Function related to statistics and probability theory
Part of a series on
Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
Background
Bayesian inference
Bayesian probability
Bayes' theorem
Bernstein–von Mises theorem
Coherence
Cox's theorem
Cromwell's rule
Principle of indifference
Principle of maximum entropy
Model building
Weak prior ... Strong prior
Conjugate prior
Linear regression
Empirical Bayes
Hierarchical model
Posterior approximation
Markov chain Monte Carlo
Laplace's approximation
Integrated nested Laplace approximations
Variational inference
Approximate Bayesian computation
Estimators
Bayesian estimator
Credible interval
Maximum a posteriori estimation
Evidence approximation
Evidence lower bound
Nested sampling
Model evaluation
Bayes factor
Model averaging
Posterior predictive
Mathematics portal
v
t
e
The likelihood function (often simply called the likelihood) is the joint probability mass (or probability density) of observed data viewed as a function of the parameters of a statistical model.[1][2][3] Intuitively, the likelihood function is the probability of observing data assuming is the actual parameter.
In maximum likelihood estimation, the arg max (over the parameter ) of the likelihood function serves as a point estimate for , while the Fisher information (often approximated by the likelihood's Hessian matrix) indicates the estimate's precision.
In contrast, in Bayesian statistics, parameter estimates are derived from the converse of the likelihood, the so-called posterior probability, which is calculated via Bayes' rule.[4]
^Casella, George; Berger, Roger L. (2002). Statistical Inference (2nd ed.). Duxbury. p. 290. ISBN 0-534-24312-6.
^Wakefield, Jon (2013). Frequentist and Bayesian Regression Methods (1st ed.). Springer. p. 36. ISBN 978-1-4419-0925-1.
^Lehmann, Erich L.; Casella, George (1998). Theory of Point Estimation (2nd ed.). Springer. p. 444. ISBN 0-387-98502-6.
^Zellner, Arnold (1971). An Introduction to Bayesian Inference in Econometrics. New York: Wiley. pp. 13–14. ISBN 0-471-98165-6.
and 25 Related for: Likelihood function information
likelihoodfunction (often simply called the likelihood) is the joint probability mass (or probability density) of observed data viewed as a function...
distribution, given some observed data. This is achieved by maximizing a likelihoodfunction so that, under the assumed statistical model, the observed data is...
measure of goodness-of-fit is the likelihoodfunction L, or its logarithm, the log-likelihood ℓ. The likelihoodfunction L is analogous to the ε 2 {\displaystyle...
In Bayesian probability theory, if, given a likelihoodfunction p ( x ∣ θ ) {\displaystyle p(x\mid \theta )} , the posterior distribution p ( θ ∣ x )...
is contained in the likelihoodfunction. A likelihoodfunction arises from a probability density function considered as a function of its distributional...
distribution resulting from applying Bayes theorem to a binomial likelihoodfunction and a prior probability, the interpretation of the addition of both...
A marginal likelihood is a likelihoodfunction that has been integrated over the parameter space. In Bayesian statistics, it represents the probability...
known, the log likelihood of an observed vector x {\displaystyle {\boldsymbol {x}}} is simply the log of the probability density function: ln L ( x )...
probability density function Kernel density estimation – EstimatorPages displaying short descriptions with no spaces Likelihoodfunction – Function related to...
goodness of fit (as assessed by the likelihoodfunction), but it also includes a penalty that is an increasing function of the number of estimated parameters...
constraints on statistical parameters based on the gradient of the likelihoodfunction—known as the score—evaluated at the hypothesized parameter value...
{\mathcal {L}}(\theta \mid x)} denotes the likelihoodfunction. Thus, the relative likelihood is the likelihood ratio with fixed denominator L ( θ ^ ∣ x...
approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihoodfunction: ln L ( μ , σ 2 ) = ∑ i = 1 n ln ...
tobit likelihoodfunction is thus a mixture of densities and cumulative distribution functions. Below are the likelihood and log likelihoodfunctions for...
In statistics, Whittle likelihood is an approximation to the likelihoodfunction of a stationary Gaussian time series. It is named after the mathematician...
extension of maximum likelihood using regularization of the weights to prevent pathological solutions (usually a squared regularizing function, which is equivalent...
lower BIC are generally preferred. It is based, in part, on the likelihoodfunction and it is closely related to the Akaike information criterion (AIC)...
(statistics), the derivative of the log-likelihoodfunction with respect to the parameter In positional voting, a function mapping the rank of a candidate to...
p ( θ | X ) {\displaystyle p(\theta |X)} . It contrasts with the likelihoodfunction, which is the probability of the evidence given the parameters: p...
engineering. In statistics (when used as a likelihoodfunction) it is known as a tobit model. This function has numerous applications in mathematics and...
respect to θ {\displaystyle \theta } of the natural logarithm of the likelihoodfunction is called the score. Under certain regularity conditions, if θ {\displaystyle...
maximum of the likelihoodfunction occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the...
}}} that was found as the maximizing argument of the unconstrained likelihoodfunction is compared with a hypothesized value θ 0 {\displaystyle \theta _{0}}...
maximum likelihood fit of all the information, but instead uses a likelihoodfunction calculated from a transformed set of data, so that nuisance parameters...
Global Information
