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A categorical formalization of epistemic uncertainty frameworks

Torgeir Aambø

TL;DR

A general categorical form of belief updating based on change of enrichment is introduced, and it is proved that Bayesian updating and possibilistic conditioning arise as examples.

Abstract

Epistemic uncertainty arises in lack of complete knowledge about the state of a system. There are multiple mathematical frameworks for measuring such uncertainty quantitatively, often referred to as imprecise probability theories. Inspired by work of Opdan, we introduce a general category theoretic definition of epistemic calculi, which we use as a foundation for modelling and studying contradictions and synergies between several philosophical epistemological concepts. We further develop an enriched category theoretic process for changing calculi, and use this to study relationships between existing examples, like possibility theory and certainty factors. Finally, we introduce a general categorical form of belief updating based on change of enrichment, and prove that Bayesian updating and possibilistic conditioning arise as examples.

A categorical formalization of epistemic uncertainty frameworks

TL;DR

A general categorical form of belief updating based on change of enrichment is introduced, and it is proved that Bayesian updating and possibilistic conditioning arise as examples.

Abstract

Epistemic uncertainty arises in lack of complete knowledge about the state of a system. There are multiple mathematical frameworks for measuring such uncertainty quantitatively, often referred to as imprecise probability theories. Inspired by work of Opdan, we introduce a general category theoretic definition of epistemic calculi, which we use as a foundation for modelling and studying contradictions and synergies between several philosophical epistemological concepts. We further develop an enriched category theoretic process for changing calculi, and use this to study relationships between existing examples, like possibility theory and certainty factors. Finally, we introduce a general categorical form of belief updating based on change of enrichment, and prove that Bayesian updating and possibilistic conditioning arise as examples.
Paper Structure (11 sections, 9 theorems, 25 equations, 1 table)

This paper contains 11 sections, 9 theorems, 25 equations, 1 table.

Table of Contents

  1. Introduction
  2. Category theory
  3. Opdans CF category
  4. Epistemic calculi
  5. Examples
  6. Extensions and properties
  7. Inconsistencies
  8. Change of calculi
  9. Enriched category theory
  10. Enriched epistemic updating
  11. Conclusion

Key Result

Theorem A

If $\EuScript{V}$ is an uncertainty calculus, then $\EuScript{V}$ cannot be closed, conservative and cancellative simultaneously. Furthermore, if $\EuScript{V}$ is complete, then being closed is equivalent to being fallible.

Theorems & Definitions (24)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • proof
  • ...and 14 more