Foundations of Probability for Data Analytics of Decision Making in Complex Systems
Abstract
The field of data analytics has led to an explosion of methods and algorithms designed to infer order and relationships amongst so-called big-data. The paradigm that a model means a differential equation, or perhaps a parametrically defined probability distribution, and validation means post-hoc verification of outputs of the model, to observational data, has been joined with an entirely alternative perspective. Now we are coming into an era when it is possible to take a data-rich perspective, whereby relationships are decided directly from data. The fields of computer science and machine learning, dynamical systems and stochastic processes, probability and statistics, and logic have been converging on a central theme of how to handle decision making regarding summarizing massive data sets. Whether order relationships are inferred from dynamical systems, by entropy concepts, by spectral methods or from operator theoretic perspectives, the underlying concept of order relationships persists across many fields. Any decision making process premised on developing an order, or at least a partial order, amongst possible outcomes must be grounded in an appropriate algebra of sets. While order and measure theory are the foundation of probability theory, alternative concepts of logical relationships are generalized in terms of lattice theory. Inclusive is the relation of nesting that is implicitly necessary in any decision making process. Cognitive modeling and principled decision making may include methods based on information theory, but especially concepts of generalized probability frameworks are increasingly relevant. A central concept in ÒlearningÓ is the idea of summarizing features of data, to ask how many tokens in a reduced description can allow suitable explanation of underlying features. Considering a set of tokens, and then causal connections between these is of particularly exciting interest. When there are hidden (unobservable) information sources, some of the causal connections between the observables can in fact be false positives due to the unobservable influence from the hidden nodes. Specifically, for example, a hidden node that directly influences a subset of observable nodes often induces a (causal) clique among these nodes. Cognitive modeling, concepts in quantum probability relations, flow of information in dynamical systems including group and robotic swarms, and causal inference all have aspects that benefit from a grounding possibly alternative from classical probability theory, but rather in the language of generalized partially ordered sets, inclusion and lattices. Also, Bayesian inference, while grounded in probability theory and Boolean algebra of sets with measure, there are strong theoretical reasons to allow a more general frame work for a lattice of assertions, beyond standard concepts of ideals and filters, beyond the axioms of Cox and beyond Jaynes. It has been said that there is a strong connection between questions and information theory. Meanwhile in computation there is the field of machine learning that while it consists of many widely disparate algorithmic designs, generally centers on concepts of using computers to process data sets to infer statements and about proposed questions and relationships, again allowing better decision making process. The goal of this workshop will be to bring together specialists from disparate fields that nonetheless have problems that touch on this same central concept, that foundations of the axioms underlying relations of inclusion are fundamental to developing a calculus of questions to formulate a framework generalizing probability theory appropriate for future decision makers.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 29, 2019
- Source ID
- W911NF1810304
Entities
People
- Erik Bollt
Organizations
- Army Contracting Command
- Clarkson University
- United States Army