# 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 scalarvalued 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 loglikelihood 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: