The unfolding problem in particle physics is to make inferences about the true particle spectrum based on smeared observations from a detector. This is an ill-posed inverse problem, where small changes in the smeared distribution can lead to large fluctuations in the unfolded distribution. The forward operator is the response matrix which models the detector response. In practice, the forward operator is rarely known analytically and is instead estimated using Monte Carlo simulation. This raises the question of how to best estimate the response matrix and what impact this estimation has on the unfolded solutions. In most analyses at the LHC, response matrix estimation is done by binning the true and smeared events and counting the propagation of events between the bins. However, this approach can result in a noisy estimate of the response matrix, especially with a small Monte Carlo sample size. Unexpectedly, we also find that the noise in the estimated response matrix can inadvertently regularize the problem. As an alternative, we propose to estimate the response matrix through the use of conditional density estimation of the response kernel in the unbinned setting followed by binning this estimator. Using simulation studies, we investigate the performance of the two approaches.

Livestock depredation by large carnivores is a global conservation challenge that fuels human-carnivore conflict and hinders coexistence with agropastoral communities. Understanding carnivore activity patterns and implementing temporal segregation are key to mitigating conflict risks. In this study, we have compiled nearly a decade (2014 to 2024) of infrared camera monitoring data from a representative region of China (the Sanjiangyuan Region), where human-large carnivore conflicts are particularly pronounced. We employed kernel and conditional circular kernel density estimation to analyze the diel activity patterns of four large carnivores and to identify seasonal variations between cold and warm periods, thereby enabling the identification of high-risk intervals for potential human-large carnivore conflicts. Our analysis showed that all four large carnivores were primarily nocturnal with species-specific peak activity times. Notably, the wolf (Canis lupus) exhibited distinct seasonal diel activity patterns, unlike the other species. The temporal risk assessment of livestock predation identified species-specific high-risk windows: for snow leopards (Panthera uncia), the high-risk periods are from 02:35 to 06:41 and from 16:00 to 21:08; for wolves, the high-risk periods are from 06:24 to 11:33 and from 16:12 to 21:24; for Eurasian lynx (Lynx lynx), the high-risk periods are from 01:42 to 06:46 and from 15:57 to 19:32; and for brown bears (Ursus arctos), the high-risk periods for intruding into pastoral dwellings, causing property damage, and posing risks to human safety are from 20:42 to 02:36. Our study established temporal management frameworks in SR for human-carnivore conflict risk mitigation, providing transferable insights for global human-wildlife conflict resolution.

Bayes factors often require numerical estimation because closed-form solutions are unavailable. In six simulation studies, we explored the reliability, bias, and computational cost of two easy-to-use and broadly applicable methods: bridge sampling and the Savage–Dickey density ratios, based on Gaussian, logspline, and spline-smoothed kernel density approximations of the posterior distribution, as well as conditional marginal density estimation. In generalized linear mixed effect models for normally and binomially distributed data, we explore the effects of the (1) number of MCMC samples from the posterior, (2) size of effects or magnitude of the Bayes factor, (3) number of participants, and (4) number of model parameters. Our findings suggest that, with enough MCMC samples, both methods yield reliable and accurate estimates across a wide range of conditions. However, with many model parameters, bridge sampling becomes computationally expensive and can be unreliable. In contrast, the Savage–Dickey density ratio scales well, remaining computationally efficient and reliable, even with many model parameters. However, Savage–Dickey density ratio requires careful consideration of posterior density estimation to mitigate bias while limiting the variability of Bayes factor estimates. We provide practical recommendations to guide researchers in selecting the most suitable estimation method for their applications.

Extracting low-dimensional summary statistics from large datasets is essential for efficient (likelihood-free) inference. We characterize three different classes of summaries and demonstrate their importance for correctly analyzing dimensionality reduction algorithms. We demonstrate that minimizing the expected posterior entropy (EPE) under the prior predictive distribution of the model provides a unifying principle that subsumes many existing methods; they are shown to be equivalent to, or special or limiting cases of, minimizing the EPE. We offer a unifying framework for obtaining informative summaries and propose a practical method using conditional density estimation to learn high-fidelity summaries automatically. We evaluate this approach on diverse problems, including a challenging benchmark model with a multi-modal posterior, a population genetics model, and a dynamic network model of growing trees. The results show that EPE-minimizing summaries can lead to posterior inference that is competitive with, and in some cases superior to, dedicated likelihood-based approaches, providing a powerful and general tool for practitioners.

. Considering a continuous random variable Y together with a continuous random vector X, I propose a nonparametric estimator f ̂ ( ⋅ | x ) for the conditional density of Y given X = x. This estimator takes the form of an exponential series whose coefficients θ ̂ x = ( θ ̂ x , 1 , … , θ ̂ x , J ) are the solution of a system of nonlinear equations that depends on an estimator of the conditional expectation E [ ϕ ( Y ) | X = x ] , where ϕ is a J-dimensional vector of basis functions. The distinguishing feature of the proposed estimator is that E [ ϕ ( Y ) | X = x ] is estimated by generalized random forest (Athey, Tibshirani, and Wager, Annals of Statistics, 2019), targeting the heterogeneity of θ ̂ x across x. I show that f ̂ ( ⋅ | x ) is uniformly consistent and asymptotically normal, allowing J→∞. I also provide a standard error formula to construct asymptotically valid confidence intervals. Results from Monte Carlo experiments are provided, and an empirical application to U.S. timber auction data illustrates how the proposed estimator can be used to estimate the conditional density of bids given auctioned object characteristics.

Abstract Key science questions, such as galaxy distance estimation and weather forecasting, often require knowing the full predictive distribution of a target variable y given complex inputs x . Despite recent advances in machine learning and physics-based models, it remains challenging to assess whether an initial model is calibrated for all x , and when needed, to reshape the densities of y toward ``instance-wise'' calibration. This paper introduces the LADaR (Local Amortized Diagnostics and Reshaping of Conditional Densities) framework and proposes a new computationally efficient algorithm (Cal-PIT) that produces interpretable local diagnostics and provides a mechanism for adjusting conditional density estimates (CDEs). Cal-PIT learns a single interpretable local probability--probability map from calibration data that identifies where and how the initial model is miscalibrated across feature space, which can be used to morph CDEs such that they are well-calibrated. We illustrate the LADaR framework on synthetic examples, including probabilistic forecasting from image sequences, akin to predicting storm wind speed from satellite imagery. Our main science application involves estimating the probability density functions of galaxy distances given photometric data, where Cal-PIT achieves better instance-wise calibration than all 11 other literature methods in a benchmark data challenge, demonstrating its utility for next-generation cosmological analyses. (Code available as a Python package here: https://github.com/lee-group-cmu/Cal-PIT)

Conditional generative models represent a significant advancement in machine learning, enabling controlled data synthesis by incorporating additional information into the generation process. In this work, we introduce a novel Schrödinger bridge-based deep generative method for learning conditional distributions. Our approach begins with a unit-time diffusion process governed by a stochastic differential equation (SDE) that evolves a fixed point at time t = 0 into a desired target conditional distribution at t = 1 . For effective implementation, we discretize the SDE using the Euler–Maruyama method, estimating the drift term nonparametrically with a deep neural network. We apply our method to both low-dimensional and high-dimensional conditional generation tasks. Numerical studies show that, although our method does not directly provide conditional density estimation, the samples generated exhibit higher quality than those from several existing methods. Furthermore, the generated samples can be effectively used to estimate the conditional density and related statistical quantities, such as the conditional mean and conditional standard deviation.

Abstract We present Mantis Shrimp , a multisurvey deep learning model for photometric redshift estimation that fuses ultraviolet (Galaxy Evolution Explorer), optical (PanSTARRS), and infrared (UnWISE) imagery. Machine learning is now an established approach for photometric redshift estimation, with generally acknowledged higher performance in areas with a high density of spectroscopically identified galaxies over template-based methods. Multiple works have shown that image-based convolutional neural networks can outperform tabular-based color/magnitude models. In comparison to tabular models, image models have additional design complexities: It is largely unknown how to fuse inputs from different instruments that have different resolutions or noise properties. The Mantis Shrimp model estimates the conditional density estimate of redshift using cutout images. The density estimates are well calibrated, and the point estimates perform well in the distribution of available spectroscopically confirmed galaxies with (bias = 1e-2), scatter (NMAD = 2.44e-2), and catastrophic outlier rate ( η >0.15 = 4.51%). We find that early-fusion approaches (e.g., resampling and stacking images from different instruments) match the performance of late-fusion approaches (e.g., concatenating latent space representations), so that the design choice ultimately is left to the user. Finally, we study how the model learns to use information across bands, finding evidence that our model successfully incorporates information from all surveys. The applicability of our model to the analysis of large populations of galaxies is limited by the speed and ease of downloading and preparing cutouts from external servers; however, our model could be useful in smaller studies such as in generating priors over redshift for stellar population synthesis.

ABSTRACTThis paper proposes the Deep Symbolic Learning (DSL) model, a deep learning‐based framework for robust regression, specifically designed when both the response and predictors are histogram‐valued variables. DSL utilizes cumulative distribution functions (CDFs) of covariate histograms within a one‐dimensional convolutional neural network (1D‐CNN) to transform the conditional density estimation problem into a multi‐class classification task, optimized using the joint binary cross‐entropy (JBCE) loss function. Extensive simulations and real‐world applications, including air quality, traffic volume, and climate data, demonstrate that the DSL model outperforms existing methods across three key evaluation metrics: CDF distance, empirical coverage of the 90% prediction interval, and average quantile loss. This work contributes to the field of symbolic data analysis and conditional density estimation.

This article introduces a unified approach to estimating the mutual density ratio, defined as the ratio between the joint density function and the product of the individual marginal density functions of two random vectors. It serves as a fundamental measure for quantifying the relationship between two random vectors. Our method uses the Bregman divergence to construct the objective function and leverages deep neural networks to approximate the logarithm of the mutual density ratio. We establish a non-asymptotic error bound for our estimator, achieving the optimal minimax rate of convergence under a bounded support condition. Additionally, our estimator mitigates the curse of dimensionality when the distribution is supported on a lower-dimensional manifold. We extend our results to overparameterized neural networks and the case with unbounded support. Applications of our method include conditional probability density estimation, mutual information estimation, and independence testing. Simulation studies and real data examples demonstrate the effectiveness of our approach. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

CPD is a statistical technique that finds the change points in data sequences where the statistical properties of the data have shifted. This technique has valuable applications in the field of economics as well as finance, and public health. In this study a CPD methodology is developed by proposing an enhanced three-step nonparametric approach based on the existing two-step method. The proposed framework couples Kernel Conditional Density Estimation with Fourier features and machine learning techniques for the precise identification and classification of change points. Data preprocessing for smoothness and noise reduction will be included, followed by KCDE-F for conditional density estimation, and then a machine-learning classifier refines the sensitivity and specificity of the detected change points. This paper identifies critical change points in Nigerian inflation dynamics using data from 2019 to 2023. The result shows that the developed the three-step procedure for change point detection presented here is not only also capable in change point detection but also in the estimation of structural breaks in time-series data. The wide applicability of this methodology is envisioned to extend beyond economics into other domains where the need for change point detection is compelling.

In this paper we present a multi-stage stochastic optimization model to solve an inventory routing problem for the collection of recyclable municipal waste. The objective is the maximization of the total expected profit of the waste collection company. The decisions are related to the selection of the bins to be visited and the corresponding routing plan in a predefined time horizon. Stochasticity in waste accumulation is modeled through scenario trees generated via conditional density estimation and dynamic stochastic approximation techniques. The proposed formulation is solved through a rolling horizon approach, providing a rigorous worst-case analysis on its performance. Extensive computational experiments are carried out on small- and large-sized instances based on real data provided by a large Portuguese waste collection company. The impact of stochasticity on waste generation is examined through stochastic measures, showing the importance of adopting a stochastic model over a deterministic formulation when addressing a waste collection problem. The performance of the rolling horizon approach is evaluated, demonstrating that this heuristic provides cost-effective solutions in short computational time. Managerial insights related to different geographical configurations of the instances and varying levels of uncertainty are finally discussed.

A precise measurement of photometric redshifts (photo-z) is crucial for the success of modern photometric galaxy surveys. Machine learning (ML) methods show great promise in this context, but suffer from covariate shift in training sets due to selection bias where interesting sources, e.g., high redshift objects, are underrepresented, and the corresponding ML models exhibit poor generalisation properties. We present an application of the StratLearn method to the estimation of photo-z (StratLearn-z), validating against simulations where we enforce the presence of covariate shift to different degrees. StratLearn is a statistically principled approach which relies on splitting the combined source and target datasets into strata, based on estimated propensity scores. The latter is the probability for an object in the dataset to be in the source set, given its observed covariates. After stratification, two conditional density estimators are fit separately within each stratum, and then combined via a weighted average. We benchmark our results against the GPz algorithm, quantifying the performance of the two algorithms with a set of metrics. Our results show that the StratLearn-z metrics are only marginally affected by the presence of covariate shift, while GPz shows a significant degradation of performance, specifically concerning the photo-z prediction for fainter objects for which there is little training data. In particular, for the strongest covariate shift scenario considered, StratLearn-z yields a reduced fraction of catastrophic errors, a factor of 2 improvement for the RMSE as well as one order of magnitude improvement on the bias. We also assess the quality of the predicted conditional redshift estimates using the probability integral transform (PIT) and the continuous rank probability score (CRPS). The PIT for StratLearn-z indicates that predictions are well-centered around the true redshift value, if conservative in their variance; the CRPS shows marked improvement at high redshifts when compared with GPz. Our julia implementation of the method, StratLearn-z, is publicly available at .

In this paper, our goal is to generate synthetic data for heterogeneous (mixed-type) tabular datasets with high machine learning utility (MLu). Since the MLu performance depends on accurately approximating the conditional distributions, we focus on devising a synthetic data generation method based on conditional distribution estimation. We introduce MaCoDE by redefining the consecutive multi-class classification task of Masked Language Modeling (MLM) as histogram-based non-parametric conditional density estimation. Our approach enables the estimation of conditional densities across arbitrary combinations of target and conditional variables. We bridge the theoretical gap between distributional learning and MLM by demonstrating that minimizing the orderless multi-class classification loss leads to minimizing the total variation distance between conditional distributions. To validate our proposed model, we evaluate its performance in synthetic data generation across 10 real-world datasets, demonstrating its ability to adjust data privacy levels easily without re-training. Additionally, since masked input tokens in MLM are analogous to missing data, we further assess its effectiveness in handling training datasets with missing values, including multiple imputations of the missing entries.

In financial econometrics, it is empirically challenging to test the predictability of lagged predictors with varying levels of persistence in predictive quantile regression. A recent double-weighted method developed by Cai, Chen, and Liao (2023) has demonstrated desirable local power properties for both non stationary and stationary predictors. In this article, we propose a strategy to improve the construction of the auxiliary variables in the double-weighted method. This improvement makes it applicable to a broader range of persistent types in empirical analysis. Furthermore, we propose a random weighted bootstrap procedure to address the challenges involved in conditional density estimation. Simulation results demonstrate the effectiveness of the proposed test in correcting size distortion at the lower and upper quantiles. Finally, we apply the proposed test to reassess the predictability of macroeconomic and financial predictors on stock returns across different quantile levels, finding fewer significant predictors at the tails compared to Cai, Chen, and Liao (2023). Our results highlight that this test serves as a more conservative inference tool for practitioners evaluating the predictability of financial returns.

Understanding the efficiency of agricultural eco-product value realization is critical for sustainable development and regional equity. Here, we present a comprehensive analysis of the spatiotemporal patterns and regional disparities in the value realization efficiency of agricultural ecological products across China’s 31 provinces from 2010 to 2022. Utilizing an advanced Super-NSBM model, we quantify three dimensions of efficiency: overall value realization, economic value conversion, and social welfare value realization. Spatial mapping and dynamic evolution analysis are conducted through Dagum Gini coefficient decomposition and conditional kernel density estimation. Our results reveal three key insights: (1) China’s agricultural eco-product value realization efficiency remains suboptimal, with a gradual upward trend. Economic value conversion outperforms social welfare value realization, which exhibits significant regional heterogeneity. A distinct east–west gradient is observed, with Western regions demonstrating notable progress despite initial inefficiencies. (2) Inter-regional disparities are narrowing, particularly between Eastern and Central regions. While polarization in Northeast China has diminished, Western regions show widening efficiency gaps and emerging polarization trends. (3) Regional differences are predominantly driven by inter-group disparities, with Eastern China exhibiting the lowest intra-regional variability. Cross-regional differences follow a U-shaped trajectory, decreasing initially before rebounding in recent years. These findings provide a robust empirical foundation for optimizing regional strategies in ecological product value conversion and offer critical insights for addressing spatial inequities in sustainable agricultural development.

Conditional cumulative distribution functions (CDFs), conditional probability density functions (PDFs), and derivatives thereof, are important parameters of interest in statistics, econometrics, and other data science disciplines. The package lpcde implements new estimation and inference methods for conditional CDFs, conditional PDFs, and derivatives thereof, employing the kernelbased local polynomial smoothing approach introduced in Cattaneo et al. (2024a).

Sequential neural posterior estimation (SNPE) techniques have been recently proposed for dealing with simulation-based models with intractable likelihoods. Unlike approximate Bayesian computation, SNPE techniques learn the posterior from sequential simulation using neural network-based conditional density estimators by minimizing a specific loss function. The SNPE method proposed by Lueckmann et al. (NeurIPS 2017) used a calibration kernel to boost the sample weights around the observed data, resulting in a concentrated loss function. However, the use of calibration kernels may increase the variances of both the empirical loss and its gradient, making the training inefficient. To improve the stability of SNPE, this paper proposes to use an adaptive calibration kernel and several variance reduction techniques. The proposed method greatly speeds up the process of training and provides a better approximation of the posterior than the original SNPE method and some existing competitors as confirmed by numerical experiments. We also managed to demonstrate the superiority of the proposed method for a high-dimensional model with a real-world dataset.

Abstract A methodology is proposed for the determination of factor-dependent bandwidths for the kernel-based estimation of the conditional density $\rho (x|z)$ underlying a set of observations. The adaptive determination of the bandwidths is based on a $z$-dependent effective number of samples and variance. The procedure extends to categorical factors, where a non-trivial ‘bandwidth’ can be designed that optimally uses across-class information while capturing class-specific traits. A hierarchy of algorithms is developed, and their effectiveness is demonstrated on synthetic and real-world data.

The beta distribution serves as a canonical tool for modeling probabilities in statistics and machine learning. However, there is limited work on flexible and computationally convenient stochastic process extensions for modeling dependent random probabilities. We propose a novel stochastic process called the logistic-beta process, whose logistic transformation yields a stochastic process with common beta marginals. Logistic-beta processes can model dependence on both discrete and continuous domains, such as space or time, and have a flexible dependence structure through correlation kernels. Moreover, its normal variance-mean mixture representation leads to effective posterior inference algorithms. We show how the proposed logistic-beta process can be used to design computationally tractable dependent Bayesian nonparametric models, including dependent Dirichlet processes and extensions. We illustrate the benefits through nonparametric binary regression and conditional density estimation examples, both in simulation studies and in a pregnancy outcome application.