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.