Algorithmic statistics: forty years later

Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of a "good model" is introduced, a natural question arises: it is possible that for some piece of data there is no good model? If yes, how often these bad ("non-stochastic") data appear "in real life"? Another, more technical motivation comes from algorithmic information theory. In this theory a notion of complexity of a finite object (=amount of information in this object) is introduced; it assigns to every object some number, called its algorithmic complexity (or Kolmogorov complexity). Algorithmic statistic provides a more fine-grained classification: for each finite object some curve is defined that characterizes its behavior. It turns out that several different definitions give (approximately) the same curve. In this survey we try to provide an exposition of the main results in the field (including full proofs for the most important ones), as well as some historical comments. We assume that the reader is familiar with the main notions of algorithmic information (Kolmogorov complexity) theory.Comment: Missing proofs adde

L'information algorithmique en physique : émergence, sophistication et localité quantique

Cette thèse explore des aspects du monde naturel par la lentille de l'information algorithmique. La notion de l'émergence, intuitivement reliée à tant de phénomènes naturels, se voit offrir une définition cadrée dans le domaine plus spécifique des statistiques algorithmiques. Capturant toutes deux l'organisation non triviale d'un objet, la sophistication et la profondeur logique sont relativisées à un objet auxiliaire puis remises en relation. Enfin, des modèles proposant une description locale des systèmes quantiques sont démontrés équivalents, ont leur coût de description quantifié et sont généralisés aux systèmes continus.This thesis explores aspects of the physical world through the lens of algorithmic information. The notion of emergence, intuitively linked to many natural phenomena, is offered a definition framed in the field of algorithmic statistics. Both capturing non-trivial organization of an object, sophistication and logical depth are compared once relativized to an auxiliary object. Finally, models proposing a local description of the quantum systems are shown equivalent, have their description cost quantified and are generalized to continuous systems