Arg max
From Wikipedia, the free encyclopedia
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:
In other words,
is the value of x for which f(x) has the largest value (for example, if f(x) is −|x|, then it attains its maximum value at x = 0).
This is well-defined only if the maximum is reached at a single value. Thus
holds if and only if x0 is the unique value of x for which f(x) is maximized. So, for example,
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;4\pi]} \cos(x) = \{0;2\pi;4\pi\}](../../../math/e/1/e/e1e80a01bdb66f81487197dd3dedc2bd.png)
since the maximum value of cos(x) is 1, which happens when x = 0, 2π or 4π.
arg min (or argmin) is defined analogously.
[edit] In LaTeX
LaTeX has no built-in \argmax command. Some people get around this by using \arg\max. 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.)
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.
