Law of total probability and Bayes' theorem in Riesz spaces

TL;DR

This paper extends fundamental probability concepts like total probability, Bayes' theorem, and inclusion-exclusion to Riesz spaces, broadening their mathematical framework using order-theoretic methods.

Contribution

It introduces a generalized notion of conditional probability in Riesz spaces and proves key probability laws within this abstract setting, unifying classical and advanced mathematical theories.

Findings

Law of total probability established in Riesz spaces

Bayes' theorem generalized to Riesz spaces

Inclusion-exclusion formula proven in Riesz spaces

Abstract

This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the aid of this concept, we establish the law of total probability and Bayes' theorem in Riesz spaces; we also prove an inclusion-exclusion formula in Riesz spaces. Several examples are provided to show that the law of total probability, Bayes' theorem and inclusion-exclusion formula in probability theory are special cases of our results.