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Likelihood Function
The likelihood function
is defined as the
probability density function of
given
, evaluated at a particular
, with
regarded as a variable.
In other words, the likelihood function
is just the PDF
of
with a particular value of
plugged in, and any parameters
in the PDF (mean and variance in this case) are treated as variables.
For the sinusoidal parameter estimation problem, given a set of
observed data samples
, for
, the likelihood
function is

(6.50) 
and the log likelihood function is

(6.51) 
We see that the maximum likelihood estimate for the parameters of a
sinusoid in Gaussian white noise is the same as the least
squares estimate. That is, given
, we must find parameters
,
, and
which minimize

(6.52) 
as we saw before in (5.33).
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