Unbiased Canonical Set-Valued Oracles Via Lattice Theory

Computer Science > Artificial Intelligence

Title:Unbiased Canonical Set-Valued Oracles Via Lattice Theory

View PDF HTML (experimental)Abstract:A non-agentic "oracle" AI that estimates probabilities of future events faces a self-reference problem: once its answer is learned and acted upon, it can change the very probability it was asked to report. One response, advocated for the Scientist AI programme, is to ask only counterfactual questions, evaluated as if the answer had no influence. We observe that such answers tend to become irrelevant the moment they are learned, precisely because their premise is then false. We therefore explore a self-referential alternative in which the oracle reports not a single probability but a credal set that is simultaneously unbiased and self-consistent with the consequences of being learned. The naive self-consistency requirement is satisfied by too many sets (including the useless answer $[0,1]$), so the problem is to single out a canonical, nontrivial member. We do so with the Knaster--Tarski fixed-point theorem on the complete lattice of closed credal sets, taking the least fixed point of a suitably defined isotone operator; a variant instead reports the least fixed point that contains every self-consistent point estimate. We prove existence, self-consistency, and nonemptiness, show that the construction collapses to the classical point answer for non-performative questions, and that for a binary event the canonical answer is, under a natural hull-factoring assumption, an interval. The development is purely lattice-theoretic and extends unchanged from a binary event $B$ to an arbitrary random variable $X$, with $P(B\mid A,C)$ replaced by the conditional law $\mathcal{L}(X\mid A,C)$. We close with open questions, including whether the interval characterization itself survives that generalization.

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