \beta ^ { \alpha } } { \Gamma ( \alpha ) } } x^ { \alpha -1 } e^ { -\beta ten }. }

Finding the utmost likelihood appraisal of β { \displaystyle \beta } for a individual watch value x { \displaystyle x } looks quite 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 base take the overtone derivative with respect to β { \displaystyle \beta } :

∂ logarithm ⁡ 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 total of independent observations x 1, …, x newton { \displaystyle x_ { 1 }, \ldots, x_ { newton } } , then the joint log-likelihood will be the union of individual log-likelihoods, and the derivative of this sum will be a kernel of derivatives of each individual log-likelihood :

∂ log ⁡ L ( α, β ∣ x 1, …, x newton ) ∂ β = ∂ log ⁡ L ( α, β ∣ x 1 ) ∂ β + ⋯ + ∂ log ⁡ L ( α, β ∣ x normality ) ∂ β = nitrogen α β − ∑ one = 1 normality adam one. { \displaystyle { \begin { aligned } & { \frac { \partial \log { \mathcal { L } } ( \alpha, \beta \mid x_ { 1 }, \ldots, x_ { north } ) } { \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 } ^ { nitrogen } x_ { i } .\end { aligned } } }

To complete the maximization operation for the joint log-likelihood, the equation is set to zero and solved for β { \displaystyle \beta } :

β ^ = α x ¯. { \displaystyle { \widehat { \beta } } = { \frac { \alpha } { \bar { ten } } }. }

here β ^ { \displaystyle { \widehat { \beta } } } denotes the maximum-likelihood estimate, and x ¯ = 1 newton ∑ one = 1 nitrogen ten one { \displaystyle \textstyle { \bar { ten } } = { \frac { 1 } { newton } } \sum _ { i=1 } ^ { n } x_ { i } } is the sample mean of the observations .

## Background and interpretation

### historic remarks

The terminus “ likelihood ” has been in use in English since at least late Middle English. [ 38 ] Its formal use to refer to a specific function in mathematical statistics was proposed by Ronald Fisher, [ 39 ] in two research papers published in 1921 [ 40 ] and 1922. [ 41 ] The 1921 newspaper introduced what is nowadays called a “ likelihood interval ” ; the 1922 paper introduced the term “ method acting of maximal likelihood “. Quoting fisher :

[ I ] newton 1922, I proposed the term ‘likelihood, ‘ in view of the fact that, with regard to [ the parameter ], it is not a probability, and does not obey the laws of probability, while at the lapp time it bears to the problem of rational option among the possible values of [ the parameter ] a relative 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.. .. ” [ 42 ]

The concept of likelihood should not be confused with probability as mentioned by Sir Ronald Fisher

I stress this because in cattiness of the vehemence that I have always laid upon the dispute between probability and likelihood there is however a tendency to treat likelihood as though it were a sort of probability. The foremost solution is therefore that there are two different measures of rational belief appropriate to unlike cases. Knowing the population we can express our incomplete cognition of, or anticipation of, the sample in terms of probability ; knowing the sample distribution we can express our incomplete cognition of the population in terms of likelihood. [ 43 ]

Fisher ‘s invention of statistical likelihood was in reaction against an earlier shape of reasoning called inverse probability. [ 44 ] His consumption of the term “ likelihood ” fixed the mean of the term within mathematical statistics. A. W. F. Edwards ( 1972 ) established the axiomatic basis for habit of the log-likelihood proportion 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 cosmopolitan treatment of the subject of statistical tell. [ 45 ]

### Interpretations under different foundations

Among statisticians, there is no consensus about what the foundation of statistics should be. There are four main paradigm that have been proposed for the initiation : frequentism, Bayesianism, likelihoodism, and AIC-based. [ 46 ] For each of the proposed foundations, the interpretation of likelihood is different. The four interpretations are described in the subsections below .

#### bayesian interpretation

In Bayesian inference, although one can speak about the likelihood of any suggestion or random variable given another random variable star : for model the likelihood of a parameter measure or of a statistical model ( see marginal likelihood ), given specified data or other evidence, [ 47 ] [ 48 ] [ 49 ] [ 50 ] the likelihood function remains the same entity, with the extra interpretations of ( i ) a conditional concentration of the data given the parameter ( since the parameter is then a random variable ) and ( two ) a measure or sum of information brought by the data about the parameter rate or even the exemplary. [ 47 ] [ 48 ] [ 49 ] [ 50 ] [ 51 ] due to the presentation of a probability structure on the parameter space or on the solicitation of models, it is possible that a parameter rate or a statistical model have a bombastic likelihood value for given data, and however have a gloomy probability, or vice versa. [ 49 ] [ 51 ] This is much the case in medical context. [ 52 ] Following Bayes ‘ Rule, the likelihood when seen as a conditional concentration can be multiplied by the prior probability concentration of the parameter and then normalized, to give a posterior probability density. [ 47 ] [ 48 ] [ 49 ] [ 50 ] [ 51 ] More broadly, the likelihood of an nameless quantity X { \displaystyle x } given another unknown quantity Y { \displaystyle Y } is proportional to the probability of

Y

{\displaystyle Y}

given

X

{\displaystyle X}

. [ 47 ] [ 48 ] [ 49 ] [ 50 ] [ 51 ]

#### Likelihoodist rendition

In frequentist statistics, the likelihood function is itself a statistic that summarizes a single sample from a population, whose account prize depends on a choice of several parameters θ 1 … θ phosphorus, where p is the count of parameters in some already-selected statistical model. The value of the likelihood serves as a digit of deservingness for the choice used for the parameters, and the argument set with maximal likelihood is the best choice, given the data available. The particular calculation of the likelihood is the probability that the detect sample distribution would be assigned, assuming that the model chosen and the values of the respective parameters θ give an accurate estimate of the frequency distribution of the population that the detect sample was drawn from. Heuristically, it makes feel that a good choice of parameters is those which render the sample actually observed the utmost possible post-hoc probability of having happened. Wilks ‘ theorem quantifies the heuristic predominate by showing that the dispute in the logarithm of the likelihood generated by the appraisal ‘s argument values and the logarithm of the likelihood generated by population ‘s “ true ” ( but unknown ) parameter values is asymptotically χ2 distributed. Each independent sample ‘s utmost likelihood estimate is a break estimate of the “ true ” parameter set describing the population sampled. consecutive estimates from many autonomous samples will cluster together with the population ‘s “ true ” set of parameter values hidden somewhere in their midst. The remainder in the logarithm of the utmost likelihood and adjacent argument 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 repair value. The χ2 distribution given by Wilks ‘ theorem converts the region ‘s log-likelihood differences into the “ assurance ” that the population ‘s “ true ” parameter set lies inside. The artwork of choosing the fix log-likelihood dispute is to make the confidence acceptably high while keeping the region acceptably little ( pin down scope of estimates ). As more data are observed, alternatively of being used to make autonomous estimates, they can be combined with the previous samples to make a individual aggregate sample, and that large sample distribution may be used for a new maximal likelihood estimate. As the size of the compound sample increases, the size of the likelihood region with the same confidence shrinks. finally, either the size of the confidence region is very about a single compass point, or the integral population has been sampled ; in both cases, the estimate parameter set is basically the same as the population parameter set .

#### AIC-based rendition

Under the AIC prototype, likelihood is interpreted within the context of information hypothesis. [ 53 ] [ 54 ] [ 55 ]

## Notes

1. ^Valavanis, Stefan (1959). “Probability and Likelihood”. Econometrics : An Introduction to Maximum Likelihood Methods. New York: McGraw-Hill. pp. 24–28. OCLC 6257066. While frequently used synonymously in common speech, the terms “ likelihood “ and “ probability “ have distinct meanings in statistics. Probability is a property of the sample, specifically how probable it is to obtain a especial sample distribution for a given value of the parameters of the distribution ; likelihood is a property of the parameter values. See

2. ^ See exponential kin § Interpretation

## far interpretation

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