Likelihood

Likelihood[dist,{x1,x2,…}]

gives the likelihood function for observations x1, x2, … from the distribution dist.

Likelihood[proc,{{t1,x1},{t2,x2},…}]

gives the likelihood function for the observations xi at time ti from the process proc.

Likelihood[proc,{path1,path2,…}]

gives the likelihood function for observations from path1, path2, … from the process proc.

Details

  • The likelihood function Likelihood[dist,{x1,x2,…}] is given by , where is the probability density function at xi, PDF[dist,xi].
  • For a scalar‐valued process proc the likelihood function Likelihood[proc,{{t1,x1},{t2,x2},…}] is given by Likelihood[SliceDistribution[proc,{t1,t2,…}],{{x1,x2,…}}].
  • For a vector-valued process proc the likelihood function Likelihood[proc,{{t1,{x1,…,z1},{t2,{x2,…,z2}},…}] is given by Likelihood[SliceDistribution[proc,{t1,t2,…}],{{x1,…,z1,x2,…,z2,…}}].
  • The likelihood function for a collection of paths Likelihood[proc,{path1,path2,…}] is given by ∏iLikelihood[proc,pathi].

Examples

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Basic Examples  (4)

Get the likelihood function for a normal distribution:

Compute a likelihood for numeric data:

Plot likelihood contours as a function of and on a log scale:

Compute the likelihood for multivariate data:

Compute the likelihood for a process:

Scope  (12)

Univariate Parametric Distributions  (2)

Compute the likelihood for a continuous distribution:

Compute the likelihood for a discrete distribution:

Plot the likelihood, assuming is unknown:

Multivariate Parametric Distributions  (2)

Obtain the log‐likelihood for a continuous multivariate distribution with unknown parameters:

Visualize the log of the likelihood surface, assuming :

For a multivariate discrete distribution:

Derived Distributions  (5)

Compute the likelihood for a truncated standard normal:

Compute the likelihood for a constructed distribution:

Visualize the likelihood contours as a function of the lower bound and :

Compute the likelihood for a product distribution:

Obtain the result as a product of the independent componentwise likelihoods:

Compute the likelihood for a copula distribution:

Plot the likelihood as a function of the kernel parameter:

Compute the likelihood for a component mixture:

Random Processes  (3)

Compute the likelihood of a continuous parametric process:

Compute the likelihood of a scalar-valued discrete parametric process:

Plot the likelihood as a function of the process parameter:

Compute the likelihood of a scalar-valued time series process:

Compute the likelihood of a vector-valued time series process:

Applications  (2)

Solve for the Poisson maximum likelihood estimate in closed form:

Compute a maximum likelihood estimate directly:

Maximize using the log of the likelihood for numeric stability:

Label the optimal point on a plot of the likelihood function:

Properties & Relations  (4)

Likelihood is a product of PDF values for the data:

The log of Likelihood is LogLikelihood:

EstimatedDistribution estimates parameters by maximizing the likelihood:

FindDistributionParameters gives the parameter estimates as rules:

Visualize the likelihood function near the optimal value:

Likelihood of a process can be computed using its slice distribution:

Use the slice distribution:

For a vector-valued process:

Use the slice distribution:

Vectorize the path values for use in the LogLikelihood of the time slice distribution:

Possible Issues  (1)

Likelihood of a continuous parametric process may be undefined:

This is due to degenerate slice distribution at time 0:

Start at positive time:

Neat Examples  (2)

Visualize isosurfaces for an exponential power likelihood:

Visualize isosurfaces for a bivariate normal likelihood:

Wolfram Research (2010), Likelihood, Wolfram Language function, https://reference.wolfram.com/language/ref/Likelihood.html (updated 2014).

Text

Wolfram Research (2010), Likelihood, Wolfram Language function, https://reference.wolfram.com/language/ref/Likelihood.html (updated 2014).

CMS

Wolfram Language. 2010. "Likelihood." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/Likelihood.html.

APA

Wolfram Language. (2010). Likelihood. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Likelihood.html

BibTeX

@misc{reference.wolfram_2024_likelihood, author="Wolfram Research", title="{Likelihood}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/Likelihood.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_likelihood, organization={Wolfram Research}, title={Likelihood}, year={2014}, url={https://reference.wolfram.com/language/ref/Likelihood.html}, note=[Accessed: 21-December-2024 ]}