Then the function
considered as a function great site
{\displaystyle \theta }
, is the likelihood function (of
{\displaystyle \theta }
, given the outcome
x
{\displaystyle x}
of
X
{\displaystyle X}
).
More examples of how to derive log-likelihood functions can be found in the
lectures on:
maximum
likelihood (ML) estimation of the parameter of the Poisson distribution
ML estimation of
the parameter of the exponential distribution
ML
estimation of the parameters of a normal linear regression model
The log-likelihood and its properties are discussed in a more detailed manner
in the lecture on
maximum likelihood
estimation. 47 His use of the term “likelihood” fixed the meaning of the term within mathematical statistics. 01, 0. Sometimes the probability of “the value
x
{\displaystyle x}
of
X
{\displaystyle i was reading this
for the parameter value
{\displaystyle \theta }
” is written as P(X = x | θ) or P(X = x; θ). 15, 0.
Dear : You’re Not Probability Density Functions
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The log-likelihood is, as the term suggests, the natural logarithm of the
likelihood.
The above can be extended in a simple way to allow consideration of distributions which contain both discrete and continuous components.
Since graphically the procedure of concentration is equivalent to slicing the likelihood surface along the ridge of values of the nuisance parameter
2
{\displaystyle \beta _{2}}
that maximizes the likelihood function, creating an isometric profile of the likelihood function for a given
1
{\displaystyle \beta _{1}}
, the result of this procedure is also known as profile likelihood. .