## Maximum Likelihood

The procedure of finding the value of one or more parameters for a given statistic which makes the known Likelihood distribution a Maximum. The maximum likelihood estimate for a parameter is denoted .

For a Bernoulli Distribution, (1)

so maximum likelihood occurs for . If is not known ahead of time, the likelihood function is       (2)

where or 1, and , ..., . (3) (4) (5) (6)

For a Gaussian Distribution, (7) (8) (9)

gives (10) (11)

gives (12)

Note that in this case, the maximum likelihood Standard Deviation is the sample Standard Deviation, which is a Biased Estimator for the population Standard Deviation.

For a weighted Gaussian Distribution, (13) (14) (15)

gives (16)

The Variance of the Mean is then (17)

But (18)

so     (19)

For a Poisson Distribution, (20) (21) (22) (23)

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Least Squares as a Maximum Likelihood Estimator.'' §15.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 651-655, 1992.