# Statistical And On-line Compression Modelling

In standard statistical modelling Reality is modelled as a family probability measures `⚠ $\{P_{\theta} \mid \theta\in\Theta\}$`

. In on-line compression modelling Reality is modelled as a 5-tuple whose key elements are the forward functions and backward kernels. With each on-line compression model `⚠ $M$`

we can associate the statistical model `⚠ $\phi(M)$`

defined as the extreme points of the probability measures on `⚠ $\mathbf{Z}^{\infty}$`

(where `⚠ $\mathbf{Z}$`

is the example space) that agree with the on-line compression model. Natural questions (some rather vague) are:

- What are the statistical models that can be obtained in this way (are of the form
`⚠ $\phi(M)$`

for some`⚠ $M$`

)? - Characterize the on-line compression models
`⚠ $M$`

for which there is no "loss of information" in replacing`⚠ $M$`

by`⚠ $\phi(M)$`

. - Does
`⚠ $\phi$`

establish a bijection between some wide and natural classes of on-line compression and statistical models?

Some work in this direction has been done by Martin-Lof, Lauritzen, and other authors for repetitive structures (models very closely related to on-line compression models). See Vovk et al. (2005), Section 8.8.

**Bibliography**

- Vladimir Vovk, Alexander Gammerman and Glenn Shafer (2005). Algorithmic learning in a random world. Springer, New York.