Scale invariance of incident size distributions in response to sizes of their causes

Incidents can be defined as low-probability, high-consequence events and lesser events of the same type. Lack of data on extremely large incidents makes it difficult to determine distributions of incident size that reflect such disasters, even though they represent the great majority of total losses. If the form of the incident size distribution can be determined, then predictive Bayesian methods can be used to assess incident risks from limited available information. Moreover, incident size distributions have generally been observed to have scale invariant, or power law, distributions over broad ranges. Scale invariance in the distributions of sizes of outcomes of complex dynamical systems has been explained based on mechanistic models of natural and built systems, such as models of self-organized criticality. In this article, scale invariance is shown to result also as the maximum Shannon entropy distribution of incident sizes arising as the product of arbitrary functions of cause sizes. Entropy is shown by simulation and derivation to be maximized as a result of dependence, diversity, abundance, and entropy of multiplicative cause sizes. The result represents an information-theoretic explanation of invariance, parallel to those of mechanistic models. For example, distributions of incident size resulting from 30 partially dependent causes are shown to be scale invariant over several orders of magnitude. Empirical validation of power law distributions of incident size is reviewed, and the Pareto (power law) distribution is validated against oil spill, hurricane, and insurance data. The applicability of the Pareto distribution, in particular, for assessment of total losses over a planning period is discussed. Results justify the use of an analytical, predictive Bayesian version of the Pareto distribution, derived previously, to assess incident risk from available data.