A Note on Inferential Decisions, Errors and Path-Dependency
Kangda K. Wren
TL;DR
The paper addresses sequential inference for binary outcomes, proving that regular inference processes cannot be informationally redundant unless they are identical up to an a priori factor, which implies path-dependency for time-homogeneous decisions unless the posterior tracks the true probabilities. It develops a Bayesian framework with log-likelihood-ratio dynamics and examines resolution properties, showing that regular inferences diverge in log-LR as outcomes are resolved, while non-resolving tests converge. By linking redundancy to Bayes' rule and functional equations, the work extends to continuous time via Ito's lemma and highlights that path-dependency is almost unavoidable when the true law is unknown and replaced by similar test measures. Practically, the paper decomposes inferential error into a path-dependent diffusive component and a path-independent bias, with implications for social dynamics and asset pricing, and suggests that understanding these components can guide better management of inference-driven systems.
Abstract
Consider the sequential inference of a binary outcome. The process of a posteriori beliefs and its objectively true conditional-probability counterpart generally differ but should lead to the same result eventually in well-defined tests. We show that unless the two are 'essentially identical', differing only by an a priori factor, time-homogeneous continuous decisions based on the former must be path-dependent with respect to state-variables based on the latter or any non-essentially-identical process. Inferential errors decompose into path-dependent and path-independent parts, whose distinct properties are relevant to error mitigation.