Statistical Hypothesis Testing

We introduce a novel characterization of phase transitions based on hypothesis testing. In our formulation, a phase transition is defined as the breakdown of statistical indistinguishability under vanishing parameter perturbations in the thermodynamic limit. This perspective provides a general, order-parameter-free framework that does not rely on model-specific insights or learning procedures. We show that conventional approaches, such as those based on the Binder parameter, can be reinterpreted as special cases within this framework. As a concrete realization, we employ a distribution-free two-sample run test and demonstrate that the critical point of the two-dimensional Ising model is accurately identified without prior knowledge of the order parameter.

Beyond conditional average treatment effects, treatments may impact the entire outcome distribution in covariate-dependent ways, for example, by altering the variance or tail risks for specific subpopulations. We propose a novel estimand to capture such conditional distributional treatment effects, and develop a doubly robust estimator that is minimax optimal in the local asymptotic sense. Using this, we develop a test for the global homogeneity of conditional potential outcome distributions that accommodates discrepancies beyond the maximum mean discrepancy (MMD), has provably valid type 1 error, and is consistent against fixed alternatives---the first test, to our knowledge, with such guarantees in this setting. We then provide a test that aggregates evidence across a grid of kernel-bandwidth choices. Furthermore, we derive exact closed-form expressions for two natural discrepancies (including the MMD), and provide a computationally efficient, permutation-free algorithm for our test.

The rapid proliferation of AI applications has intensified debate on effective regulation of these black-box services. Effective regulation must balance two competing goals: (1) deterring non-compliant providers from entering the market, while (2) retaining compliant ones. We call this ideal the perfect market outcome (PMO). Regulators face two compounding obstacles that make PMO difficult to achieve: providers hold private information and can act strategically to evade compliance, while any evidence drawn or derived from a finite sample carries statistical uncertainty in proving non-compliance. As this information asymmetry and statistical uncertainty is inherent to any effective regulation, we formalise them through a mechanism design framework that explicitly accounts for such statistical uncertainty. This yields a sharp characterisation: a mechanism achieves PMO if and only if the set of non-compliant evidence distributions forms a closed, convex set of probability measures, known in imprecise probability as a credal set. This result serves as a diagnostic tool to determine whether PMO is achievable under a given regulation. We further show that PMO-achieving mechanisms can be constructed from a collection of hypothesis tests, and validate our theoretical contributions through experiments on spurious-feature and fairness-based regulations.

Empirical investigations into unintended model behavior often show that the algorithm is predicting another outcome than what was intended. These exposés highlight the need to identify when algorithms predict unintended quantities - ideally before deploying them into consequential settings. We propose a falsification framework that provides a principled statistical test for discriminant validity: the requirement that an algorithm predict intended outcomes better than impermissible ones. Drawing on falsification practices from causal inference, econometrics, and psychometrics, our framework compares calibrated prediction losses across outcomes to assess whether the algorithm exhibits discriminant validity with respect to a specified impermissible proxy. In settings where the target outcome is difficult to observe, multiple permissible proxy outcomes may be available; our framework accommodates both this setting and the case with a single permissible proxy. Throughout we use nonparametric hypothesis testing methods that make minimal assumptions on the data-generating process. We illustrate the method in an admissions setting, where the framework establishes discriminant validity with respect to gender but fails to establish discriminant validity with respect to race. This demonstrates how falsification can serve as an early validity check. We also provide analysis in a criminal justice setting, where we highlight the limitations of our framework and emphasize the need for complementary approaches to assess other aspects of construct validity and external validity.

Models with intractable normalizing constants are widely used in statistics and machine learning. Assessing the adequacy of such models poses significant challenges: obtaining samples from the fitted model often requires sophisticated sampling algorithms. Moreover, model fitting sometimes requires iterative numerical optimization, making bootstrap procedures that require repeated refitting computationally expensive. In this paper, we leverage the kernel-based testing framework to develop a general semiparametric goodness-of-fit test based on the kernelized Stein discrepancy. We establish the consistency and the asymptotic null distribution of the test statistic under general nuisance estimation. To produce a level-$α$ test, we propose a novel influence-adjusted wild bootstrap that requires neither refitting the model nor sampling from it. We prove the consistency of the proposed bootstrap test procedure under the null and the alternative, and characterize its limiting power under contiguous local alternatives. Across simulations ranging from classical normality testing to models with intractable likelihoods, the proposed test delivers competitive or superior power at a computational cost orders of magnitude lower than that of existing approaches. We illustrate the method by assessing the adequacy of a protein signaling network model for reverse-phase protein array data from lung adenocarcinoma tumors. As a complementary insight, we show that the SKSD test can be regarded as a nonparametric score test under exponentially tilted models, connecting score-based and distance-based goodness-of-fit testing.

Existing two-sample testing techniques, particularly those based on choosing a kernel for the Maximum Mean Discrepancy (MMD), often assume equal sample sizes from the two distributions. Applying these methods in practice can require discarding valuable data, unnecessarily reducing test power. We address this long-standing limitation by extending the theory of generalized U-statistics and applying it to the usual MMD estimator, resulting in new characterization of the asymptotic distributions of the MMD estimator with unequal sample sizes (particularly outside the proportional regimes required by previous partial results). This generalization also provides a new criterion for optimizing the power of an MMD test with unequal sample sizes. Our approach preserves all available data, enhancing test accuracy and applicability in realistic settings. Along the way, we give much cleaner characterizations of the variance of MMD estimators, revealing something that might be surprising to those in the area: while zero MMD implies a degenerate estimator, it is sometimes possible to have a degenerate estimator with nonzero MMD as well; we give a construction and a proof that it does not happen in common situations.

We study the problem of selecting the best heterogeneous treatment effect (HTE) estimator from a collection of candidates in settings where the treatment effect is fundamentally unobserved. We cast estimator selection as a multiple testing problem and introduce a ground-truth-free procedure based on a cross-fitted, exponentially weighted test statistic. A key component of our method is a two-way sample splitting scheme that decouples nuisance estimation from weight learning and ensures the stability required for valid inference. Leveraging a stability-based central limit theorem, we establish asymptotic familywise error rate control under mild regularity conditions. Empirically, our procedure provides reliable error control while substantially reducing false selections compared with commonly used methods across ACIC 2016, IHDP, and Twins benchmarks, demonstrating that our method is feasible and powerful even without ground-truth treatment effects.

AI-Scientist systems risk manufacturing spurious discoveries through uncontrolled multiple testing. We present a functional architecture that enforces statistical rigor at two levels: a Haskell embedded domain-specific language (the Research monad) that makes it impossible to test a hypothesis without updating the error budget, and a declarative scaffold, backed by an OS-level sandbox, that makes validation data physically absent from the environment in which LLM-generated code runs. We ground the design in a machine-checked Lean4 formalization of LORD++ online false-discovery-rate (FDR) control: we derive its error budget and prove both marginal and full FDR control, then close the gap to the implementation by verifying the budget's wealth invariant over IEEE754 arithmetic in SPARK/Ada. To our knowledge this is the first verified chain from theorem to floating-point implementation for an online FDR procedure. In simulation, the architecture holds the false discovery rate near 1% against a 5% target, where a naive approach reaches 41%. In end-to-end case studies, a valid test avoids the false discoveries a flawed one produces, yet still finds real effects when the data allow. An adversarial evaluation confirms that generated code cannot read the held-out data even when given its exact path.

Social scientists are increasingly turning to unstructured datasets to unlock new empirical insights, e.g., estimating descriptive statistics of or causal effects on quantitative measures derived from text, audio, or video data. In many settings, unsupervised analysis is of primary interest, in that the researcher does not want to (or cannot) manually pre-specify all important aspects of the unstructured data to measure; they are interested in "discovery." This paper proposes a general and flexible framework for pursuing such discovery from unstructured data in a statistically principled way. The framework leverages recent methods from the literature on AI interpretability to map unstructured data points to high-dimensional, sparse, and interpretable "concept embeddings"; computes statistics from these concept embeddings for testing interpretable, concept-by-concept hypotheses; performs selective inference on these hypotheses using algorithms validated by new results in high-dimensional central limit theory, producing a selected set ("discoveries"); and both generates and evaluates human-interpretable natural language descriptions of these discoveries. The proposed framework has few researcher degrees of freedom, is robust to data snooping and other post-selection inference concerns, and facilitates fast and inexpensive sensitivity analysis and replication. Applications to recent descriptive and causal analyses of unstructured data in empirical economics are explored.

Generative networks are perfect tools to enhance the speed and precision of LHC simulations. Especially when generating events beyond the size of the training dataset, it is important to understand their statistical precision. We present two complementary methods to estimate the amplification factor without large holdout datasets. Averaging amplification uses Bayesian networks or ensembling to estimate amplification from the precision of integrals over given phase-space volumes. Differential amplification uses hypothesis testing to quantify amplification without any resolution loss. Applied to state-of-the-art event generators, both methods indicate that amplification is already possible in specific regions of phase space.

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.