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.

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