Background
Suppose that there is an underlying signal {x(t)}, of which an observed signal {r(t)} is available. The observed signal r is related to x via a transformation that may be nonlinear and may involve attenuation, and would usually involve the incorporation of random noise. The statistical parameters of this transformation are assumed known. The problem to be solved is to use the observations {r(t)} to create a good estimate of {x(t)}.
Maximum likelihood sequence estimation is formally the application of maximum likelihood to this problem. That is, the estimate of {x(t)} is defined to be sequence of values which maximize the functional
where p(r|x) denotes the conditional joint probability density function of the observed series {r(t)} given that the underlying series has the values {x(t)}.
In contrast, the related method of maximum a posteriori estimation is formally the application of the Maximum a posteriori (MAP) estimation approach. This is more complex than maximum likelihood sequence estimation and requires a known distribution (in Bayesian terms, a prior distribution) for the underlying signal. In this case the estimate of {x(t)} is defined to be sequence of values which maximize the functional
where p(x|r) denotes the conditional joint probability density function of the underlying series {x(t)} given that the observed series has taken the values {r(t)}. Bayes' theorem implies that
In cases where the contribution of random noise is additive and has a multivariate normal distribution, the problem of maximum likelihood sequence estimation can be reduced to that of a least squares minimization.
Read more about this topic: Maximum Likelihood Sequence Estimation
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