LogLikelihood

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`LogLikelihood`

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LogLikelihood[dist,{x₁,x₂,…}]

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

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LogLikelihood[proc,{{t₁,x₁},{t₂,x₂},…}]

gives the log-likelihood function for the observations x_i at time t_i from the process proc.

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LogLikelihood[proc,{path₁,path₂,…}]

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

Details

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

Examples

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Basic Examples (4)Summary of the most common use cases

Get the log‐likelihood function for a normal distribution:

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-od6gxj

Out[2]=2

Compute a log‐likelihood for numeric data:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-0cu12

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-dyiizg

Out[2]=2

Plot log‐likelihood contours as a function of and :

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-d57rdp

Out[3]=3

Compute the log‐likelihood for multivariate data:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-ekvidp

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-gf9yr4

Out[2]=2

Compute the log-likelihood for a process:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-qpn81m

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-8ozx37

Out[2]=2

Scope (12)Survey of the scope of standard use cases

Univariate Parametric Distributions (2)

Compute the log‐likelihood for a continuous distribution:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-eu39i7

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-j3wpsc

Out[2]=2

Compute the log‐likelihood for a discrete distribution:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-fgnof8

Out[1]=1

Plot the log‐likelihood, assuming is unknown:

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-cjpa2p

Out[2]=2

Multivariate Parametric Distributions (2)

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

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-ego0yg

In[14]:=14

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-c6uc1g

Out[14]=14

Visualize the log‐likelihood surface, assuming :

In[15]:=15

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-evkacj

Out[15]=15

For a multivariate discrete distribution with known parameters:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-5gnmf

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-bdihyc

Out[2]=2

Derived Distributions (5)

Compute the log‐likelihood for a truncated standard normal:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-cnhaze

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-bl5hmb

Out[3]=3

Plot the log‐likelihood contours as a function of the truncation points:

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-lgny14

Out[4]=4

Compute the log‐likelihood for a constructed distribution:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-xjacs

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-d6iz7v

Out[2]=2

Compute the log‐likelihood for a product distribution:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-siutcl

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-ove1pe

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-iqkewk

Out[3]=3

Obtain the result as a sum of the independent componentwise log‐likelihoods:

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-djj9ap

Out[4]=4

Compute the log‐likelihood for a copula distribution:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-bcoybi

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-kajyvr

Plot the log‐likelihood as a function of the kernel parameter:

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-cirhpa

Out[3]=3

Compute the log‐likelihood for a component mixture:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-wcap1b

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-g5qjhs

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-dsbx4n

Out[3]=3

Random Processes (3)

Compute the log-likelihood of a continuous parametric process:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-cs75ib

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-p0tbtg

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-mqlbiq

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-eld3yv

Out[2]=2

Plot the log‐likelihood as a function of the process parameter:

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-52xkly

Out[3]=3

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

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-zh9md3

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-ymrsh8

Out[2]=2

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

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-k36by3

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-h5mb9e

Out[4]=4

Applications (4)Sample problems that can be solved with this function

Visualize the log‐likelihood surface for a distribution of two parameters:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-hh5i95

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-m7xxjk

Out[2]=2

Visualize as contours of equal log‐likelihood:

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-e00bj2

Out[2]=2

Show log-likelihood functions with mixed continuous and discrete parameters:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-bvhwer

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-7k5cjp

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-j5amgy

Out[3]=3

Solve for the Poisson maximum log-likelihood estimate in closed form:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-j96ax1

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-i1myd3

Out[2]=2

Compute a maximum log-likelihood estimate directly:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-crbq1v

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-exyi3y

Maximize:

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-e0bzeb

Out[3]=3

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

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-lepzyu

Out[4]=4

Estimate the variance of the MLE estimator as the reciprocal of the expectation of second derivative of the log-likelihood function with respect to parameters:

In[5]:=5

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-ncqygf

Out[5]=5

Properties & Relations (5)Properties of the function, and connections to other functions

LogLikelihood is the sum of logs of PDF values for data:

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-fs61wa

In[6]:=6

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-hiudw6

Out[6]=6

In[7]:=7

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-5kxsv

Out[7]=7

LogLikelihood is the log of Likelihood:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-bnand1

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-jyhke

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-q3x8b

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-fmqao1

Out[4]=4

EstimatedDistribution estimates parameters by maximizing the log‐likelihood:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-cvbqgt

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-i7a7vi

Out[2]=2

FindDistributionParameters gives the parameter estimates as rules:

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-pyzmqq

Out[3]=3

Visualize the log‐likelihood function near the optimal value:

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-kudrs0

Out[4]=4

Log-likelihood of a process can be computed using its slice distribution:

In[1]:=1

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-sdxem7

In[2]:=2

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-bb7exr

Out[2]=2

Use the slice distribution:

In[3]:=3

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-ydisrd

In[4]:=4

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-c3qtc7

Out[4]=4

For a vector-valued process:

In[5]:=5

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-pw7f0w

In[6]:=6

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-1exu07

Out[6]=6

Use the slice distribution:

In[7]:=7

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-n1azj1

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

In[8]:=8

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https://wolfram.com/xid/0hyxiu9ag5ft2nmp-gk1vja

Out[8]=8

Given a sample from a distribution , the difference between values of LogLikelihood for  and the MLE estimated distribution doubled is random and follows ChiSquareDistribution with degrees of freedom equal to the number of distribution parameters:

In[1]:=1