Next, we apply our proposed fitting method (see Section 2) on this data set.

The widespread use of black box prediction methods has sparked an increasing interest in algorithm/model-agnostic approaches for quantifying goodness-of-fit, with direct ties to specification testing, model selection and variable importance assessment. A commonly used framework involves defining a predictiveness criterion, applying a cross-fitting procedure to estimate the predictiveness, and utilizing the difference in estimated predictiveness between two models as the test statistic. However, even after standardization, the test statistic typically fails to converge to a non-degenerate distribution under the null hypothesis of equal goodness, leading to what is known as the degeneracy issue. To addresses this degeneracy issue, we present a simple yet effective device, Zipper. It draws inspiration from the strategy of additional splitting of testing data, but encourages an overlap between two testing data splits in predictiveness evaluation. Zipper binds together the two overlapping splits using a slider parameter that controls the proportion of overlap. Our proposed test statistic follows an asymptotically normal distribution under the null hypothesis for any fixed slider value, guaranteeing valid size control while enhancing power by effective data reuse. Finite-sample experiments demonstrate that our procedure, with a simple choice of the slider, works well across a wide range of settings.

Learning the probability distribution of high-dimensional data is a challenging problem. To solve this problem, we formulate a deep energy adversarial network (DEAN), which casts the energy model learned from real data into an optimization of a goodness-of-fit (GOF) test statistic. DEAN can be interpreted as a GOF game between two generative networks, where one explicit generative network learns an energy-based distribution that fits the real data, and the other implicit generative network is trained by minimizing a GOF test statistic between the energy-based distribution and the generated data, such that the underlying distribution of the generated data is close to the energy-based distribution. We design a two-level alternative optimization procedure to train the explicit and implicit generative networks, such that the hyper-parameters can also be automatically learned. Experimental results show that DEAN achieves high quality generations compared to the state-of-the-art approaches.

The kernel Maximum Mean Discrepancy~(MMD) is a popular multivariate distance metric between distributions. The usual kernel-MMD test statistic (for two-sample testing) is a degenerate U-statistic under the null, and thus it has an intractable limiting null distribution. Hence, the standard approach for designing a level-$(1-\alpha)$ two-sample test using this statistic involves selecting the rejection threshold as the $(1-\alpha)$-quantile of the permutation distribution. The resulting nonparametric test has finite-sample validity but suffers from large computational cost, since the test statistic must be recomputed for every permutation. We propose the cross-MMD, a new quadratic time MMD test statistic based on sample-splitting and studentization. We prove that under mild assumptions, it has a standard normal limiting distribution under the null. Importantly, we also show that the resulting test is consistent against any fixed alternative, and when using the Gaussian kernel, it has minimax rate-optimal power against local alternatives. For large sample-sizes, our new cross-MMD provides a significant speedup over the MMD, for only a slight loss in power.

Goodness-of-fit (GoF) tests are fundamental for assessing model adequacy. Score-based tests are appealing because they require fitting the model only once under the null. However, extending them to powerful nonparametric alternatives is difficult due to the lack of suitable score functions. Through a class of exponentially tilted models, we show that the resulting score-based GoF tests are equivalent to the tests based on integral probability metrics (IPMs) indexed by a function class. When the class is rich, the test is universally consistent. This simple yet insightful perspective enables reinterpretation of classical distance-based testing procedures-including those based on Kolmogorov-Smirnov distance, Wasserstein-1 distance, and maximum mean discrepancy-as arising from score-based constructions. Building on this insight, we propose a new nonparametric score-based GoF test through a special class of IPM induced by kernelized Stein's function class, called semiparametric kernelized Stein discrepancy (SKSD) test. Compared with other nonparametric score-based tests, the SKSD test is computationally efficient and accommodates general nuisance-parameter estimators, supported by a generic parametric bootstrap procedure. The SKSD test is universally consistent and attains Pitman efficiency. Moreover, SKSD test provides simple GoF tests for models with intractable likelihoods but tractable scores with the help of Stein's identity and we use two popular models, kernel exponential family and conditional Gaussian models, to illustrate the power of our method. Our method achieves power comparable to task-specific normality tests such as Anderson-Darling and Lilliefors, despite being designed for general nonparametric alternatives.

The advantages of adaptive experiments have led to their rapid adoption in economics, other fields, as well as among practitioners. However, adaptive experiments pose challenges for causal inference. This note suggests a BOLS (batched ordinary least squares) test statistic for inference of treatment effects in adaptive experiments. The statistic provides a precision-equalizing aggregation of per-period treatment-control differences under heteroskedasticity. The combined test statistic is a normalized average of heteroskedastic per-period z-statistics and can be used to construct asymptotically valid confidence intervals. We provide simulation results comparing rejection rates in the typical case with few treatment periods and few (or many) observations per batch.

When using an LLM through an API provider, users expect the served model to remain consistent over time, a property crucial for the reliability of downstream applications and the reproducibility of research. Existing audit methods are too costly to apply at regular time intervals to the wide range of available LLM APIs. This means that model updates are left largely unmonitored in practice. In this work, we show that while LLM log probabilities (logprobs) are usually non-deterministic, they can still be used as the basis for cost-effective continuous monitoring of LLM APIs. We apply a simple statistical test based on the average value of each token logprob, requesting only a single token of output. This is enough to detect changes as small as one step of fine-tuning, making this approach more sensitive than existing methods while being 1,000x cheaper. We introduce the TinyChange benchmark as a way to measure the sensitivity of audit methods in the context of small, realistic model changes. LLM API providers typically offer version-pinned endpoints, signaling to users that a given endpoint will serve a consistent model. Users of APIs tend to rely on this consistency: developers want to avoid unexpected regressions in their applications; researchers seek reproducibility in their experiments; regulators perform initial compliance assessments, and assume that the API will keep serving the same model afterward (Y an & Zhang, 2022).

Watermarking aims to embed hidden signals in generated text that can be reliably detected when given access to a secret key. Open-weight language models pose acute challenges for such watermarking schemes because the inference-time interventions that dominate contemporary approaches cannot be enforced once model weights are public. Existing watermaking techniques for open-weight models, such as the recently proposed GaussMark, typically rely on small modifications to model weights, which can yield signals detectable to those equipped with a secret key, but achieving detection power comparable to inference-time watermarks generally requires weight perturbations that noticeably reduce generation quality. We introduce MarkTune, a theoretically principled, on-policy fine-tuning framework that treats the GaussMark signal as a reward while simultaneously regularizing against degradation in text quality. We derive MarkTune as an improvement on GaussMark and demonstrate that MarkTune consistently improves the quality-detectability trade-off over GaussMark by steering finer-grained, watermark-aware weight updates within the model's representation space while preserving generation quality. Empirically, we show that MarkTune pushes the quality-detectability frontier of GaussMark close to that of inference-time watermarking, remains robust to paraphrasing and fine-tuning attacks, and exhibits strong generalization: a model fine-tuned on one dataset retains substantial watermark detection power on unseen datasets. Together, these results establish MarkTune as a general strategy for embedding robust, high-quality watermarks into open-weight LMs.

Two-sample tests have been extensively employed in various scientific fields and machine learning such as evaluation on the effectiveness of drugs and A/B testing on different marketing strategies to discriminate whether two sets of samples come from the same distribution or not. Kernel-based procedures for hypothetical testing have been proposed to efficiently disentangle high-dimensional complex structures in data to obtain accurate results in a model-free way by embedding the data into the reproducing kernel Hilbert space (RKHS). While the choice of kernels plays a crucial role for their performance, little is understood about how to choose kernel especially for small datasets. Here we aim to construct a hypothetical test which is effective even for small datasets, based on the theoretical foundation of kernel-based tests using maximum mean discrepancy, which is called MMD-FUSE. To address this, we enhance the MMD-FUSE framework by incorporating quantum kernels and propose a novel hybrid testing strategy that fuses classical and quantum kernels. This approach creates a powerful and adaptive test by combining the domain-specific inductive biases of classical kernels with the unique expressive power of quantum kernels. We evaluate our method on various synthetic and real-world clinical datasets, and our experiments reveal two key findings: 1) With appropriate hyperparameter tuning, MMD-FUSE with quantum kernels consistently improves test power over classical counterparts, especially for small and high-dimensional data. 2) The proposed hybrid framework demonstrates remarkable robustness, adapting to different data characteristics and achieving high test power across diverse scenarios. These results highlight the potential of quantum-inspired and hybrid kernel strategies to build more effective statistical tests, offering a versatile tool for data analysis where sample sizes are limited.

For testing two vector random variables for independence, we propose testing whether the distance of one vector from an arbitrary center point is independent from the distance of the other vector from another arbitrary center point by a univariate test. We prove that under minimal assumptions, it is enough to have a consistent univariate independence test on the distances, to guarantee that the power to detect dependence between the random vectors increases to one with sample size. If the univariate test is distribution-free, the multivariate test will also be distribution-free.