Posterior Samples Research Articles

Semantic information guided diffusion posterior sampling for remote sensing image fusion

The task of image fusion for optical images and SAR images is to integrate valuable information from source images. Recently, owing to powerful generation, diffusion models, e.g., diffusion denoising probabilistic model and score-based diffusion model, are flourished in image processing, and there are some effective attempts in image fusion by scholars’ progressive explorations. However, the diffusion models for image fusion suffer from inevitable SAR speckle that seriously shelters from effective information in the same location of optical image. Besides, these methods integrate pixel-level features without information for high-level tasks, e.g., target detection and image classification, which leads fused images are insufficient and their application accuracies are low, for high-level tasks. To tackle these hurdles, we propose the semantic information guided diffusion posterior sampling for image fusion. Firstly, we employ the SAR-BM3D as preprocessing to despeckle. Then, the sampling model is established with fidelity, regularization and semantic information guidance term. The first two terms are obtained by the variational diffusion method via variational inference and first-order stochastic optimization. The last term is served by cross entropy loss between annotation and classification result from FLCNet we design. Finally, the experiments validate the feasibility and superiority of the proposed method on WHU-OPT-SAR dataset and DDHRNet dataset.

Variational inference for acceleration of SN Ia photometric distance estimation with BayeSN

Abstract Type Ia supernovae (SNe Ia) are standarizable candles whose observed light curves can be used to infer their distances, which can in turn be used in cosmological analyses. As the quantity of observed SNe Ia grows with current and upcoming surveys, increasingly scalable analyses are necessary to take full advantage of these new datasets for precise estimation of cosmological parameters. Bayesian inference methods enable fitting SN Ia light curves with robust uncertainty quantification, but traditional posterior sampling using Markov Chain Monte Carlo (MCMC) is computationally expensive. We present an implementation of variational inference (VI) to accelerate the fitting of SN Ia light curves using the BayeSN hierarchical Bayesian model for time-varying SN Ia spectral energy distributions (SEDs). We demonstrate and evaluate its performance on both simulated light curves and data from the Foundation Supernova Survey with two different forms of surrogate posterior – a multivariate normal and a custom multivariate zero-lower-truncated normal distribution – and compare them with the Laplace Approximation and full MCMC analysis. To validate of our variational approximation, we calculate the pareto-smoothed importance sampling (PSIS) diagnostic, and perform variational simulation-based calibration (VSBC). The VI approximation achieves similar results to MCMC but with an order-of-magnitude speedup for the inference of the photometric distance moduli. Overall, we show that VI is a promising method for scalable parameter inference that enables analysis of larger datasets for precision cosmology.

Pop-cosmos: Scaleable Inference of Galaxy Properties and Redshifts with a Data-driven Population Model

We present an efficient Bayesian method for estimating individual photometric redshifts and galaxy properties under a pretrained population model (pop-cosmos) that was calibrated using purely photometric data. This model specifies a prior distribution over 16 stellar population synthesis (SPS) parameters using a score-based diffusion model, and includes a data model with detailed treatment of nebular emission. We use a GPU-accelerated affine-invariant ensemble sampler to achieve fast posterior sampling under this model for 292,300 individual galaxies in the COSMOS2020 catalog, leveraging a neural network emulator (Speculator) to speed up the SPS calculations. We apply both the pop-cosmos population model and a baseline prior inspired by Prospector-α, and compare these results to published COSMOS2020 redshift estimates from the widely used EAZY and LePhare codes. For the ∼12,000 galaxies with spectroscopic redshifts, we find that pop-cosmos yields redshift estimates that have minimal bias (∼10−4), high accuracy (σ MAD = 7 × 10−3), and a low outlier rate (1.6%). We show that the pop-cosmos population model generalizes well to galaxies fainter than its r < 25 mag training set. The sample we have analyzed is ≳3× larger than has previously been possible via posterior sampling with a full SPS model, with average throughput of 15 GPU-sec per galaxy under the pop-cosmos prior, and 0.6 GPU-sec per galaxy under the Prospector prior. This paves the way for principled modeling of the huge catalogs expected from upcoming Stage IV galaxy surveys.

Model uncertainty quantification of a degradation model of miter gates using normalizing flow-based likelihood-free inference

Degradation modeling is essential for failure prognostics of miter gates and subsequent risk-informed maintenance or operational decision-making. Typically, degradation models are formulated based on certain assumptions and simplifications and may not fully capture the complexity of underlying degradation mechanisms. Even minor errors in such models may lead to significant discrepancies in failure prognostics due to error accumulation over time. Aiming to improve the accuracy of the degradation model and ensure reliable failure prognostics, this article proposes a degradation model updating framework for miter gates using normalizing flow-based likelihood-free inference, accounting for both model parameter uncertainty and model form uncertainty (i.e., model bias). The proposed work consists of two phases: the offline training phase and the online updating phase. During the offline training phase, the unknown model bias term is modeled as a random noise with uncertain standard deviation. Synthetic data are then generated using a physical model based on prior knowledge of the uncertain model parameters and model bias. The synthetic data are utilized to train an inference model, which has a summary network for data compression and a conditional invertible neural network for parameter estimation. In the online updating phase, the trained inference model uses strain observations to obtain posterior samples. The Gaussian mixture model and dual particle filter techniques are utilized to recursively update posterior samples, thereby improving the estimation accuracy. Subsequently, model bias is inversely estimated using the degradation model, which uses the updated model parameters and the gap length from the inference model. Following that, a regression model is constructed to correct the biases inherent in the simplified degradation model. This process is implemented iteratively over time to perform continuous model updating and failure prognostics. Results of a case study demonstrate the efficacy of the proposed framework in reducing both model parameter uncertainty and recovering model biases and thereby improving the accuracy of failure prognostics of miter gates.

Constrained cosmological simulations of the Local Group using Bayesian hierarchical field-level inference

We present a novel approach based on Bayesian field-level inference that provides representative initial conditions for simulation of the Local Group (LG) of galaxies and its neighbourhood, constrained by present-day observations. We extended the Bayesian Origin Reconstruction from Galaxies ( algorithm with a multi-resolution approach, allowing us to reach the smaller scales needed to apply the constraints. Our data model simultaneously accounts for observations of mass tracers within the dark haloes of the Milky Way (MW) and M31, for their observed separation and relative velocity, and for the quiet surrounding Hubble flow, represented by the positions and velocities of 31 galaxies at distances between one and four megaparsec. Our approach delivers representative posterior samples of realisations that are statistically and simultaneously consistent with all of these observations, leading to significantly tighter mass constraints than found if the individual datasets are considered separately. In particular, we estimate the virial masses of the MW and M31 to be $ (M_ 200c /M_ and $12.33 respectively, their sum to be $ 200c /M_ and the enclosed mass within spheres of radius $R$ to be $ (M(R)/M_ and $12.96 for $R=1$ Mpc and $3$ Mpc, respectively. The M31-MW orbit is nearly radial for most of our realisations, and most of them feature a dark matter sheet aligning approximately with the supergalactic plane, despite the surrounding density field not being used explicitly as a constraint. High-resolution, high-fidelity resimulations from initial conditions identified using the approximate simulations of our inference scheme continue to satisfy the observational constraints, demonstrating a route to future high-resolution, full-physics simulations of ensembles of LG look-alikes, all of which closely mirror the observed properties of the real system and its immediate environment.

Bayesian material flow analysis for systems with multiple levels of disaggregation and high dimensional data

AbstractMaterial flow analysis (MFA) is used to quantify and understand the life cycles of materials from production to end of use, which enables environmental, social, and economic impacts and interventions. MFA is challenging as available data are often limited and uncertain, leading to an under‐determined system with an infinite number of possible stocks and flows values. Bayesian statistics is an effective way to address these challenges by principally incorporating domain knowledge, quantifying uncertainty in the data, and providing probabilities associated with model solutions. This paper presents a novel MFA methodology under the Bayesian framework. By relaxing the mass balance constraints, we improve the computational scalability and reliability of the posterior samples compared to existing Bayesian MFA methods. We propose a mass‐based, child and parent process framework to model systems with disaggregated processes and flows. We show posterior predictive checks can be used to identify inconsistencies in the data and aid noise and hyperparameter selection. The proposed approach is demonstrated in case studies, including a global aluminum cycle with significant disaggregation, under weakly informative priors and significant data gaps to investigate the feasibility of Bayesian MFA. We illustrate that just a weakly informative prior can greatly improve the performance of Bayesian methods, for both estimation accuracy and uncertainty quantification.

Noise suppression in photon-counting computed tomography using unsupervised Poisson flow generative models

Deep learning (DL) has proven to be important for computed tomography (CT) image denoising. However, such models are usually trained under supervision, requiring paired data that may be difficult to obtain in practice. Diffusion models offer unsupervised means of solving a wide range of inverse problems via posterior sampling. In particular, using the estimated unconditional score function of the prior distribution, obtained via unsupervised learning, one can sample from the desired posterior via hijacking and regularization. However, due to the iterative solvers used, the number of function evaluations (NFE) required may be orders of magnitudes larger than for single-step samplers. In this paper, we present a novel image denoising technique for photon-counting CT by extending the unsupervised approach to inverse problem solving to the case of Poisson flow generative models (PFGM)++. By hijacking and regularizing the sampling process we obtain a single-step sampler, that is NFE = 1. Our proposed method incorporates posterior sampling using diffusion models as a special case. We demonstrate that the added robustness afforded by the PFGM++ framework yields significant performance gains. Our results indicate competitive performance compared to popular supervised, including state-of-the-art diffusion-style models with NFE = 1 (consistency models), unsupervised, and non-DL-based image denoising techniques, on clinical low-dose CT data and clinical images from a prototype photon-counting CT system developed by GE HealthCare.

Exploring Estimation Procedures for Reducing Dimensionality in Psychological Network Modeling.

To understand psychological data, it is crucial to examine the structure and dimensions of variables. In this study, we examined alternative estimation algorithms to the conventional GLASSO-based exploratory graph analysis (EGA) in network psychometric models to assess the dimensionality structure of the data. The study applied Bayesian conjugate or Jeffreys' priors to estimate the graphical structure and then used the Louvain community detection algorithm to partition and identify groups of nodes, which allowed the detection of the multi- and unidimensional factor structures. Monte Carlo simulations suggested that the two alternative Bayesian estimation algorithms had comparable or better performance when compared with the GLASSO-based EGA and conventional parallel analysis (PA). When estimating the multidimensional factor structure, the analytically based method (i.e., EGA.analytical) showed the best balance between accuracy and mean biased/absolute errors, with the highest accuracy tied with EGA but with the smallest errors. The sampling-based approach (EGA.sampling) yielded higher accuracy and smaller errors than PA; lower accuracy but also lower errors than EGA. Techniques from the two algorithms had more stable performance than EGA and PA across different data conditions. When estimating the unidimensional structure, the PA technique performed the best, followed closely by EGA, and then EGA.analytical and EGA.sampling. Furthermore, the study explored four full Bayesian techniques to assess dimensionality in network psychometrics. The results demonstrated superior performance when using Bayesian hypothesis testing or deriving posterior samples of graph structures under small sample sizes. The study recommends using the EGA.analytical technique as an alternative tool for assessing dimensionality and advocates for the usefulness of the EGA.sampling method as a valuable alternate technique. The findings also indicated encouraging results for extending the regularization-based network modeling EGA method to the Bayesian framework and discussed future directions in this line of work. The study illustrated the practical application of the techniques to two empirical examples in R.

Regional Shopping Objectives in British Grocery Retail Transactions Using Segmented Topic Models

ABSTRACTUnderstanding the customer behaviours behind transactional data has high commercial value in the grocery retail industry. Customers generate millions of transactions every day, choosing and buying products to satisfy specific shopping needs. Product availability may vary geographically due to local demand and local supply, thus driving the importance of analysing transactions within their corresponding store and regional context. Topic models provide a powerful tool in the analysis of transactional data, identifying topics that display frequently‐bought‐together products and summarising transactions as mixtures of topics. We use the segmented topic model (STM) to capture customer behaviours that are nested within stores. STM not only provides topics and transaction summaries but also topical summaries at the store level that can be used to identify regional topics. We summarise the posterior distribution of STM by post‐processing multiple posterior samples and selecting semantic modes represented as recurrent topics, and employ Gaussian process regression to model topic prevalence across British territory while accounting for spatial autocorrelation. We implement our methods on a dataset of transactional data from a major UK grocery retailer and demonstrate that shopping behaviours may vary regionally and nearby stores tend to exhibit similar regional demand.

Estimation and Analysis of Trigonometric Models under Bayesian Approach

In this study, we explore three innovative trigonometric models within the Bayesian framework, utilizing the inverse Weibull distribution as our foundation. These models—namely the Sine inverse Weibull, Cosine inverse Weibull, and Tan inverse Weibull—are crafted from distinct distribution families. We employ both maximum likelihood estimation and Markov Chain Monte Carlo (MCMC) simulation techniques to estimate parameters, drawing upon a comprehensive dataset. By scrutinizing posterior samples numerically and graphically, we evaluate the efficacy of our models, generating Bayes estimates for parameters, examining reliability and hazard functions, and establishing credible intervals. Furthermore, we assess the predictive capacity of all three models through posterior predictive checks. We also conduct comparative analyses, pitting our models against competing ones using real-world data. Notably, our results reveal that the proposed trio of models exhibit strikingly similar performance in terms of fitting the data.

The future of cosmological likelihood-based inference: accelerated high-dimensional parameter estimation and model comparison

We advocate for a new paradigm of cosmological likelihood-based inference, leveraging recent developments in machine learning and its underlying technology, to accelerate Bayesian inference in high-dimensional settings. Specifically, we combine (i) emulation, where a machine learning model is trained to mimic cosmological observables, e.g. CosmoPower-JAX; (ii) differentiable and probabilistic programming, e.g. JAX and NumPyro, respectively; (iii) scalable Markov chain Monte Carlo (MCMC) sampling techniques that exploit gradients, e.g. Hamiltonian Monte Carlo; and (iv) decoupled and scalable Bayesian model selection techniques that compute the Bayesian evidence purely from posterior samples, e.g. the learned harmonic mean implemented in harmonic. This paradigm allows us to carry out a complete Bayesian analysis, including both parameter estimation and model selection, in a fraction of the time of traditional approaches. First, we demonstrate the application of this paradigm on a simulated cosmic shear analysis for a Stage IV survey in 37- and 39-dimensional parameter spaces, comparing ΛCDM and a dynamical dark energy model ( w0waCDM). We recover posterior contours and evidence estimates that are in excellent agreement with those computed by the traditional nested sampling approach while reducing the computational cost from 8 months on 48 CPU cores to 2 days on 12 GPUs. Second, we consider a joint analysis between three simulated next-generation surveys, each performing a 3x2pt analysis, resulting in 157- and 159-dimensional parameter spaces. Standard nested sampling techniques are simply unlikely to be feasible in this high-dimensional setting, requiring a projected 12 years of compute time on 48 CPU cores; on the other hand, the proposed approach only requires 8 days of compute time on 24 GPUs. All packages used in our analyses are publicly available.

UIEDP: Boosting underwater image enhancement with diffusion prior

Underwater image enhancement (UIE) aims to generate clear images from low-quality underwater images. Due to the unavailability of clear reference images, researchers often synthesize them to construct paired datasets for training deep models. However, these synthesized images may sometimes lack quality, adversely affecting training outcomes. To address this issue, we propose to boost UIE with Diffusion Prior (UIEDP). It is a novel framework treating UIE as a posterior distribution sampling process of clear images conditioned on degraded underwater inputs. Specifically, UIEDP combines a pre-trained diffusion model capturing natural image priors with any existing UIE algorithm, leveraging the latter to guide conditional generation. The diffusion prior mitigates the drawbacks of inferior synthetic images, resulting in higher-quality image generation. Extensive experiments have demonstrated that our UIEDP yields significant improvements across various metrics, especially no-reference image quality assessment. And the generated enhanced images also exhibit a more natural appearance.

Information-directed policy sampling for episodic Bayesian Markov decision processes

We consider finite-stage Markov Decision Processes (MDPs) under incomplete information, where the decision-maker only knows that the true transition probability and reward matrices belong to given, finite sets. The decision-maker interacts with the system over a finite number of episodes. The first episode begins with a probabilistic belief about the true probability and reward matrices. This belief is updated at the end of each episode using observed events. The goal is to maximize the expected total reward earned over all episodes. In the resulting model-based episodic Bayesian MDP, it suffices to only consider (the known) policies that are optimal to each one of the possible probability and reward matrices. Nevertheless, the decision-maker should execute policies that provide information about the true probabilities and rewards (exploration), but also exploit this knowledge to increase rewards. We propose a framework called Information-Directed Policy Sampling (IDPS). In each episode, the decision-maker balances the exploitation-exploration trade-off by executing a randomized policy that minimizes a so-called convex information ratio. We derive a regret bound that is independent of state- and action-space cardinalities when the set of matrices is exogenously determined. Numerical experiments show IDPS outperforming a state-of-the-art approach called Posterior Sampling.

Scalable Bayesian uncertainty quantification with data-driven priors for radio interferometric imaging

Abstract Next-generation radio interferometers like the Square Kilometer Array have the potential to unlock scientific discoveries thanks to their unprecedented angular resolution and sensitivity. One key to unlocking their potential resides in handling the deluge and complexity of incoming data. This challenge requires building radio interferometric (RI) imaging methods that can cope with the massive data sizes and provide high-quality image reconstructions with uncertainty quantification (UQ). This work proposes a method coined QuantifAI to address UQ in RI imaging with data-driven (learned) priors for high-dimensional settings. Our model, rooted in the Bayesian framework, uses a physically motivated model for the likelihood. The model exploits a data-driven convex prior potential, which can encode complex information learned implicitly from simulations and guarantee the log-concavity of the posterior. We leverage probability concentration phenomena of high-dimensional log-concave posteriors to obtain information about the posterior, avoiding MCMC sampling techniques. We rely on convex optimisation methods to compute the MAP estimation, which is known to be faster and better scale with dimension than MCMC strategies. QuantifAI allows us to compute local credible intervals and perform hypothesis testing of structure on the reconstructed image. We propose a novel fast method to compute pixel-wise uncertainties at different scales, which uses 3 and 6 orders of magnitude less likelihood evaluations than other UQ methods like length of the credible intervals and Monte Carlo posterior sampling, respectively. We demonstrate our method by reconstructing RI images in a simulated setting and carrying out fast and scalable UQ, which we validate with MCMC sampling. Our method shows an improved image quality and more meaningful uncertainties than the benchmark method based on a sparsity-promoting prior. QuantifAI’s source code is available from https://github.com/astro-informatics/QuantifAI.

Augmentation Samplers for Multinomial Probit Bayesian Additive Regression Trees

The multinomial probit (MNP) (Imai and van Dyk, 2005) framework is based on a multivariate Gaussian latent structure, allowing for natural extensions to multilevel modeling. Unlike multinomial logistic models, MNP does not assume independent alternatives. Kindo et al. (2016) proposed multinomial probit BART (MPBART) to accommodate Bayesian additive regression trees (BART) formulation in MNP. The posterior sampling algorithms for MNP and MPBART are collapsed Gibbs samplers. Because the collapsing augmentation strategy yields a geometric rate of convergence no greater than that of a standard Gibbs sampling step, it is recommended whenever computationally feasible (Liu, 1994a; Imai and van Dyk, 2005). While this strategy necessitates simple sampling steps and a reasonably fast converging Markov chain, the complexity of the stochastic search for posterior trees may undermine its benefit. We address this problem by sampling posterior trees conditional on the constrained parameter space and compare our proposals to that of Kindo et al. (2016), who sample posterior trees based on an augmented parameter space. We also compare to the approach by Sparapani et al. (2021) that specified the multinomial model in terms of conditional probabilities. In terms of MCMC convergence and posterior predictive accuracy, our proposals are comparable to the conditional probability approach and outperform the augmented tree sampling approach. We also show that the theoretical mixing rates of our proposals are guaranteed to be no greater than the augmented tree sampling approach. Appendices and codes for simulations and demonstrations are available online.

A Diffusion‐Based Uncertainty Quantification Method to Advance E3SM Land Model Calibration

AbstractCalibrating land surface models and accurately quantifying their uncertainty are crucial for improving the reliability of simulations of complex environmental processes. This, in turn, advances our predictive understanding of ecosystems and supports climate‐resilient decision‐making. Traditional calibration methods, however, face challenges of high computational costs and difficulties in accurately quantifying parameter uncertainties. To address these issues, we develop a diffusion‐based uncertainty quantification (DBUQ) method. Unlike conventional generative diffusion methods, which are computationally expensive and memory‐intensive, DBUQ innovates by formulating a parameterized generative model and approximates this model through supervised learning, which enables quick generation of parameter posterior samples to quantify its uncertainty. DBUQ is effective, efficient, and general‐purpose, making it suitable for site‐specific ecosystem model calibration and broadly applicable for parameter uncertainty quantification across various earth system models. In this study, we applied DBUQ to calibrate the Energy Exascale Earth System Model land model at the Missouri Ozark AmeriFlux forest site. Results indicated that DBUQ produced accurate parameter posterior distributions similar to those from Markov Chain Monte Carlo sampling but with 30 times less computing time. This significant improvement in efficiency suggests that DBUQ can enable rapid, site‐level model calibration at a global scale, enhancing our predictive understanding of climate impacts on terrestrial ecosystems.

Advancements in multi-wavelength data analysis: The ThreeML framework for high-energy astronomy

The Multi-Mission Maximum Likelihood framework (ThreeML) is a Python-based software package designed for multi-wavelength data analysis in high-energy astronomy. Integrating X-ray and gamma-ray data from various instruments, along with measurements at lower wavelengths, is essential for unlocking the full potential of observational data. However, the lack of standardization and unique challenges posed by each instrument often complicate the process of combining data from multiple sources. ThreeML addresses these challenges with its flexible, plugin-based structure, allowing for the seamless inclusion of data from diverse observatories in their native formats. Leveraging astromodels, a versatile modeling framework, ThreeML enables separate handling of source modeling and data access from likelihood optimization, facilitating a flexible combination of both aspects. Moreover, in addition to frequentist maximum likelihood analysis, ThreeML supports Bayesian analysis through posterior distribution sampling.

CT reconstruction using diffusion posterior sampling conditioned on a nonlinear measurement model.

Recently, diffusion posterior sampling (DPS), where score-based diffusion priors are combined with likelihood models, has been used to produce high-quality computed tomography (CT) images given low-quality measurements. This technique permits one-time, unsupervised training of a CT prior, which can then be incorporated with an arbitrary data model. However, current methods rely on a linear model of X-ray CT physics to reconstruct. Although it is common to linearize the transmission tomography reconstruction problem, this is an approximation to the true and inherently nonlinear forward model. We propose a DPS method that integrates a general nonlinear measurement model. We implement a traditional unconditional diffusion model by training a prior score function estimator and apply Bayes' rule to combine this prior with a measurement likelihood score function derived from the nonlinear physical model to arrive at a posterior score function that can be used to sample the reverse-time diffusion process. We develop computational enhancements for the approach and evaluate the reconstruction approach in several simulation studies. The proposed nonlinear DPS provides improved performance over traditional reconstruction methods and DPS with a linear model. Moreover, as compared with a conditionally trained deep learning approach, the nonlinear DPS approach shows a better ability to provide high-quality images for different acquisition protocols. This plug-and-play method allows the incorporation of a diffusion-based prior with a general nonlinear CT measurement model. This permits the application of the approach to different systems, protocols, etc., without the need for any additional training.

A double Pólya-Gamma data augmentation scheme for a hierarchical Negative Binomial - Binomial data model

A double Pólya-Gamma data augmentation scheme is developed for posterior sampling from a Bayesian hierarchical model of total and categorical count data. The scheme applies to a Negative Binomial - Binomial (NBB) hierarchical regression model with logit links and normal priors on regression coefficients. The approach is shown to be very efficient and in most cases out-performs the Stan program. The hierarchical modeling framework and the Pólya-Gamma data augmentation scheme are applied to human mitochondrial DNA data.

A Projection Approach to Local Regression with Variable-Dimension Covariates

Incomplete covariate vectors are known to be problematic for estimation and inferences on model parameters, but their impact on prediction performance is less understood. We develop an imputation-free method that builds on a random partition model admitting variable-dimension covariates. Cluster-specific response models further incorporate covariates via linear predictors, facilitating estimation of smooth prediction surfaces with relatively few clusters. We exploit marginalization techniques of Gaussian kernels to analytically project response distributions according to any pattern of missing covariates, yielding a local regression with internally consistent uncertainty propagation that uses only one set of coefficients per cluster. Aggressive shrinkage of these coefficients regulates uncertainty due to missing covariates. The method allows in- and out-of-sample prediction for any missingness pattern, even if the pattern in a new subject’s incomplete covariate vector was not seen in the training data. We develop an MCMC algorithm for posterior sampling that improves a computationally expensive update for latent cluster allocation. Finally, we demonstrate the model’s effectiveness for nonlinear point and density prediction under various circumstances by comparing with other recent methods for regression of variable dimensions on synthetic and real data. Supplemental materials for this article are available online.