Likelihood function

In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model. The likelihood of a set of parameter values, θ, given outcomes x, is equal to the probability of those observed outcomes given those parameter values, that is ${\displaystyle {\mathcal {L}}(\theta |x)=P(x|\theta )}$.

Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics. In informal contexts, "likelihood" is often used as a synonym for "probability." But in statistical usage, a distinction is made depending on the roles of the outcome or parameter. Probability is used when describing a function of the outcome given a fixed parameter value. For example, if a coin is flipped 10 times and it is a fair coin, what is the probability of it landing heads-up every time? Likelihood is used when describing a function of a parameter given an outcome. For example, if a coin is flipped 10 times and it has landed heads-up 10 times, what is the likelihood that the coin is fair?

Definition

The likelihood function is defined differently for discrete and continuous probability distributions.

Discrete probability distribution

Let X be a random variable with a discrete probability distribution p depending on a parameter θ. Then the function

${\displaystyle {\mathcal {L}}(\theta |x)=p_{\theta }(x)=P_{\theta }(X=x),\,}$

considered as a function of θ, is called the likelihood function (of θ, given the outcome x of X). Sometimes the probability on the value x of X for the parameter value θ is written as ${\displaystyle P(X=x|\theta )}$; often written as ${\displaystyle P(X=x;\theta )}$ to emphasize that this value is not a conditional probability, because θ is a parameter and not a random variable.

Continuous probability distribution

Let X be a random variable with a continuous probability distribution with density function f depending on a parameter θ. Then the function

${\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}$

considered as a function of θ, is called the likelihood function (of θ, given the outcome x of X). Sometimes the density function for the value x of X for the parameter value θ is written as ${\displaystyle f(x|\theta )}$, but should not be considered as a conditional probability density.

The actual value of a likelihood function bears no meaning. Its use lies in comparing one value with another. For example, one value of the parameter may be more likely than another, given the outcome of the sample. Or a specific value will be most likely: the maximum likelihood estimate. Comparison may also be performed in considering the quotient of two likelihood values. That is why ${\displaystyle {\mathcal {L}}(\theta |x)}$ is generally permitted to be any positive multiple of the above defined function ${\displaystyle {\mathcal {L}}}$. More precisely, then, a likelihood function is any representative from an equivalence class of functions,

${\displaystyle {\mathcal {L}}\in \left\lbrace \alpha \;P_{\theta }:\alpha >0\right\rbrace ,\,}$

where the constant of proportionality α > 0 is not permitted to depend upon θ, and is required to be the same for all likelihood functions used in any one comparison. In particular, the numerical value ${\displaystyle {\mathcal {L}}(\theta |x)}$ alone is immaterial; all that matters are maximum values of ${\displaystyle {\mathcal {L}}}$, or likelihood ratios, such as those of the form

${\displaystyle {\frac {{\mathcal {L}}(\theta _{2}|x)}{{\mathcal {L}}(\theta _{1}|x)}}={\frac {\alpha P(X=x|\theta _{2})}{\alpha P(X=x|\theta _{1})}}={\frac {P(X=x|\theta _{2})}{P(X=x|\theta _{1})}},}$

that are invariant with respect to the constant of proportionality α.

For more about making inferences via likelihood functions, see also the method of maximum likelihood, and likelihood-ratio testing.

Log-likelihood

For many applications, the natural logarithm of the likelihood function, called the log-likelihood, is more convenient to work with. Because the logarithm is a monotonically increasing function, the logarithm of a function achieves its maximum value at the same points as the function itself, and hence the log-likelihood can be used in place of the likelihood in maximum likelihood estimation and related techniques. Finding the maximum of a function often involves taking the derivative of a function and solving for the parameter being maximized, and this is often easier when the function being maximized is a log-likelihood rather than the original likelihood function.

For example, some likelihood functions are for the parameters that explain a collection of statistically independent observations. In such a situation, the likelihood function factors into a product of individual likelihood functions. The logarithm of this product is a sum of individual logarithms, and the derivative of a sum of terms is often easier to compute than the derivative of a product. In addition, several common distributions have likelihood functions that 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.

In phylogenetics the log-likelihood ratio is sometimes termed support and the log-likelihood function support function.[1] However, given the potential for confusion with the mathematical meaning of 'support' this terminology is rarely used outside this field.

Example: the gamma distribution

The gamma distribution has two parameters α and β. The likelihood function is

${\displaystyle {\mathcal {L}}(\alpha ,\beta \,|\,x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}$.

Finding the maximum likelihood estimate of β for a single observed value x looks rather daunting. Its logarithm is much simpler to work with:

${\displaystyle \log {\mathcal {L}}(\alpha ,\beta \,|\,x)=\alpha \log \beta -\log \Gamma (\alpha )+(\alpha -1)\log x-\beta x.\,}$

Maximizing the log-likelihood first requires taking the partial derivative with respect to β:

${\displaystyle {\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \,|\,x)}{\partial \beta }}={\frac {\alpha }{\beta }}-x}$.

If there are a number of independent random samples x1, ..., xn, 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:

${\displaystyle {\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \,|\,x_{1},\ldots ,x_{n})}{\partial \beta }}={\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \,|\,x_{1})}{\partial \beta }}+\cdots +{\frac {\partial \log {\mathcal {L}}(\alpha ,\beta \,|\,x_{n})}{\partial \beta }}={\frac {n\alpha }{\beta }}-\sum _{i=1}^{n}x_{i}.}$

To complete the maximization procedure for the joint log-likelihood, the equation is set to zero and solved for β:

${\displaystyle {\hat {\beta }}={\frac {\alpha }{\bar {x}}}.}$

Here ${\displaystyle {\hat {\beta }}}$ denotes the maximum-likelihood estimate, and ${\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$ is the sample mean of the observations.

Likelihood function of a parameterized model

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

${\displaystyle x\mapsto f(x\mid \theta ),\!}$

where θ is the parameter, the likelihood function is

${\displaystyle \theta \mapsto f(x\mid \theta ),\!}$

written

${\displaystyle {\mathcal {L}}(\theta \mid x)=f(x\mid \theta ),\!}$

where x is the observed outcome of an experiment. In other words, when f(x | θ) is viewed as a function of x with θ fixed, it is a probability density function, and when viewed as a function of θ with x fixed, it is a likelihood function.

This is not the same as the probability that those parameters are the right ones, given the observed sample. Attempting to interpret the likelihood of a hypothesis given observed evidence as the probability of the hypothesis is a common error, with potentially disastrous consequences in medicine, engineering or jurisprudence. See prosecutor's fallacy for an example of this.

From a geometric standpoint, if we consider f (xθ) as a function of two variables then the family of probability distributions can be viewed as a family of curves parallel to the x-axis, while the family of likelihood functions are the orthogonal curves parallel to the θ-axis.

Likelihoods for continuous distributions

The use of the probability density instead of a probability in specifying the likelihood function above may be justified in a simple way. Suppose that, instead of an exact observation, x, the observation is the value in a short interval (xj−1xj), with length Δj, where the subscripts refer to a predefined set of intervals. Then the probability of getting this observation (of being in interval j) is approximately

${\displaystyle {\mathcal {L}}_{\text{approx}}(\theta \mid x{\text{ in interval }}j)=f(x_{*}\mid \theta )\Delta _{j},\!}$

where x* can be any point in interval j. Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is approximately

${\displaystyle {\mathcal {L}}_{\text{approx}}(\theta \mid x{\text{ in interval }}j)=f(x_{*}\mid \theta ),\!}$

and then, on considering the lengths of the intervals to decrease to zero,

${\displaystyle {\mathcal {L}}(\theta \mid x)=f(x\mid \theta ).\!}$

Likelihoods for mixed continuous–discrete distributions

The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components. Suppose that the distribution consists of a number of discrete probability masses pk(θ) and a density f(x | θ), where the sum of all the p's added to the integral of f is always one. Assuming that it is possible to distinguish an observation corresponding to one of the discrete probability masses from one which corresponds to the density component, the likelihood function for an observation from the continuous component can be dealt with as above by setting the interval length short enough to exclude any of the discrete masses. For an observation from the discrete component, the probability can either be written down directly or treated within the above context by saying that the probability of getting an observation in an interval that does contain a discrete component (of being in interval j which contains discrete component k) is approximately

${\displaystyle {\mathcal {L}}_{\text{approx}}(\theta \mid x{\text{ in interval }}j{\text{ containing discrete mass }}k)=p_{k}(\theta )+f(x_{*}\mid \theta )\Delta _{j},\!}$

where ${\displaystyle x_{*}\ }$ can be any point in interval j. Then, on considering the lengths of the intervals to decrease to zero, the likelihood function for an observation from the discrete component is

${\displaystyle {\mathcal {L}}(\theta \mid x)=p_{k}(\theta ),\!}$

where k is the index of the discrete probability mass corresponding to observation x.

The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation x, but not with the parameter θ.

Example 1

The likelihood function for estimating the probability of a coin landing heads-up without prior knowledge after observing HH
The likelihood function for estimating the probability of a coin landing heads-up without prior knowledge after observing HHT

Let ${\displaystyle p_{\text{H}}}$ be the probability that a certain coin lands heads up (H) when tossed. So, the probability of getting two heads in two tosses (HH) is ${\displaystyle p_{\text{H}}^{2}}$. If ${\displaystyle p_{\text{H}}=0.5}$, then the probability of seeing two heads is 0.25.

${\displaystyle P({\text{HH}}|p_{\text{H}}=0.5)=0.25.}$

Another way of saying this is that the likelihood that ${\displaystyle p_{\text{H}}=0.5}$, given the observation HH, is 0.25, that is

${\displaystyle {\mathcal {L}}(p_{\text{H}}=0.5|{\text{HH}})=P({\text{HH}}|p_{\text{H}}=0.5)=0.25.}$

But this is not the same as saying that the probability that ${\displaystyle p_{\text{H}}=0.5}$, given the observation HH, is 0.25. The likelihood that ${\displaystyle p_{\text{H}}=1}$, given the observation HH, is 1, but it is not true that the probability that ${\displaystyle p_{\text{H}}=1}$, given the observation HH, is 1. Two heads in a row does not prove that the coin always comes up heads, because two heads in a row is possible for any ${\displaystyle p_{\text{H}}>0}$.

The likelihood function is not a probability density function. The integral of a likelihood function is not in general 1. In this example, the integral of the likelihood over the interval [0, 1] in ${\displaystyle p_{\text{H}}}$ is 1/3, demonstrating that the likelihood function cannot be interpreted as a probability density function for ${\displaystyle p_{\text{H}}}$.

Example 2

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Consider a jar containing N lottery tickets numbered from 1 through N. If you pick a ticket randomly then you get positive integer n, with probability 1/N if n ≤ N and with probability zero if n > N. This can be written

${\displaystyle P(n|N)={\frac {[n\leq N]}{N}}}$

where the Iverson bracket [n ≤ N] is 1 when n ≤ N and 0 otherwise. When considered a function of n for fixed N this is the probability distribution, but when considered a function of N for fixed n this is a likelihood function. The maximum likelihood estimate for N is N0 = n (by contrast, the unbiased estimate is 2n − 1).

This likelihood function is not a probability distribution, because the total

${\displaystyle \sum _{N=1}^{\infty }P(n|N)=\sum _{N}{\frac {[N\geq n]}{N}}=\sum _{N=n}^{\infty }{\frac {1}{N}}}$

is a divergent series.

Suppose, however, that you pick two tickets rather than one.

The probability of the outcome {n1n2}, where n1 < n2, is

${\displaystyle P(\{n_{1},n_{2}\}|N)={\frac {[n_{2}\leq N]}{\binom {N}{2}}}.}$

When considered a function of N for fixed n2, this is a likelihood function. The maximum likelihood estimate for N is N0 = n2.

This time the total

${\displaystyle \sum _{N=1}^{\infty }P(\{n_{1},n_{2}\}|N)=\sum _{N}{\frac {[N\geq n_{2}]}{\binom {N}{2}}}={\frac {2}{n_{2}-1}}}$

is a convergent series, and so this likelihood function can be normalized into a probability distribution.

If you pick 3 or more tickets, the likelihood function has a well defined mean value, which is larger than the maximum likelihood estimate. If you pick 4 or more tickets, the likelihood function has a well defined standard deviation too.

Relative likelihood

Relative likelihood function

Suppose that the maximum likelihood estimate for θ is ${\displaystyle {\hat {\theta }}}$. Relative plausibilities of other θ values may be found by comparing the likelihood of those other values with the likelihood of ${\displaystyle {\hat {\theta }}}$. The relative likelihood of θ is defined[2][3] as ${\displaystyle {\mathcal {L}}(\theta |x)/{\mathcal {L}}({\hat {\theta }}|x).}$

A 10% likelihood region for θ is

${\displaystyle \{\theta :{\mathcal {L}}(\theta |x)/{\mathcal {L}}({\hat {\theta }}|x)\geq 0.10\},}$

and more generally, a p% likelihood region for θ is defined[2][3] to be

${\displaystyle \{\theta :{\mathcal {L}}(\theta |x)/{\mathcal {L}}({\hat {\theta }}|x)\geq p/100\}.}$

If θ is a single real parameter, a p% likelihood region will typically comprise an interval of real values. In that case, the region is called a likelihood interval.[2][3][4]

Likelihood intervals can be compared to confidence intervals. If θ is a single real parameter, then under certain conditions, a 14.7% likelihood interval for θ will be the same as a 95% confidence interval.[2] 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).[4]

The idea of basing an interval estimate on the relative likelihood goes back to Fisher in 1956 and has been used by many authors since then.[4] A likelihood interval can be used without claiming any particular coverage probability; as such, it differs from confidence intervals.

Relative likelihood of models

The definition of relative likelihood can be generalized to compare different statistical models. This generalization is based on AIC (Akaike information criterion), or sometimes AICc (Akaike Information Criterion with correction).

Suppose that, for some dataset, we have two statistical models, M1 and M2. Also suppose that AIC(M1) ≤ AIC(M2). Then the relative likelihood of M2 with respect to M1 is defined[5] to be

exp((AIC(M1)−AIC(M2))/2)

To see that this is a generalization of the earlier definition, suppose that we have some model M with a (possibly multivariate) parameter θ. Then for any θ, set M2 = M(θ), and also set M1 = M(${\displaystyle {\hat {\theta }}}$). The general definition now gives the same result as the earlier definition.

Likelihoods that eliminate nuisance parameters

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 being marginal, conditional and profile likelihoods.[6][7]

These approaches are useful because standard likelihood methods can become unreliable or fail entirely when there are many nuisance parameters or when the nuisance parameters are high-dimensional. This is particularly true when the nuisance parameters can be considered to be "missing data"; they represent a non-negligible fraction of the number of observations and this fraction does not decrease when the sample size increases. Often these approaches can be used to derive closed-form formulae for statistical tests when direct use of maximum likelihood requires iterative numerical methods. These approaches find application in some specialized topics such as sequential analysis.

Conditional likelihood

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.

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

{{#invoke:main|main}} 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.

Profile likelihood

It is often possible to write some parameters as functions of other parameters, thereby reducing the number of independent parameters. (The function is the parameter value which maximizes the likelihood given the value of the other parameters.) This procedure is called concentration of the parameters and results in the concentrated likelihood function, also occasionally known as the maximized likelihood function, but most often called the profile likelihood function.

For example, consider a regression analysis model with normally distributed errors. The most likely value of the error variance is the variance of the residuals. The residuals depend on all other parameters. Hence the variance parameter can be written as a function of the other parameters.

Unlike conditional and marginal likelihoods, profile likelihood methods can always be used, even when the profile likelihood cannot be written down explicitly. However, the profile likelihood is not a true likelihood, as it is not based directly on a probability distribution, and this leads to some less satisfactory properties. Attempts have been made to improve this, resulting in modified profile likelihood.

The idea of 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. In the case of parameter estimation in partially observed systems, the profile likelihood can be also used for identifiability analysis.[8] Results from profile likelihood analysis can be incorporated in uncertainty analysis of model predictions.[9]

Partial likelihood

A partial likelihood is a factor component of the likelihood function that isolates the parameters of interest.[10] It is a key component of the proportional hazards model.

Historical remarks

In English, "likelihood" has been distinguished as being related to, but weaker than, "probability" since its earliest uses. The comparison of hypotheses by evaluating likelihoods has been used for centuries, for example by John Milton in Aeropagitica (1644): "when greatest likelihoods are brought that such things are truly and really in those persons to whom they are ascribed".

In the Netherlands Christiaan Huygens used the concept of likelihood in his book "Van rekeningh in spelen van geluck" ("On Reasoning in Games of Chance") in 1657.

In Danish, "likelihood" was used by Thorvald N. Thiele in 1889.[11][12][13]

In English, "likelihood" appears in many writings by Charles Sanders Peirce, where model-based inference (usually abduction but sometimes including induction) is distinguished from statistical procedures based on objective randomization. Peirce's preference for randomization-based inference is discussed in "Illustrations of the Logic of Science" (1877–1878) and "A Theory of Probable Inference" (1883)".

"probabilities that are strictly objective and at the same time very great, although they can never be absolutely conclusive, ought nevertheless to influence our preference for one hypothesis over another; but slight probabilities, even if objective, are not worth consideration; and merely subjective likelihoods should be disregarded altogether. For they are merely expressions of our preconceived notions" (7.227 in his Collected Papers).

"But experience must be our chart in economical navigation; and experience shows that likelihoods are treacherous guides. Nothing has caused so much waste of time and means, in all sorts of researchers, as inquirers' becoming so wedded to certain likelihoods as to forget all the other factors of the economy of research; so that, unless it be very solidly grounded, likelihood is far better disregarded, or nearly so; and even when it seems solidly grounded, it should be proceeded upon with a cautious tread, with an eye to other considerations, and recollection of the disasters caused." (Essential Peirce, volume 2, pages 108–109)

Like Thiele, Peirce considers the likelihood for a binomial distribution. Peirce uses the logarithm of the odds-ratio throughout his career. Peirce's propensity for using the log odds is discussed by Stephen Stigler.{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

In Great Britain, "likelihood" was popularized in mathematical statistics by R.A. Fisher in 1922:[14] "On the mathematical foundations of theoretical statistics". In that paper, Fisher also uses the term "method of maximum likelihood". Fisher argues against inverse probability as a basis for statistical inferences, and instead proposes inferences based on likelihood functions. Fisher's use of "likelihood" fixed the terminology that is used by statisticians throughout the world.

Notes

1. Edwards, A.W.F. 1972. Likelihood. Cambridge University Press, Cambridge (expanded edition, 1992, Johns Hopkins University Press, Baltimore). ISBN 0-8018-4443-6
2. Kalbfleisch J.G. (1985) Probability and Statistical Inference, Springer (§9.3.)
3. Sprott D.A. (2000) Statistical Inference in Science, Springer (chap.2)
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5. Burnham K. P. & Anderson D.R. (2002), Model Selection and Multimodel Inference, §2.8 (Springer).
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11. Steffen L. Lauritzen, Aspects of T. N. Thiele’s Contributions to Statistics. Bulletin of the International Statistical Institute, 58, 27–30, 1999.
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