Criticality of mostly informative samples: A Bayesian model selection approach

Abstract

We discuss a Bayesian model selection approach to high-dimensional data in the deep under-sampling regime. The data is based on a representation of the possible discrete states s, as defined by the observer, and it consists of M observations of the state. This approach shows that, for a given sample size M, not all states observed in the sample can be distinguished. Rather, only a partition of the sampled states s can be resolved. Such a partition defines an emergent classification qs of the states that becomes finer and finer as the sample size increases, through a process of symmetry breaking between states. This allows us to distinguish between the resolution of a given representation of the observer defined states s, which is given by the entropy of s, and its relevance, which is defined by the entropy of the partition qs. Relevance has a nonmonotonic dependence on resolution, for a given sample size. In addition, we characterise most relevant samples and we show that they exhibit power law frequency distributions, generally taken as signatures of criticality. This suggests that criticality reflects the relevance of a given representation of the states of a complex system, and does not necessarily require a specific mechanism of self-organisation to a critical point.