#01Nov 10, 2026
cs.LG
The Unused Input: The Initial State as a Conditioning Interface for Recurrent TSFMs
Bernhard Voggenberger, Christian Ganhör, Levente Zólyomi and 2 more
Time series foundation models (TSFMs) forecast unseen series without task-specific training. Increasingly they are built on recurrent backbones, whose constant per-step cost makes long-context inference cheap, and which, unlike transformers, carry a state. A recurrent backbone computes its prediction from two inputs: the observed context window and the initial recurrent state. In practice only the first is utilised and the initial state is unused by convention. We ask whether the initial state can be utilised in recurrent TSFMs. Using the xLSTM-based zero-shot forecaster TiRex as a case study, we freeze the backbone and train only a small head that writes a context vector into the initial state, so that any change in accuracy is attributable to the injected state rather than to adapted weights. A spectral summary of the input window, learned once and applied unchanged to every series, lowers mean GiftEval MASE by 0.175%, with every seed improving, which neither a parameter-matched LoRA adapter nor a learnable static state reproduces. An embedding of text attached to each series, evaluated on six MoTime datasets, reveals two roles for the state. As a per-dataset adaptation surface, a static state lowers MASE by 0.04-6.96% of the zero-shot baseline (8-70% of the gain of full fine-tuning). As a channel for exogenous information, the text-conditioned state lowers it by a further 0.15-0.41% of the zero-shot baseline on four of six datasets and, combined with a LoRA adapter, exceeds full fine-tuning, which has no access to the text, on one of them. In TiRex, the effect concentrates in the mLSTM blocks and is strongest at short context. The results are preliminary and the improvements small, but they point to the initial state as a direction for further research.
#02Nov 10, 2026
cs.LG
Causal Concept Explanations for Deep Neural Models
Josh Rountree, Pulkit Verma, Oswin So and 2 more
Concept-based explanations in neural models are important for the development, certification, and adoption of autonomous systems. Yet, interpretability remains correlational. Without methods to guide and quantify causal relationships between explanations, a concept explanation can be provided without being necessary or sufficient for the observed output thereby appearing correct while misleading. We introduce Causal Concept Wrapper Network (CCW-Net), the first use of interventional mediation analysis from fields such as economics, medicine, and epidemiology as a training objective to align causal effects of concept-based explanations with a model's output. We further apply mediation analysis as an evaluation tool to quantify, per concept label combination, how much of any model's output is causally attributable to each concept explanation. As a training objective, CCW-Net shapes a model's concept representations such that they are both necessary and sufficient for the outputs they are credited with and independent of other concepts' contributions. Counterfactual concept representations anchor a set of causal alignment criteria: necessity, sufficiency, and independence. Necessity drives irrelevant concept effects toward zero, sufficiency ensures relevant concept representations fully reconstruct a model's output, and independence enforces the total effect decomposes linearly across concepts. We evaluate CCW-Net across aircraft control, driving, and image classification. CCW-Net provides a step toward concept-based explanations that are not merely interpretable, but quantifiably causal, supporting safer evaluation, certification, and deployment of transparent and trustworthy neural models.
#03Nov 10, 2026
math.AP
Bayesian inverse problems and epidemic-threshold uncertainty for a heterogeneous SIS reaction-diffusion model
Duc-Lam Duong
We study Bayesian inference of persistence from early observations of a heterogeneous susceptible--infected--susceptible (SIS) reaction--diffusion model with standard incidence. We show that the leading prevalence dynamics depend on the unknown coefficients only through the difference between the transmission and recovery rates and the infected diffusivity. The same pair determines the principal eigenvalue whose sign governs persistence, so early prevalence can inform the threshold even when the individual rates remain uncertain. We prove global weak well-posedness and quantitative parameter stability for the SIS system and use the resulting observation-map estimates to establish Bayesian well-posedness. For an initial infection of size $\varepsilon$, we derive a uniform second-order expansion that describes how the separate rates enter prevalence. Incidence depends explicitly on transmission at leading order. When the noise standard deviation is proportional to $\varepsilon$, we prove $O(\varepsilon)$ convergence in Hellinger distance of the nonlinear posteriors to their low-prevalence limits for both prevalence alone and joint prevalence and incidence data. We bound the principal eigenvalue using the growth rate of the infected $L^2$ norm between two times. This rate gives a lower bound for the full nonlinear system, and the upper bound separates spatial transients from nonlinear saturation. Uniform spectral estimates and bounds on infection size and measurement error give sufficient conditions for persistence and extinction. We also bound the error in posterior persistence probabilities under forward and eigenvalue approximation, accounting for posterior mass near the threshold. Numerical examples illustrate threshold inference and the predicted \(O(\varepsilon)\) convergence of the nonlinear posteriors to their low-prevalence limits in Hellinger distance.
#04Nov 10, 2026
stat.ML
A Theory of Cross-Validation Variance
Nathaniel S. O'Connell, Jaime L. Speiser
Cross-validation estimates predictive performance by repeatedly splitting the data into training and validation sets, fitting the model on the first and evaluating it on the second. Every observation influences the estimate in two roles, through its own evaluation when it is held out and through the models it helps fit when it trains. The random splits determine its weight in each role. From this representation, we prove an exact finite-sample identity for the variance of the cross-validated estimate. For a given splitting scheme with a common training set size, the variance is a function of these weights, with constants of the model, the metric, and the population. It covers metrics averaged over observations, such as mean squared error, and over case--control pairs, such as the AUC. The identity shows which schemes to use. The variance decomposes into sampling variance, which depends only on the training size, and variance due to the random fold splitting, which arises when the weights change from one draw of the splits to the next. K-fold cross-validation, whose weights are equal across observations by design, is therefore variance optimal at first order; Monte Carlo and bootstrap cross-validation are not. The first-order variance of K-fold does not depend on K, and under reasonable conditions on the model leave-one-out has the smallest variance of any K-fold scheme. For the AUC the same holds for leave-pair-out, which evaluates every case--control pair. Finally, we show the cross-validated estimate is asymptotically normal with the identity's full variance. Together, these results provide a general theoretical foundation for inference on cross-validated performance and establish principles for selecting variance-optimal cross-validation designs.
#05Nov 10, 2026
stat.ME
One Curve, Different Weights: AUC, the Brier Score, Decision-Weighted AUC, and the Policy Youden Index
Yegor Baranovski
AUC and the Brier score are standard summaries of a binary risk score, yet they can rank models differently. We introduce a weighted AUC, AUC$_π$, which weights each pair of observations by how differently a decision maker with probability of acting $π$ treats them. We show that all three are built on weighted averages of one curve, the Youden index $J(t) = \mathrm{TPR}(t) - \mathrm{FPR}(t)$ across cutoffs $t$, and that the averages differ in the weights they put on the cutoffs. Each weights a cutoff by how much something rises across it: for AUC, the distribution of all scores; for the Brier score, the true probability $\mathbb{E}[Y \mid S]$; for AUC$_π$, the decision maker's probability of acting. Up to normalization, AUC$_π$ is the Brier skill score of the ideal forecast with its weight moved to the cutoffs the decision maker cares about. We also show that weighting pairs by $|π(s) - π(s')|$ is, up to scale, the only pair weight fixed in advance that makes a pair-weighted AUC an average of the Youden curve for every score distribution. As a ranking measure, AUC weights every model the same way. Hand (2009) shows that it can also be read as an average over cost thresholds, and in that reading each model weights the thresholds differently. The numerator of AUC$_π$, which we call the policy Youden index, is the expected share of class 1 a fixed policy acts on minus the expected share of class 0. It judges every model by the same policy and so avoids this; the full ratio AUC$_π$ does not.