A Tale of Two Variances: When Single-Seed Benchmarks Fail in Bayesian Deep Learning

In limited-data settings, a single endpoint mean of an evaluation metric such as the Continuous Ranked Probability Score (CRPS) is itself a random variable, yet it is routinely reported as if it were a stable property of the method. We study when this practice fails. Using 50 independent repetitions across six regression datasets, we show that CRPS variance trajectories differ substantially across methods and are not always well described by a smooth power-law decay. Methods with a learned heteroscedastic variance head, namely MAP and Deep Ensembles, can develop pronounced, reproducible variance peaks at intermediate training sizes on real datasets, whereas MC Dropout and Bayes by Backprop typically show smooth variance contraction. These peaks have direct practical consequences: at the variance peak on Seoul Bike, the relative RMSE of a single-seed MAP estimate reaches 93.6\%, and the probability of falling within \(\pm 10\%\) of the repeated-run mean drops to 5.9\%. We show that local CRPS variance provides a direct signal of single-seed estimation error, with Spearman correlations above 0.96 on every real dataset. Power-law fit quality and monotonicity together provide compact method-level summaries of trajectory regularity. Finally, replacing the standard heteroscedastic objective with \(β\)-NLL substantially reduces the irregular behavior, consistent with the view that the heteroscedastic training objective contributes to the instability. Practitioners should report trajectory summaries alongside endpoint means and concentrate repeated evaluation in high-variance regions.

Unstable Rankings in Bayesian Deep Learning Evaluation

Standard evaluations of Bayesian deep learning methods assume that metric estimates are reliable, but we show this assumption fails under data scarcity. Method rankings are not only unreliable at small $n$, but also dataset-dependent in ways that point estimates cannot reveal: the same method comparison yields $P(\mathrm{MCD} \prec \mathrm{Ensemble}) = 1.000$ at $n = 50$ on one dataset and remains below $0.95$ even at $n = 500$ on another. Across the datasets we consider, no universal sample size threshold exists, which is precisely why dataset-specific posterior inference is necessary. To address this, we use a Bayesian hierarchical model with method-specific variances to treat evaluation metrics as random variables across data realizations, and we use a predictive Minimum Detectable Difference curve to assess whether an observed gap would be detectable at a given training size. Across six Bayesian deep learning methods and five regression datasets, our results show that uncertainty-aware evaluation is necessary in low-data settings, because current evidence for method superiority and predictive detectability at the same training size can diverge substantially. Our framework provides practitioners with principled tools to determine whether their evaluation data is sufficient before drawing conclusions about method superiority.

How Order-Sensitive Are LLMs? OrderProbe for Deterministic Structural Reconstruction

Large language models (LLMs) excel at semantic understanding, yet their ability to reconstruct internal structure from scrambled inputs remains underexplored. Sentence-level restoration is ill-posed for automated evaluation because multiple valid word orders often exist. We introduce OrderProbe, a deterministic benchmark for structural reconstruction using fixed four-character expressions in Chinese, Japanese, and Korean, which have a unique canonical order and thus support exact-match scoring. We further propose a diagnostic framework that evaluates models beyond recovery accuracy, including semantic fidelity, logical validity, consistency, robustness sensitivity, and information density. Experiments on twelve widely used LLMs show that structural reconstruction remains difficult even for frontier systems: zero-shot recovery frequently falls below 35%. We also observe a consistent dissociation between semantic recall and structural planning, suggesting that structural robustness is not an automatic byproduct of semantic competence.