# Relative likelihood

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Statistical model tool

In [statistics](/source/Statistics), when [selecting](/source/Model_selection) a [statistical model](/source/Statistical_model) for given data, the **relative likelihood** compares the relative plausibilities of different candidate models or of different values of a [parameter](/source/Statistical_parameter) of a single model.

## Relative likelihood of parameter values

Assume that we are given some data x for which we have a statistical model with parameter θ. Suppose that the [maximum likelihood estimate](/source/Maximum_likelihood_estimate) for θ is θ ^ {\displaystyle {\hat {\theta }}} . Relative 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[1][2][3][4][5] L ( θ ∣ x ) L ( θ ^ ∣ x ) , {\displaystyle {\frac {~{\mathcal {L}}(\theta \mid x)~}{~{\mathcal {L}}({\hat {\theta }}\mid x)~}},} where L ( θ ∣ x ) {\displaystyle {\mathcal {L}}(\theta \mid x)} denotes the [likelihood function](/source/Likelihood_function). Thus, the relative likelihood is the [likelihood ratio](/source/Likelihood_ratio_test) with fixed denominator L ( θ ^ ∣ x ) {\displaystyle {\mathcal {L}}({\hat {\theta }}\mid x)} .

The function θ ↦ L ( θ ∣ x ) L ( θ ^ ∣ x ) {\displaystyle \theta \mapsto {\frac {~{\mathcal {L}}(\theta \mid x)~}{~{\mathcal {L}}({\hat {\theta }}\mid x)~}}} is the *relative likelihood function*.

### 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.[1][3][6]

{ θ : L ( θ ∣ x ) L ( θ ^ ∣ x ) ≥ p 100 } . {\displaystyle \left\{\theta :{\frac {{\mathcal {L}}(\theta \mid x)}{{\mathcal {L}}({\hat {\theta \,}}\mid x)}}\geq {\frac {p}{100}}\right\}.}

If θ is a single real parameter, a p% likelihood region will usually comprise an [interval](/source/Interval_(mathematics)) of real values. If the region does comprise an interval, then it is called a *likelihood interval*.[1][3][7]

Likelihood intervals, and more generally likelihood regions, are used for [interval estimation](/source/Interval_estimation) within likelihood-based statistics ("likelihoodist" statistics): They are similar to [confidence intervals](/source/Confidence_interval) in frequentist statistics and [credible intervals](/source/Credible_interval) in Bayesian statistics. Likelihood intervals are interpreted directly in terms of relative likelihood, not in terms of [coverage probability](/source/Coverage_probability) (frequentism) or [posterior probability](/source/Posterior_probability) (Bayesianism).

Given a model, likelihood intervals can be compared to confidence intervals. If θ is a single real parameter, then under certain conditions, a 14.65% likelihood interval (about 1:7 likelihood) for θ will be the same as a 95% confidence interval (19/20 coverage probability).[1][6] In a slightly different formulation suited to the use of log-likelihoods (see [Wilks' theorem](/source/Likelihood-ratio_test#Asymptotic_distribution:_Wilks'_theorem)), the [test statistic](/source/Test_statistic) is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a [chi-squared distribution](/source/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).[6][7]

## Relative likelihood of models

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

Suppose that for some given data we have two statistical models, *M*1 and *M*2. Also suppose that AIC(*M*1) ≤ AIC(*M*2). Then the *relative likelihood* of *M*2 with respect to *M*1 is defined as follows.[8]

exp ⁡ ( AIC ⁡ ( M 1 ) − AIC ⁡ ( M 2 ) 2 ) {\displaystyle \exp \left({\frac {\operatorname {AIC} (M_{1})-\operatorname {AIC} (M_{2})}{2}}\right)}

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 *M*2 = *M*(θ), and also set *M*1 = *M*( θ ^ {\displaystyle {\hat {\theta }}} ). The general definition now gives the same result as the earlier definition.

## See also

- [Statistical model selection](/source/Statistical_model_selection)

- [Statistical model specification](/source/Statistical_model_specification)

- [Statistical model validation](/source/Statistical_model_validation)

## Notes

1. ^ [***a***](#cite_ref-Kalbfleisch_1-0) [***b***](#cite_ref-Kalbfleisch_1-1) [***c***](#cite_ref-Kalbfleisch_1-2) [***d***](#cite_ref-Kalbfleisch_1-3) [Kalbfleisch, J.G.](/source/James_G._Kalbfleisch) (1985), *Probability and Statistical Inference*, Springer, §9.3

1. **[^](#cite_ref-2)** [Azzalini, A.](/source/Adelchi_Azzalini) (1996), [*Statistical Inference — Based on the likelihood*](https://books.google.com/books?id=hyN6gXHvSo0C), [Chapman & Hall](/source/Chapman_%26_Hall), §1.4.2, [ISBN](/source/ISBN_(identifier)) [9780412606502](https://en.wikipedia.org/wiki/Special:BookSources/9780412606502)

1. ^ [***a***](#cite_ref-Sprott_3-0) [***b***](#cite_ref-Sprott_3-1) [***c***](#cite_ref-Sprott_3-2) Sprott, D.A. (2000), *Statistical Inference in Science*, Springer, chap. 2

1. **[^](#cite_ref-4)** Davison, A.C. (2008), *Statistical Models*, [Cambridge University Press](/source/Cambridge_University_Press), §4.1.2

1. **[^](#cite_ref-5)** Held, L.; Sabanés Bové, D.S. (2014), *Applied Statistical Inference — Likelihood and Bayes*, Springer, §2.1

1. ^ [***a***](#cite_ref-Rossi2018_6-0) [***b***](#cite_ref-Rossi2018_6-1) [***c***](#cite_ref-Rossi2018_6-2) Rossi, R.J. (2018), *Mathematical Statistics*, [Wiley](/source/Wiley_(publisher)), p. 267

1. ^ [***a***](#cite_ref-Hudson_7-0) [***b***](#cite_ref-Hudson_7-1) Hudson, D.J. (1971), "Interval estimation from the likelihood function", *[Journal of the Royal Statistical Society, Series B](/source/Journal_of_the_Royal_Statistical_Society%2C_Series_B)*, **33** (2): 256–262, [doi](/source/Doi_(identifier)):[10.1111/j.2517-6161.1971.tb00877.x](https://doi.org/10.1111%2Fj.2517-6161.1971.tb00877.x)

1. **[^](#cite_ref-8)** Burnham, K. P.; Anderson, D. R. (2002), *Model Selection and Multimodel Inference: A practical information-theoretic approach*, Springer, §2.8

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