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Encyclopedia > Arg max

In mathematics, arg max (or argmax) stands for the argument of the maximum, that is to say, the value of the given argument for which the value of the given expression attains its maximum value: Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... The factual accuracy of this article is disputed. ... The largest and the smallest element of a set are called extreme values, or extreme records. ...

$arg max_x f(x) in {x | forall y : (y neq x Longrightarrow f(y) < f(x))}$

This is well-defined only if the maximum is reached at a single value. Thus

$x_0 = arg max_x f(x)$

holds if and only if x0 is the unique value of x for which f(x) is maximized. So, for example, It has been suggested that this article or section be merged with Logical biconditional. ...

$arg max_{x in Bbb{R}} x(10-x) = 5$

since the maximum value of x(10 − x) is 25, which happens when x = 5.

However, in case the maximum is reached at many values, arg max can be extended to value a set of solutions.

Then, we have for example

$arg max_{x in [0;4pi]} cos(x) = {0;2pi;4pi}$

since the maximum value of cos(x) is 1, which happens when x = 0, 2π or 4π.

arg min (or argmin) is defined analogously.

LaTeX has no built-in argmax command. Some people get around this by using argmax. This is undesirable, because a subscripted variable will appear centered beneath the word "max", instead of centered beneath the whole word. (This can be seen in the examples above, since the typesetting capabilities of Wikipedia are limited.) The LaTeX logo, typeset with LaTeX , written as LaTeX in plain text, is a document preparation system for the (TeX) typesetting program. ...

One of following commands can be used to define an argmax command in LaTeX, respectively with or without a small space between arg and max:

` DeclareMathOperator*{argmax}{arg,max} DeclareMathOperator*{argmax}{argmax} `

Place one of these lines in the header of your LaTeX document. It can then be used as follows:

` hat{e} = argmax_{e} Pr(e | f) `

This gives a much nicer appearance. Note, remember to include the line `usepackage{amsmath}` in the preamble so that the command `DeclareMathOperator` is available.

Results from FactBites:

 Arg max - encyclopedia article about Arg max. (485 words) , arg max (or argmax) stands for the argument of the maximum, that is to say, the value of the given argument A parameter is a measurement or value on which something else depends. For example, a parametric equaliser is a tone control circuit that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. However, in case the maximum is reached at many values, arg max can be extended to value a set of solutions.
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