Minimum Viable Evidence and Value-optimal Evidence: Decision-theoretic Sufficiency, Error-controlled Testing, and Categorical Structure

Minimum viable evidence (MVE) is the decision-theoretic status of evidence that is sufficientto justify acting, stopping, delaying, regulating, deploying, or redesigning while being minimalrelative to an acquisition or informativeness order. This paper develops MVE as a calculus ofposterior risk, robust risk, regret, assurance, sequential stopping, value of information, partialidentification, and evidence portfolios. It also gives an error theory: a faulty MVE-drivenaction is an MVE-induced decision error, with false viability and false non-viability serving asthe Type I and Type II analogues for an actionability gate. For a fixed decision specification,the signed MVE statistic is the scaled slack to the viability boundary, and evidence is viableexactly when this statistic is nonnegative. Calibrating the signed statistic against a nonviabilitynull yields MVE p-values, e-values, confidence-sequence tests, loss-weighted critical values,power functions, and sample-size formulae for positive viability slack.The paper then asks whether MVE is the strongest decision-theoretic way to do what itclaims to do. The answer is qualified but important: MVE is a rigorous certificate, but itis not generally an optimal controller. A more powerful method is value-optimal evidenceacquisition (VOE), which jointly optimizes the experiment, stopping time, and terminal actionby minimizing expected terminal loss plus acquisition cost, or its robust analogue. The Bellmanstopping region stops precisely when the immediate terminal decision value dominates optimalcontinuation value. Any fixed MVE hitting rule is weakly dominated by this Bellman policy andis strictly dominated whenever it stops while continuation has positive net value or continueswhen stopping has lower expected loss. MVE coincides with VOE only when the declaredviability region equals the Bellman stopping region. Likewise, in the formal testing problem,thresholding signed MVE slack is not generally most powerful; decision-value likelihood-ratiotests, least-favorable likelihood ratios, or growth-optimal e-processes dominate except whenthe likelihood ratio is monotone in, or sufficient with respect to, the signed MVE statistic. Thepaper concludes by categorifying the resulting objects using experiments as Markov kernels,garblings as morphisms, viability and error events as subobjects, MVE as a minimal object ina cost-enriched viable subcategory, and VOE as a Bellman-optimal policy on the belief-stateprocess.