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Encyclopedia > Maximum

The largest and the smallest element of a set are called extreme values, or extreme records.

For a differentiable function f, if f(x0) is an extreme value for the set of all values f(x), and if x0 is in the interior of the domain of f, then (x0,f(x0)) is a stationary point.

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In the case of a general partial order one should not confuse a least element (smaller than all other) and a minimal element (nothing is smaller). Likewise, a greatest element of a poset is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if mb (for any b in A) then m = b.

Any least element or greatest element of a poset will be unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable.

In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element and the maximal element will also be the greatest element.

If a chain is finite then it will always have a maximum (maximal element, greatest element) and a minimum (minimal element, least element). If a chain is infinite then it need not have a maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum.

If an infinite chain S is bounded, then the closure Cl(S) of the set will have a minimum and a maximum, which are the greatest lower bound and the least upper bound of the set S, and which either belong to S or are accumulation points of S.

Compare: extreme point. Results from FactBites:

 Maximum Entropy (932 words) In ``A maximum entropy approach to natural language processing'' (Computational Linguistics 22:1, March 1996), the appendix describes an approach to computing the gain of a single feature f. This link is to the Maximum Entropy Modeling Toolkit, for parameter estimation and prediction for maximum entropy models in discrete domains. We present a maximum-likelihood approach for automatically constructing maximum entropy models and describe how to implement this approach efficiently, using as examples several problems in natural language processing.
 Maximum likelihood - Wikipedia, the free encyclopedia (1244 words) Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution of a given data set. Maximum likelihood estimators achieve minimum variance (as given by the Cramer-Rao lower bound) in the limit as the sample size tends to infinity. Maximum likelihood estimation is related to generalized method of moments.
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