# Gauss Linear Model

The Gauss statistical model says that the `⚠ $(x_n,y_n)\in\mathbb{R}^p\times\mathbb{R}$`

are generated as follows:

- there are no restrictions on the way
`⚠ $x_n$`

are generated; - given
`⚠ $x_n$`

, the`⚠ $y_n$`

are generated from`⚠ $y_n = w\cdot x_n + \xi_n$`

, where`⚠ $w\in\mathbb{R}^p$`

is a vector of parameters,`⚠ $\xi_n$`

is distributed as`⚠ $N(0,\sigma^2)$`

, and`⚠ $\sigma>0$`

is another parameter.

See Section 8.5 of Vovk et al. (2005) and Vovk et al. (2009) for the formulation of this model as an on-line compression model.

The most basic version of this model is where there are no `⚠ $x$`

s, and the model is `⚠ $y_n\sim N(0,\sigma^2)$`

. The summary of `⚠ $y_1,\ldots,y_n$`

is `⚠ $t_n:=y_1^2+\cdots+y_n^2$`

and the Gauss repetitive structure postulates that the distribution of `⚠ $y_1,\ldots,y_n$`

is uniform on the sphere of radius `⚠ $t_n^{1/2}$`

. Borel (1914) noticed that the Gauss statistical model (used by Maxwell as a model in statistical physics) is equivalent to the Gauss repetitive structure (used for a similar purpose by Gibbs). For further historical comments, see Vovk et al. (2005), Section 8.8, and Diaconis and Freedman (1987), Section 6.

**Bibliography**

- Persi Diaconis and David Freedman (1987). A dozen de Finetti-style results in search of a theory.
*Annales de l'Institut Henri Poincare B*23:397-423. - Vladimir Vovk, Alexander Gammerman and Glenn Shafer (2005). Algorithmic learning in a random world. Springer, New York.
- Vladimir Vovk, Ilia Nouretdinov, and Alexander Gammerman (2009). On-line predictive linear regression.
*Annals of Statistics*37:1566-1590.