Hypothesis of linear regression

In statistics, the linear regression problem can be formalized precisely, although one seldom uses this formalization in most practical cases.

Given the mathematical formalization of the statistical regression problem, let $ThetasubseteqGamma$ be a set of coefficients. The hypothesis of the linear regression is:

$exists (eta^0,cdots,eta^p)in heta^{p+1}:$ $mathbb{E}(Y|X_1,cdots,X_p)=eta^0 + sum_{j=1}^p eta^j X_j$

and the metric used is:

$forall f,gin F, d(f,g) = mathbb{E} [(f-g)^2]$

We therefore want to minimize $mathbb{E} [(Y-f(X_1,cdots,X_p))^2]$ , which means that

$f(X_1,cdots,X_p)=mathbb{E}(Y|X_1,cdots,X_p) = eta^0 + sum_{j=1}^p eta^j X_j$

Hence, we only need to find $eta^0,cdots,eta^p$ .

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