The Inference Report

We study the problem of learning directed acyclic graphs from observational data in nonlinear models and provide finite-sample, nonasymptotic statistical guarantees in these models. Although widely studied, statistical guarantees for gradient-based methods in nonlinear models are missing from the literature. Under a generalized linear model for each node, we analyze a global nonsmooth objective combining an exact acyclicity constraint with a quasi-minimax concave penalty, which can be optimized via standard gradient-based solvers. We establish finite-sample guarantees under this model, including the most widely used logistic and Poisson models. Experiments corroborate structure recovery accuracy and computational performance, including both standardized data and negative controls as baselines.

Causal inference under treatment interference requires assumptions to reduce the exponentially large number of potential outcomes. We develop hierarchical finite-population models for potential outcomes by separating these assumptions into three layers: interference neighborhoods, exposure mappings, and outcome models. Given an interference neighborhood and exposure mapping, we show that potential outcomes admit a nonparametric decomposition into direct, spillover, and treatment--spillover interaction components. This decomposition provides a common coordinate system for nonparametric, semiparametric, and parametric models of finite-population potential outcomes. For a given potential-outcome model, we define two classes of estimands. Average causal effects (ACEs), such as direct and total treatment effects, contrast potential outcomes under fixed treatment--exposure conditions. Expected causal effects (ECEs), such as expected direct and indirect effects, contrast expectations of potential outcomes under a specified allocation policy. For both classes, we bound the variance of the canonical Horvitz--Thompson estimator by three terms: a diagonal weight term, a term from overlapping interference neighborhoods, and a residual design-dependence term. Our upper and lower bounds show that under graph-unaware designs, nonparametric design-unbiased estimation of ACEs can require sample sizes growing exponentially in the interference degree, whereas polynomial growth can suffice for design-aligned ECEs. Under neighborhood interference, we construct graph-aware independent-set designs with variance bounds polynomial in the maximum degree for direct and total treatment effects on every graph, whereas graph cluster randomization can incur worst-case variance exponential in the degree.

In this article, we study deep nonparametric regression trained via stochastic gradient Langevin dynamics (SGLD), a variant of stochastic gradient descent that injects carefully calibrated Gaussian noise during optimization. Using a PAC-Bayesian analysis tailored to deep ReLU networks, we establish an excess-risk bound that simultaneously accounts for network approximation error, statistical complexity, and the algorithmic cost accumulated along the SGLD trajectory. Under a Bernstein condition and within a stable SGLD regime, the excess-risk bound contains a stochastic term whose coefficient is inversely proportional to the variance of injected noise, while the estimator retains the optimal rate for nonparametric regression under the Hölder smoothness condition up to logarithmic factors. This result provides an approximation-error-aware, rate-optimal statistical perspective on the widely observed generalization benefits of moderate SGLD noise. For per-sample clipped SGLD, we further establish an exact f-DP guarantee and the corresponding asymptotic $μ$-GDP guarantee for the proposed estimator. We establish statistical convergence rates and differential privacy guarantees for DNN estimators trained using SGLD, while explicitly accounting for both network approximation error and the effect of injected noise in the excess-risk bound. Extensive simulation studies and analyses of two real datasets examine the finite-sample predictive performance and privacy-utility behavior of the proposed procedure.

An experiment can be informative about the parameters of a working model and uninformative about whether that model is wrong. Adaptive design can reinforce this mismatch by repeatedly selecting settings where an incorrect model fits. We bound the power of any valid adequacy test in terms of the experimenter's own sampling distribution, then distinguish information absent from the selected data from information lost by the detector. A computable diagnostic measures unresolved error relative to a declared expansion of the working model. It motivates a training-free rule that reserves part of the design budget for falsification. The empirical record is mixed: the independently specified primary study fails detection eligibility, and a later numerator diagnostic fails its improvement criteria. After the frozen rule fails on fresh continuous-null panels, a disclosed secondary horizon-calibrated revision detects 35.8% of misspecified campaigns versus 31.7% for a class-separation control at T = 240, but has no resolved advantage at T = 480. In a corrected 289-unit measured replay it improves on random design and trails class-discrimination EIG. A conditional futility bound explains why remaining time matters without proving the gate constant or its power. These results support separating parameter learning, model checking and detector performance; they do not establish a universally superior design policy or prospective scientific discovery.

Importance sampling is a building block of modern statistical computing, but the weighted estimate is prone to high or infinite variance. Moreover, its empirical standard error is often unreliable, leaving importance sampling without practical uncertainty quantification. Motivated by multilevel regression and poststratification (MRP) in survey sampling, we propose post-stratified importance sampling (PIS) for estimating target expectations from proposal draws. PIS reformulates arbitrarily high-dimensional weighting into two one-dimensional probabilistic modeling problems: regression for the conditional mean given the importance ratio and marginal density estimation for the importance ratio. Different modeling choices yield a family of PIS estimators, encompassing ordinary importance sampling as a special case. We develop a model-based implementation that combines one-dimensional Gaussian process regression for the conditional mean and score-informed heavy-tailed density estimation for the marginal law, using both the observed weights and the available density evaluations. We study the oracle gains of PIS and establish consistency under general and model-specific conditions. Simulations and applications to cross-validation demonstrate improvements in estimation accuracy and uncertainty quantification.

Making predictions in an unseen environment from data collected across multiple training environments is challenging. This challenge is particularly pronounced when the target-generating mechanism varies across environments. In this work, we introduce environment-label separation, a form of partial environmental invariance that decomposes the target conditional distribution into an environment-invariant component and a label-invariant component. We show that this partial invariance enables estimation of the target conditional distribution in an unseen environment when using labeled training data together with unlabeled test features. Within the structural causal model framework, we provide sufficient conditions for environment-label separation. Building on these conditions, we develop an algorithm, dIMP-Classifier, which jointly minimizes an estimate of the test risk and encourages invariance across environments. We further provide finite-sample guarantees that characterize the test risk in terms of deviations from invariance, estimation error, and sample complexity. Experiments on simulated data and the real-world light tunnel Causal Chamber data demonstrate that the proposed approach can perform under interventions on the target mechanism and consistently outperforms the considered standard and invariance-based baselines.