Distributional changes can be invisible to means and covariances yet appear in skewness, asymmetric interactions, or other third-order structure. We develop a nonparametric change-point method that retains every degree-at-most-three coefficient of a density while avoiding direct density estimation. For observations in $[-1,1]^d$, we construct a symmetric order-three Legendre feature tensor $H_3(X)\in\Sym^3(\R^{d+1})$ such that $A(f)=\E_fH_3(X)$ is an exact isometric encoding of the degree-three density projection: $\|A(f)-A(g)\|_{\F}=\|P_3(f-g)\|_{L^2}$. Instead, fixed tensor contractions are degree-three polynomial chaoses with $ψ_{2/3}$ tails. The two terms have the characteristic order-three tensor scaling and match the powers in sharp concentration results for simple random tensors. For a coordinate-orthogonal specialization, the bound improves to $\sqrt{\log d}$ and enables a prefix-sum implementation in hundreds of dimensions. We derive the exact population tent shape and localization margin, introduce a seeded shortest-interval algorithm with a padded local recentering step, and prove exact recovery by induction: null recursive segments remain inactive, every undetected change retains a balanced isolating interval, and the shortest active seed contains exactly one change before recentering. A two-way cross-fitted scalar refinement attains $O_{\Pp}(κ^{-2})$ localization in the small-jump regime, matching a Le Cam lower bound on a pure cubic family whose degree-two projection jump is exactly zero. Reproducible experiments at $d\in\{20,50,100,200\}$ and a three-change $d=100$ sequence demonstrate the intended high-dimensional regime without materializing a $(d+1)^3$ tensor.
We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.
Many functional data analyses reduce random functions to scalar summaries or conditional mean curves. This is limiting when we wish to understand how covariates affect the distribution of entire functional responses, including their shape, timing, or variability. We study the problem of estimating conditional laws of functional outcomes and show that these objects can be estimated and evaluated in a practical nonparametric framework. To do this, we introduce functional distributional random forests, which estimate each conditional law as a covariate-dependent distribution over sampled functions by training a random forest to minimize a kernel-based maximum mean discrepancy within the leaf nodes of the decision tree. This supports inference on arbitrary functionals of the conditional distribution while keeping predictive samples tied to realistic curves. We consider a variety of kernels defined on function spaces, including Sobolev and operator-induced kernels. We also provide conditions for consistency of our estimator and develop scoring rules for comparing it to baseline estimators. In simulations, our method recovers distributional changes that are missed by baseline methods. In an application to NHANES accelerometer data, it identifies interesting covariate-associated changes in both median activity profiles and predictive dispersion.
We study the problem of sequential change detection over a general class of probability distributions ($\mathcal P$), where both the pre-change and post-change distributions are unknown and belong to $\mathcal P$. We do not assume a pre-specified partition of $\mathcal P$ into pre- and post-change families. We propose a general class of sequential change detectors obtained by aggregating point-null e-processes over possible changepoints and taking an infimum over candidate no-change distributions. The weights in the aggregation scheme determine whether they attain average run length (ARL) control and probability-of-false-alarm (PFA) control. Under suitable assumptions, we prove that our methods achieve first-order asymptotically optimal detection delay. Concrete examples include sub-Gaussian and bounded mean changes, Gaussian mean changes with unknown variance, as well as changes in Markov transition matrices.