# Probabilistic Inference for Complex Systems

### Satellite Session

2016 Conference on Complex Systems

- Satellite Session, Tuesday Morning September 20, 2016
- Conference on Complex Systems, Amsterdam, Holland, September 19-22, 2016
- Kenric P. Nelson and Mark A. Kon, Boston University, Boston, MA

Correspondence: kenricpn at bu dot edu, 781-645-8564

**Abstracts**

*Three practical entropy concepts for complex processes*

*Rudolf Hanel*

It is well known that the concept of entropy does not possess a unique generalization to non-equilibrium processes. This differentiation of the entropy concept however need not be a draw-back. To describe the topology of complex networks one requires more measures than to describe regular lattices. Similarly we require more measures for describing path-dependent non-equilibrium processes than we would for describing systems or processes in equilibrium. So far we have identified three distinct entropy related concepts that have a practical meaning in the context of complex path-dependent processes; entropy production, the extensive entropy, and the maximum entropy functional or generalized divergence. The first telling us how compressible a sequence is, the second, how the phase-space of the process grows, and the third, for performing statistical inference, providing a starting point for the information theory of complex systems. In equilibrium systems those three measures yield identical results, while in path-dependent systems the concepts are distinct. In this talk we will briefly introduce and discuss these three concepts.

*Estimation of generalized entropies and similarity measures*

*Francesc Font-Clos, Martin Gerlach, Eduardo Altmann*

Estimating entropy measures from finite-size samples is a problem of great interest in several fields. We will show that when the underlying distributions are heavy tailed, even very large sample sizes lead to considerable biases in the estimations. This poses a problem for studies based on estimations of entropy-based distances between symbolic sequences. As a solution, we will propose a family of similarity measures based on the generalized entropy of order alpha, for which the bias in the estimations decays much faster. If time allows, I will discuss some applications to the study of the evolution of the English vocabulary over time.

*A unifying perspective on probability scoring rules and calibration curves*

*Kenric Nelson*

The emergence of machine learning capabilities for deriving patterns from data has highlighted the need to effectively manage and assess probabilistic inferences. A variety of methods are used to evaluate the performance of probabilistic inferences or forecasts. Examples include calibration curves, scoring rules, and when applied to decisions such metrics as receiver operating characteristic. Despite several decades of research in this area, a consensus of appropriate metrics is still allusive. A perspective on the issue is provided here by showing that local scoring rules can be translated back to the probability domain and thereby represent mean forecasts. This clarifies the interpretation of the scoring rule and enables the performance metric to be visualized in conjunction with probability calibration curves. The approach draws upon generalizations of information theory developed from the study of complex nonlinear systems.

*Networks of influence: transmission of information in systems of cooperative decision makers*

*Malgorzata Turalska, Bruce J. West*

In a society interconnected by family ties, friendships, acquaintances or work relations it is unavoidable that a person’s behaviors or decisions depend on the choices made by other people. The surrounding social network influences the opinions we hold, the products we buy and the activities we pursue. In this context we study a decision making model (DMM) incorporating imitation as a sole mean of opinion and information sharing across the social network. We investigate how such a system of stochastic two-state agents, communicating through local interactions, is topologically complex and is manifesting temporal complexity through an inverse power-law probability distribution function in the switching times between the two critical states of consensus. We demonstrate how the dynamics of a single network element can be tied through fractional master equation to the behavior of the entire system and we discuss possible application of this approach to control of complex systems.