# Talagrand's concentration inequality

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In probability theory, Talagrand's concentration inequality is an isoperimetric-type inequality for product probability spaces.[1][2] It was first proved by the French mathematician Michel Talagrand.[3] The inequality is one of the manifestations of the concentration of measure phenomenon.[2]

## Statement

The inequality states that if ${\displaystyle \Omega =\Omega _{1}\times \Omega _{2}\times \cdots \times \Omega _{n}}$ is a product space endowed with a product probability measure and ${\displaystyle A}$ is a subset in this space, then for any ${\displaystyle t\geq 0}$

${\displaystyle \Pr[A]\cdot \Pr \left[{\overline {A_{t}}}\right]\leq e^{-t^{2}/4}\,,}$

where ${\displaystyle {\overline {A_{t}}}}$ is the complement of At where this is defined by

${\displaystyle A_{t}=\{x\in \Omega ~:~\rho (A,x)\leq t\}}$

and where ${\displaystyle \rho }$ is Talagrand's convex distance defined as

${\displaystyle \rho (A,x)=\max _{\alpha ,\|\alpha \|_{2}\leq 1}\ \min _{y\in A}\ \sum _{i~:~x_{i}\neq y_{i}}\alpha _{i}}$
${\displaystyle \|\alpha \|_{2}=\left(\sum _{i}\alpha _{i}^{2}\right)^{1/2}}$

## Explanation

Probability deals with uncertainty. The inference of a future event or process requires the identification of key variables related to it. Certain assumptions about the nature of those variables (random variables) and their functional relationship can be drawn. Yet there are cases where the functional relationships of such variables are not directly available. But in spite of such difficulty Talagrand's concentration inequality helps us to describe some properties (in probabilistic terms) related to the variables an their functional representation. The authors [4] clearly describe what the Talagrand's concentration inequality implies under certain assumptions-"..if a function of many independent random variables does not depend too much on any of the variables then it is concentrated in the sense that with high probability, it is close to its expected value..."- and by such implication, we can for example (under many other possible applications), quantify the empirical risk of the model we are using to forecast some future event.

## References

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4. Boucheron, Stephanie;Lugosi,Gábor;jMassart,Pascal (2013).Concentration Inequalities: A Nonasymptotic Theory of Independence.OUP Oxford.ASIN: B00BPS6QQI.