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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9.18 EXPECTATION–MAXIMIZATION ALGORITHM
The expectation–maximization (EM) algorithm is a recursive technique for finding the ML estimate in situations where there are unobservable factors influencing the samples. In conventional ML estimation, the likelihood function 
or, equivalently, the log-likelhood function 
is used to estimate θ from 
. Suppose the samples are incomplete in the sense that there are other random variables correlated with {Xn}, but these are not directly measurable. Let those random variables be denoted by 
.
Consider the joint pdf 
for measurable X and unmeasurable Y. If Y is also measurable, then the ML estimator for θ would be the function of X and Y that maximizes the joint pdf. However, since only X is observable, it is possible to estimate θ by maximizing only 
, which can be expressed as
(9.273)
The influence of the nonmeasurable Y on the estimate is ignored in conventional ...
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