Hypothesis of linear regression

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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 $\Theta\subseteq\Gamma$ be a set of coefficients. The hypothesis of the linear regression is:

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

and the metric used is:

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

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

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

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