density estimation

This paper addresses the low accuracy of density estimation for spatially aggregated data (e.g., choropleth maps) caused by missing precise geographic coordinates. We propose a correlation-weighted density estimation method that integrates auxiliary information—such as satellite remote sensing data—into a measurement error model–enhanced kernel density estimation framework, where spatial correlation structures drive the weighting and fusion process to mitigate information loss from aggregation. Our key innovation lies in the first systematic modeling of spatial correlation between aggregated data and heterogeneous auxiliary variables, enabling end-to-end joint inference. Experiments on population density in Bavaria and hare distribution in Lower Saxony, Germany, demonstrate an average 28.6% reduction in estimation error compared to conventional methods. The approach significantly improves spatial density reconstruction accuracy for imprecisely geolocated data, with direct applicability to privacy-constrained domains such as demographic analysis and ecological monitoring.

Bayesian density and clustering modeling inherently face a tension between interpretability and flexibility: overparameterized models achieve high fit accuracy but lack transparency. This paper proposes a posterior-projection summarization framework that, grounded in decision theory, optimally compresses the high-dimensional posterior predictive distribution into a low-dimensional parametric density and clustering representation—preserving the original model’s fit fidelity while quantifying uncertainty. Our approach is the first to systematically integrate nonparametric modeling, posterior dimensionality reduction, and Bayesian uncertainty propagation, thereby jointly optimizing statistical interpretability and fitting fidelity. Experiments on synthetic and real-world datasets demonstrate that the resulting summaries enjoy theoretical guarantees and practical utility: fitting loss remains tightly controlled, and model transparency—as well as downstream interpretability for analysis—is substantially enhanced.

This work proposes a semiparametric density estimation approach that addresses the inefficiency of traditional kernel density estimation in high-dimensional settings or when the underlying distribution deviates substantially from assumed parametric forms. The method multiplies a parametric initial guess—such as a normal distribution—by a nonparametric kernel-based correction factor, thereby preserving robustness against departures from the parametric family while significantly enhancing local estimation accuracy. By integrating parametric priors with nonparametric adjustments, the framework incorporates a tailored bandwidth selection strategy and naturally extends to nonparametric regression. Theoretical analysis and extensive simulations demonstrate that, even when the true density markedly departs from normality, the proposed estimator consistently outperforms conventional kernel density estimators across various Gaussian mixture models, particularly excelling in high-dimensional scenarios.

This paper addresses the instability and curse of dimensionality in density-ratio estimation for high-dimensional causal inference. We propose a direct density-ratio learning method grounded in a unified framework integrating Bregman divergences and the Riesz representation theorem. Unlike conventional two-stage density estimation, our approach formulates the density ratio as a regression problem under the Riesz representation, using a Bregman divergence as the loss function, and incorporates classification-inspired objectives and data augmentation to mitigate estimation bias under unobserved intervention distributions. To our knowledge, this is the first work to theoretically unify Bregman divergences, Riesz regression, and probabilistic classification, while supporting diverse model classes—including gradient boosting, neural networks, and kernel methods. Extensive simulations demonstrate that different Bregman divergences and augmentation strategies enhance robustness. A publicly available Python package enables flexible implementation.

This study addresses the fundamental statistical problem of conditional density estimation by systematically comparing classical nonparametric approaches—such as single-index models, basis expansion methods (e.g., FlexCode), and DeepCDE—with modern generative models, including conditional GANs and conditional denoising diffusion probabilistic models. For the first time, these methods are evaluated within a unified and reproducible framework using metrics like mean squared error and Wasserstein distance to assess their accuracy, flexibility, and computational cost in estimating conditional means and standard deviations. The analysis clarifies the performance boundaries and practical applicability of each approach, offering both theoretical insights and actionable guidance for selecting appropriate methods in predictive modeling, uncertainty quantification, and probabilistic inference tasks.

This work addresses the challenge that traditional discrete probability distributions rely on manually derived analytical forms, hindering the automatic discovery of interpretable models. We propose Symbolic Density Estimation (SDE), a novel framework that, for the first time, integrates structural priors, evolutionary search, and validity-aware parameter inference to automatically discover closed-form probability mass functions within a structured symbolic space composed of elementary mathematical operations. SDE accommodates complex distributional features such as zero-inflation and finite mixtures. We introduce the first systematic benchmark dataset for this task and demonstrate that SDE accurately recovers all target distribution families. On real-world data, SDE discovers concise, interpretable mixture models that achieve superior goodness-of-fit compared to standard methods.

This study investigates the construction of Bayesian predictive inference methods with favorable asymptotic properties and offers a novel Bayesian interpretation of kernel density estimation. Focusing on two classes of predictive rules—classical kernel density estimators and their recursive variants—the work systematically examines their weak almost sure convergence under sequential observations by integrating nonparametric kernel methods, stochastic process theory, and weak convergence analysis. The analysis reveals that the classical estimator converges weakly almost surely to a probability measure with compact support, whereas its recursive counterpart converges to one with non-compact support. Beyond establishing weak almost sure convergence for both schemes, this research extends the theoretical foundations of kernel methods within Bayesian predictive inference and provides a fresh Bayesian perspective on kernel density estimation.

This work proposes the AI-Kolmogorov framework, which introduces symbolic regression systematically into probability density estimation to address the Symbolic Density Estimation (SymDE) problem. The method decomposes complex distributions through clustering or probabilistic graphical models in a multi-stage process, sequentially integrating support set estimation, nonparametric density estimation, and symbolic regression to construct interpretable analytic expressions of probability densities. Evaluated on synthetic mixture models, multivariate normal distributions, and exotic distributions from high-energy physics, the framework successfully recovers or uncovers their underlying mathematical structures, enabling both interpretable modeling and structural discovery for complex probability distributions.

This study addresses the challenge of enhancing density estimation performance by integrating the structural advantages of parametric models while preserving the flexibility of nonparametric methods. The authors propose a locally parametrized nonparametric density estimator that, for each point \(x\), estimates an optimal local parameter \(\hat{\theta}(x)\) via kernel-smoothed likelihood, yielding an estimator of the form \(f(x, \hat{\theta}(x))\). This approach achieves near-full-likelihood efficiency under correct model specification and retains nonparametric robustness under misspecification, effectively serving as a semiparametric realization of higher-order kernel methods. Theoretical analysis and empirical experiments demonstrate that the proposed estimator exhibits variance comparable to classical kernel density estimation but substantially reduced bias, leading to significantly improved accuracy in neighborhoods of well-specified parametric models.