Maximum likelihood method

If measurements y have been performed, and p(y|x) is the normalized ( ) probability density of y as function of parameters x, then the parameters x can be estimated by maximizing the joint probability density for the m measurements y_j (assumed to be independent)

is called the likelihood function . L is a measure for the probability of observing the particular sample y at hand, given x. Maximizing L by varying x amounts to interpreting L as function of x, given the measurements y.

If p(y|x) is a normal distribution, and if its variance is independent of the parameters x, then the maximum-likelihood method is identical to the least squares method.

The general problem is often solved numerically by minimization of , (see Blobel84, Press95, Bishop95).