Dutch Books for Language Models

People increasingly use language models to support life decisions. Many such decisions involve a probabilistic forecast: How likely is a major life event, a natural disaster, or an economic outcome? Users of language models may implicitly trust that these forecasts fall out of a coherent world model. In this paper, we evaluate the coherence of language model probabilistic forecasts through a procedure that builds on a theorem due to de Finetti. We elicit forecasts from language models across events generated from stock returns data. We then use linear programs to compute the largest Dutch-book profit - the profit an arbitrageur could guarantee by betting against model-generated probabilities - which we use as a measure of incoherence. Our procedure does not require outcome labels, so we can evaluate coherence even in settings where outcomes are not observed or have not yet resolved. We find substantial evidence of incoherence in language model forecasts. Such incoherence increases when there are richer logical relationships between events, and irrelevant contextual details can increase incoherence by an order of magnitude. We conclude by discussing how alternative training strategies may improve probabilistic coherence.

Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if $k$ verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in $k$, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable $α\sim G$ (de Finetti), the exact cascade posterior is $\ell_k = \ell_0 - \ln m_k$, with $m_k$ the $k$-th moment of $G$. Then: (i) $\ell_k$ is concave in $k$ for every non-degenerate $G$ -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta$(a,b)$ latents, failure decays polynomially, $1-r_k \asymp k^{-b}$, with correlation parameter $ρ_v = 1/(a+b+1)$; (iii) a blind-spot atom of mass $1-π$ at $α=1$ caps the evidence extractable from any number of gates at $-\ln(1-π)$ nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary ($β\sim H$) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of $G$ and $H$, with closed-form crossover $k^\dagger$. The mechanism is survivorship: errors surviving gates are the high-$α$ ones. The theory is measurable: $R$ repeated verdicts per instance identify the first $R$ moments of $G$, so two verdicts identify $ρ_v$; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at $k=5$ and ~3000x at $k=10$; the correlated fit at $R=8$ tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

Bounded Difference Concentration for Infinitely Exchangeable Sequences with Applications to AI Benchmark Uncertainty

We consider the concentration properties of functions of infinitely exchangeable random variables. By conditioning on the de Finetti directing measure, we show that the deviation of any function with bounded-difference constants $c_1, \dots, c_n$ decomposes into a conditional sampling fluctuation and a latent mixture fluctuation. When this latent mixture is $σ_{\mathrm{mix}}^2$-subgaussian, we establish a concentration inequality with an effective variance proxy of $\frac{1}{4}\sum_i c_i^2 + σ_{\mathrm{mix}}^2$. Crucially, we demonstrate that for zero-sum linear contrasts, such as the difference between a subsample mean and a full population mean, the latent mixture term cancels exactly. This cancellation yields a tight, mixture-free Hoeffding-type bound that provides a direct de Finetti mechanism for the infinite-extendibility limit of recent finite-exchangeable concentration results. We apply this framework to quantify uncertainty in composite AI benchmarks, such as MMLU, where question items naturally exhibit exchangeable dependence across domains. Our results provide both a domain-stratified hierarchical model for bounding the uncertainty of accuracy scores, and a distribution-free, cost-saving statistical guarantee for accurately estimating full benchmark scores from random subsets.