Briefly: the linear regression model. We suppose we can explain or predict **y** using a vector of variables **x**. As in Gauß’ estimation theory, **y** is supposed to be unobservable, and thus has to be estimated. The assumption that **y** depends on **x** is expressed this way: the posterior distribution **Prob{ Y | X }** is different from the prior distribution **Prob{ Y }**.

The minimization of variance of the difference between [our estimation of **Y** given **X**] and [**Y**] leads to a unique solution: the conditional expectation.

The linear hypothesis says that the estimated value should be an affine expression of **X**. Moreover, the affine parameters which minimise the variance of the error are given by:

The above linear model coincides with the optimal conditional expectation model when **X,Y** are Gaussian.

Michel Grabisch, in *Modeling Data by the Choquet Integral*

(liberally edited)

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