November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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9.16 MAXIMUM LIKELIHOOD ESTIMATION
Consider estimating parameter θ where its prior distribution 
is not known and cannot be used in MAP estimation. For this condition, we can develop an estimator similar to the MAP estimator which does not require any information about the prior distribution.
Definition: Likelihood Function The likelihood function associated with N random variables 
is the conditional pdf
(9.257) 
The log-likelihood function is
(9.258) 
where 
usually is the natural logarithm.
The likelihood function is a function of the unknown parameter θ for specific outcomes of the samples 
. We are interested in the value of θ for which 
is maximum; it is the “most likely” value for particular outcomes. The log-likelihood function is usually preferred because the logarithm often simplifies ...
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