Likelihood function - Wikipedia, the free encyclopedia
That is, the likelihood function for B is the equivalence class of functions ... [edit] Likelihood function of a parameterized model ...
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The Likelihood Function
Likelihood Function. Statistical Power. Lotka-Volterra Competition. Lotka-Volterra Predation ... this by taking the likelihood function and finding its maximum ...
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Maximum likelihood - Wikipedia, the free encyclopedia
The first and second derivatives of the log-likelihood function must be defined. ... The likelihood function (defined below) takes one of three values: ...
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Maximum Likelihood Function
This article discusses maximum likelihood function. ... For complete data, the likelihood function is a product of the pdf functions, ...
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PlanetMath: likelihood function
The likelihood function is a map given by ... This is version 10 of likelihood function, born on 2004-07-08, modified 2006-09-23. ...
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Can the Likelihood Value be Greater Than 0?
... the set of values that maximize the likelihood function is used as the estimated... greater than 1, which causes the Ln-likelihood function of Eqn. ...
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The Binomial Likelihood Function
parameters, while the likelihood function gives the relative likelihoods for ... Note, too, that the log-likelihood function is in the negative quadrant because ...
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Tools - Estimation Methods - Marginal Maximum Likelihood - Details
... to test items, then the value of the likelihood function for individual i is: ... quadrature points (q q), one can re-write the individual likelihood function as: ...
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Maximum Likelihood Procedures
( beta hat)_K so as to maximize the likelihood function. ... Also, for most purposes it is easier to work with the log of the likelihood function. ...
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Maximum Likelihood
... the maximum likelihood problem is to maximize a function of several ... Therefore, assuming that the likelihood function is differentiable, we can find ...
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Likelihood as a solitary term is a shorthand for likelihood function. In non-technical usage, "likelihood" is a synonym for "probability", but throughout this article only the technical definition is used. Informally, if "probability" allows us to predict unknown outcomes based on known parameters, then "likelihood" allows us to determine unknown parameters based on known outcomes.

In a sense, likelihood works backwards from probability: given B, we use the conditional probability Pr(A]:

P(B \mid A) = \frac{P(A \mid B)\;P(B)}{P(A)}.\!

In statistics, a likelihood function is a conditional probability function (mathematics) considered as a function of its second argument with its first argument held fixed, thus:

b\mapsto P(A \mid B=b), \!

and also any other function proportional to such a function.That is, the likelihood function for B is the equivalence class of functions

L(b \mid A) = \alpha \; P(A \mid B=b) \!

for any constant of proportionality \alpha > 0.Thus the numerical value L(b | A) is immaterial; all that matters are ratios of the form

\frac{L(b_2 | A)}{L(b_1 | A)}, \!

since these are invariant with respect to the constant of proportionality.

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

Concentrated likelihood For a likelihood function of more than one parameter, it is sometimes possible to write some parameters as functions of other parameters, thereby reducing the number of independent parameters.(The function is the parameter value which maximises the likelihood given the value of the other parameters.)This procedure is called concentration of the parameters and results in the concentrated likelihood function.

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

Historical remarks Some early thoughts on likelihood were made in a book by Thorvald N. Thiele published in 1889Steffen L. Lauritzen, Aspects of T. N. Thiele's Contributions to Statistics (1999)..The first paper where the full idea of the "likelihood" appears was written by R.A. Fisher in 1922Ronald A. Fisher. "On the mathematical foundations of theoretical statistics". Philosophical Transactions of the Royal Society, A, 222:309-368 (1922). ("Likelihood" is discussed in section 6.): "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.

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

x\mapsto f(x\mid\theta), \!

where θ is the parameter (in the case of discrete distributions, the probability density functions are probability "mass" functions) the likelihood function is

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.

Note: 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 real-world consequences in medicine, engineering or jurisprudence. See prosecutor's fallacy for an example of this.

Example For example, if I toss a coin, with a probability pH of landing heads up ('H'), the probability of getting two heads in two trials ('HH') is pH2. If pH = 0.5, then the probability of seeing two heads is 0.25.

In symbols, we can say the above as

P(\mbox{HH} \mid p_H = 0.5) = 0.25

Another way of saying this is to reverse it and say that "the likelihood of pH = 0.5, given the observation 'HH', is 0.25", i.e.,

L(p_H=0.5 \mid \mbox{HH}) = P(\mbox{HH}\mid p_H=0.5) =0.25.

But this is not the same as saying that the probability of pH = 0.5, given the observation, is 0.25.

To take an extreme case, on this basis we can say "the likelihood of pH = 1 given the observation 'HH' is 1". But it is clearly not the case that the probability of pH = 1 given the observation is 1: the event 'HH' can occur for any pH > 0 (and often does, in reality, for pH roughly 0.5). If the probability of pH = 1 given the observation is 1, it means that pH must and can only be equal 1 for event 'HH' to occur which is obviously not true.

The likelihood function is not a probability density function – for example, the integral of a likelihood function is not in general 1. In this example, the integral of the likelihood density over the interval 1 in pH is 1/3, demonstrating again that the likelihood density function cannot be interpreted as a probability density function for pH. On the other hand, given any particular value of pH, e.g. pH = 0.5, the integral of the probability density function over the domain of the random variables is 1.

See also

• Bayes factor
• Bayesian inference
• conditional probability
• likelihood principle
• likelihood-ratio test
• maximum likelihood
• principle of maximum entropy
• score (statistics)

Notes

References

• A. W. F. Edwards (1972). Likelihood: An account of the statistical concept of likelihood and its application to scientific inference, Cambridge University Press. Reprinted in 1992, expanded edition, Johns Hopkins University Press.

Likelihood function - Wikipedia, the free encyclopedia
In statistics, the likelihood function (often simply the likelihood) is a function of the parameters of a statistical model that plays a key role in statistical inference.

Likelihood function in crystallography
Likelihood function in crystallography ... Comparison of maximum likelihood Up: No Title Previous: Experiment and model

Maximum likelihood - Wikipedia, the free encyclopedia
The likelihood function (defined below) takes one of three values: We see that the likelihood is maximized when p =2/3, and so this is our maximum likelihood estimate for p.

Strathprints - The asymptotic convexity of the negative likelihood ...
We prove the convexity of the negative likelihood function in the asymptotic sense for GARCH models. This property provides assurance for the convergence of numerical optimization ...

Likelihood Function -- from Wolfram MathWorld
A likelihood function L(a) is the probability or probability density for the occurrence of a sample configuration x_1, ..., x_n given that the probability density f(x;a) with ...

Application of the likelihood function in phylogenetic analysis.
Pathogen Evolution Virus and Pathogen Evolution Molecular Evolution Molecular Evolution and Phylogenetic Analyses Evolutionary Inference

The Likelihood Function
Site Index: Introduction. Density Independence. Density Dependence. Age-Structured Population Growth. Binomial Sampling. Likelihood Function

Likelihood Function Estimation - What does LIFE stand for? Acronyms ...
Acronym Definition; LIFE: Lifetime Television (cable network channel) LIFE: Losartan Intervention for Endpoint Reduction in Hypertension (drug trial) LIFE: Life and Health ...

PlanetMath: likelihood function
In other words, the likelikhood function is functionally the same in form as a probability density function. However, the emphasis is changed from the to the.

DEVISING THE LIKELIHOOD FUNCTION
devising the likelihood function ... prior phase probability distribution up: a likelihood function incorporating prior phase information previous: the method of maximum likelihood