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In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model , the observed data is most probable.
The general algorithm involves message passing and is substantially similar to the belief propagation algorithm (which is the generalization of the forward-backward algorithm). With an algorithm called iterative Viterbi decoding , one can find the subsequence of an observation that matches best (on average) to a given hidden Markov model.
It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior density over the quantity one wants to estimate. MAP estimation is therefore a regularization of maximum likelihood estimation, so is not a well-defined statistic of the Bayesian posterior ...
For example, a maximum-likelihood estimate is the point where the derivative of the likelihood function with respect to the parameter is zero; thus, a maximum-likelihood estimator is a critical point of the score function. [8]
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.
Assume that the conditional distribution of X, given that the label Y takes the value r is given by (=) =,, …, where "" means "is distributed as", and where denotes a probability distribution. A classifier is a rule that assigns to an observation X = x a guess or estimate of what the unobserved label Y = r actually was.
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).
The LOCAL algorithm is an improvement of the GLOBAL algorithm presented in Mau, Newton and Larget (1999) [14] in which all branch lengths are changed in every cycle. The LOCAL algorithms modifies the tree by selecting an internal branch of the tree at random. The nodes at the ends of this branch are each connected to two other branches.