Algorithmic Randomness, Exchangeability, and the Principal Principle
Jeffrey A. Barrett, Eddy Keming Chen
TL;DR
Problem: provide a non-circular justification of the Principal Principle linking credence to objective chance. Approach: replace standard probabilistic laws with a statistical constraining law $L^\star$ that enforces Martin-Löf randomness, and adopt exchangeable credences that encode inductive symmetry; together they yield the PP via de Finetti's representation. Method: show that for an agent with $Cr$ exchangeable and committed to $L^\star$, $Cr(\cdot|L^\star)=β_{0.5}(\cdot)$ and hence $Cr(A|L^\star)=β_{0.5}(A)$ with resiliency under admissible information; extend to partial exchangeability and finite histories with vanishing error bounds. Significance: provides a metaphysically neutral, non-circular foundation for chance-credence alignment and clarifies how global nomic constraints govern single-case beliefs.
Abstract
We introduce a framework uniting algorithmic randomness with exchangeable credences to address foundational questions in philosophy of probability and philosophy of science. To demonstrate its power, we show how one might use the framework to derive the Principal Principle -- the norm that rational credence should match known objective chance -- without circularity. The derivation brings together de Finetti's exchangeability, Martin-Löf randomness, Lewis's and Skyrms's chance-credence norms, and statistical constraining laws (arXiv:2303.01411). Laws that constrain histories to algorithmically random sequences naturally pair with exchangeable credences encoding inductive symmetries. Using the de Finetti representation theorem, we show that this pairing directly entails the Principal Principle of this framework. We extend the proof to partial exchangeability and provide finite-history bounds that vanish in the infinite limit. The Principal Principle thus emerges as a mathematical consequence of the alignment between nomological constraints and inductive learning. This reveals how algorithmic randomness and exchangeability can illuminate foundational questions about chance, frequency, and rational belief.