# Statistical And On-line Compression Modelling

In standard statistical modelling Reality is modelled as a family probability measures . 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 we can associate the statistical model defined as the extreme points of the probability measures on (where 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 for some )?
- Characterize the on-line compression models for which there is no "loss of information" in replacing by .
- Does 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.