A rapidly growing range of sequential data tasks, such as identifying trend reversals in financial markets, auto-segmenting video and audio recordings, detecting changes in movement direction from motion sensors cannot be fully addressed without detection of distributional shifts in time-ordered data. We consider a sequential change-point detection problem where the conditional density switches at an unknown time, yet neither the pre- nor post-change distribution admits a closed-form. Classical likelihood-ratio statistics are inapplicable in this settings. A conditional diffusion model, trained on pre-change-point data with a frozen context encoder, defines a deterministic bijection via the probability flow ODE. Pre-change observations are mapped onto standard Gaussian latent variables. Post-change observations, processed through the same frozen map, deviate from this reference. We employ the Maximum Mean Discrepancy as the test statistic, derive closed-form expressions for its components under the Gaussian null, and establish its asymptotic distribution as a degenerate U-statistic. Afterwards we apply an online detection procedure of Shiryaev--Roberts to the resulting statistic with exact threshold calibration. The method detects arbitrary distributional shifts, including covariance rotations and higher-order structural breaks, without parametric assumptions on either regime.
Chase Mathis, Ian Waudby-Smithstat.ME math.ST stat.ML
Anytime-valid inference enables analysts to continuously monitor their data and stop experiments early. However, the majority of these methods incur a certain conservativeness by remaining valid on infinite time horizons. In practice, a bound on the horizon may be imposed due to budgetary, practical, or ethical constraints. In this paper, we ask the question: "Is it possible to obtain sharper large-sample anytime-valid inference by forgoing validity beyond some finite time horizon?". We provide a positive answer to this question by proposing a family of statistical objects that we call "confidence horizons". These objects can be viewed as large-sample confidence sequences on bounded time horizons, or alternatively as group sequential repeated confidence intervals with a maximal number of interim peeking times. We make explicit connections to the group sequential boundaries of Pocock [1977], O'Brien--Fleming [1979], and Wang--Tsiatis [1987]. We derive closed-form distribution functions of certain statistics which can be used to calculate the asymptotic quantiles of confidence horizons exactly, sidestepping the repeated integration typically employed in group sequential methods. We illustrate the use of confidence horizons for treatment effect estimation in sequentially randomized experiments under adaptive Neyman allocation.
Wang~\cite{Wang2026} introduced the Stochastic-Oracle Turing Machine (SOTM) framework and defined token complexity as the minimum expected cost of interacting with a stochastic oracle needed to attain a specified solution quality for a task. This paper develops an analogous notion for certifying the reliability of a stochastic oracle on a given domain. Certification token complexity is the minimum expected token cost required, with controlled error probability, to distinguish oracles that meet a target reliability level from those that fall below a lower reliability threshold. We construct an SPRT-based certification SOTM that queries the oracle, computes binary correctness scores, and stops when the accumulated log-likelihood evidence crosses a decision threshold. The SOTM halts almost surely, satisfies the desired two-sided error guarantee over the reliability regions to be certified, and yields an explicit upper bound on certification token complexity in terms of the reliability thresholds, the error bound, and the expected per-turn token cost. We then establish a matching information-theoretic lower bound: even with adaptive queries, every error-bounded certification SOTM must incur the same leading-order expected token cost as the SPRT-based construction as the prescribed error bound tends to zero. Together, these bounds characterize the leading-order certification token complexity in the small-error regime.
Token-level hallucination detectors are evaluated as classifiers, by AUC over all tokens, yet a streaming monitor is judged by its reaction time: the number of tokens that pass between the onset of a hallucination and the alarm. We formulate hallucination onset detection as a quickest change detection problem. A first-order Markov model of the latent faithful/hallucinated state, validated on RAGTruth, places the task inside classical change-point theory and yields Lorden's lower bound on detection delay: about 1.3 tokens at a false-alarm rate of 0.01. We then show that a causal recurrent labeler acts as a CUSUM with a learned increment. Among the onsets it catches it detects in 11-13 tokens, against 31 for a linear per-token baseline, though at this false-alarm budget every detector catches under a third of onsets and the recall-honest delay is 56-66 tokens: low-false-alarm onset detection is hard. A controlled decomposition attributes the speed advantage mostly to a better per-token score rather than to temporal accumulation. An information-rate optimality theorem of Donsker-Varadhan type explains the remaining order-of-magnitude gap: the learned score realizes only 1/4.5 of the divergence the features carry, a deficit that recalibration cannot remove, with the remainder a finite-horizon effect. Classification metrics conceal this delay structure; sequential analysis makes it measurable.