A Note on Inferential Decisions, Errors and Path-Dependency

TL;DR

This paper examines how sequential binary testing decisions are influenced by the path-dependent nature of belief processes, highlighting differences between a posteriori and conditional probabilities and their implications for error analysis.

Contribution

It demonstrates that unless belief processes are essentially identical, decision paths depend on the specific process used, affecting inferential errors and their mitigation.

Findings

Belief processes generally differ but converge in well-defined tests.

Path-dependency arises unless processes are 'essentially identical'.

Error components can be decomposed into path-dependent and independent parts.

Abstract

Consider the sequential testing of binary outcomes. The a posteriori belief process and its objective conditional-probability counterpart generally differ but converge to the same result in well-defined tests. We show that unless the two processes are 'essentially identical', differing only by an a priori factor, time-homogeneous continuous decisions based on the former are path-dependent with respect to state-variables based on the latter or any other non-essentially-identical processes. Inferential error decomposes into a path-dependent and a path-independent component, whose distinct properties are relevant to error mitigation.