Localization for MCMC: sampling high-dimensional posterior distributions with local structure
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Computer Science↗
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Localization for MCMC: sampling high-dimensional posterior densities with local structure
Presentation to be given at the SIAM Conference on Uncertainty Quantification
97 MATHEMATICS AND COMPUTING↗
BEYONDPLANCK X. Planck Low Frequency Instrument frequency maps with sample-based error propagation
We present Planck Low Frequency Instrument (LFI) frequency sky maps derived within the BEYONDPLANCK framework. This framework draws samples from a global posterior distribution that includes instrumental, astrophysical, and cosmological parameters, and the main product is an entire ensemble of frequency sky map samples, each of which corresponds to one possible realization of the various modeled instrumental systematic corrections, including correlated noise, time-variable gain, as well as far sidelobe and bandpass corrections. This ensemble allows for computationally convenient end-to-end propagation of low-level instrumental uncertainties into higher-level science products, including astrophysical component maps, angular power spectra, and cosmological parameters. We show that the two dominant sources of LFI instrumental systematic uncertainties are correlated noise and gain fluctuations, and the products presented here support – for the first time – full Bayesian error propagation for these effects at full angular resolution. We compared our posterior mean maps with traditional frequency maps delivered by the Planck Collaboration, and find generally good agreement. The most important quality improvement is due to significantly lower calibration uncertainties in the new processing, as we find a fractional absolute calibration uncertainty at 70 GHz of Δg 0 /g 0 = 5 x 10 -5 , which is nominally 40 times smaller than that reported by Planck 2018. However, we also note that the original Planck 2018 estimate has a nontrivial statistical interpretation, and this further illustrates the advantage of the new framework in terms of producing self-consistent and well-defined error estimates of all involved quantities without the need of ad hoc uncertainty contributions. We describe how low-resolution data products, including dense pixel-pixel covariance matrices, may be produced from the posterior samples directly, without the need for computationally expensive analytic calculations or simulations. We conclude that posterior-based frequency map sampling provides unique capabilities in terms of low-level systematics modeling and error propagation, and may play an important role for future Cosmic Microwave Background (CMB) B-mode experiments aiming at nanokelvin precision.
79 ASTRONOMY AND ASTROPHYSICS↗
Calibration and uncertainty quantification for Davis Equation of State models for the High Explosive PBX 9501 products
Abstract This paper investigates the uncertainty in the parameters used in the calibration of an Davis Equation Of State (EOS) for the detonation products of the High Explosive PBX 9501. The procedure sought to make use of all available information about this HE to inform the best set of calibration parameters as well as the uncertainty in these parameters. The procedure made use of historical experimental data, the results from thermo‐chemical modeling as well as data on the best isentrope function fit to cylinder test experimental data. Combining all these heterogeneous data sources together in a Bayesian calibration, yielded a posterior mean and covariance. Sampling from the posterior distribution and evaluating an important Quantity Of Interest (QOI) in the EOS model, the detonation speed of a one‐inch rate stick, produced a distribution which showed variations which were in agreement with experiments. The uncertainty in the EOS was reported as eleven sets of model calibrations which spanned the range of this QOI.
45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗
The completed SDSS-IV extended baryon oscillation spectroscopic survey: geometry and growth from the anisotropic void–galaxy correlation function in the luminous red galaxy sample
ABSTRACT We present an analysis of the anisotropic redshift-space void–galaxy correlation in configuration space using the Sloan Digital Sky Survey extended Baryon Oscillation Spectroscopic Survey (eBOSS) Data Release 16 luminous red galaxy (LRG) sample. This sample consists of LRGs between redshifts 0.6 and 1.0, combined with the high redshift z > 0.6 tail of the Baryon Oscillation Spectroscopic Survey Data Release 12 CMASS sample. We use a reconstruction method to undo redshift-space distortion (RSD) effects from the galaxy field before applying a watershed void-finding algorithm to remove bias from the void selection. We then perform a joint fit to the multipole moments of the correlation function for the growth rate fσ8 and the geometrical distance ratio DM/DH, finding $f\sigma _8(z_\rm {eff})=0.356\pm 0.079$ and $D_M/D_H(z_\rm {eff})=0.868\pm 0.017$ at the effective redshift $z_\rm {eff}=0.69$ of the sample. The posterior parameter degeneracies are orthogonal to those from galaxy clustering analyses applied to the same data, and the constraint achieved on DM/DH is significantly tighter. In combination with the consensus galaxy BAO and full-shape analyses of the same sample, we obtain fσ8 = 0.447 ± 0.039, DM/rd = 17.48 ± 0.23, and DH/rd = 20.10 ± 0.34. These values are in good agreement with the ΛCDM model predictions and represent reductions in the uncertainties of $13{{\ \rm per\ cent}}$, $23{{\ \rm per\ cent}}$, and $28{{\ \rm per\ cent}}$, respectively, compared to the combined results from galaxy clustering, or an overall reduction of 55 per cent in the allowed volume of parameter space.
Nadathur, Seshadri↗
Probabilistic Inference of Low-Surface-Brightness Galaxy Morphological Parameters Using Simulation-Based Inference
Low-surface-brightness galaxies (LSBGs) are diffuse, often dark-matter-dominated systems whose faintness makes their structural parameters difficult to measure reliably in wide-field imaging surveys. Robust parameter inference, including uncertainty quantification, is important for population studies and for comparisons with models of galaxy formation, as future surveys are expected to produce increasingly large samples of diffuse galaxies. In practice, LSBG profile modeling is sensitive to sky- background errors, masking choices, contaminating background sources, and the computational cost of obtaining posterior-level uncertainties for large samples. Motivated by these questions, we develop a simulation-based inference (SBI) framework for estimating posterior distributions of LSBG morphological parameters from simulated galaxy images. Using PyImfit, we generate DES-like single-Sersic profile LSBG images with known position angle, ellipticity, Sersic index, effective surface brightness, and effective radius. We then train a normalizing-flow-based neural posterior estimator using the sbi package to infer these parameters from the simulated images. For isolated simulated galaxies, the SBI posterior recovers the true input parameters, produces posterior predictive residuals consistent with the assumed noise model, and shows good empirical calibration in a DES-motivated test regime. We also compare SBI with PyImfit-based MCMC inference and find broadly comparable posterior constraints, while SBI enables substantially faster posterior sampling after training. Finally, we test robustness to compact background contaminants. A model trained only on isolated galaxies produces undercovered posteriors on contaminated images, whereas training on simulations with variable contaminant positions and fluxes improves calibration across contaminated test sets. These results demonstrate the promise of SBI for scalable, uncertainty-aware LSBG morphology inference, while emphasizing that posterior reliability strongly depends on whether training simulations include relevant observational complications.
Batbayar, Bilguun [U. Chicago (main)]↗
Stochastic modeling and statistical calibration with model error and scarce data
This paper introduces a procedure to assess the predictive accuracy of stochastic models subject to model error and sparse data. Model error is introduced as uncertainty on the coefficients of appropriate polynomial chaos expansions (PCE). The error associated with finite sample size allows us to conceive of these coefficients as statistics of the data that we describe as random variables whose influence on output quantities of interest is evaluated through the extended polynomial chaos expansion (EPCE). A Bayesian data assimilation scheme is introduced to update these expansions by considering the resulting nested chaos expansion as a hierarchical probabilistic model. Stochastic models of quantities of interest (QoI) are thus constructed and efficiently evaluated. Here, the Metropolis–Hastings Markov chain Monte Carlo procedure is used to sample the posterior. Two illustrative analytical and numerical problems are used to demonstrate the proposed approach.
Bayesian inference↗
Differentially Private Synthesis and Sharing of Network Data Via Bayesian Exponential Random Graph Models
Abstract Network data often contain sensitive relational information. One approach to protecting sensitive information while offering flexibility for network analysis is to share synthesized networks based on the information in originally observed networks. We employ differential privacy (DP) and exponential random graph models (ERGMs) and propose the DP-ERGM method to synthesize network data. We apply DP-ERGM to two real-world networks. We then compare the utility of synthesized networks generated by DP-ERGM, the DyadWise Randomized Response (DWRR) approach, and the Synthesis through Conditional distribution of Edge given nodal Attribute (SCEA) approach. In general, the results suggest that DP-ERGM preserves the original information significantly better than two other approaches in network structural statistics and inference for ERGMs and latent space models. Furthermore, DP-ERGM satisfies node DP through modeling the global network structure with ERGM, a stronger notion of privacy than the edge DP under which DWRR and SCEA operate.
graph synthesis↗
Bayesian Physics Informed Spatio-Temporal Network for Streamflow Data Imputation
Reliable reconstruction of incomplete streamflow records is critical for improving hydrological forecasting, flood preparedness, and water resource management. However, large observational gaps and uncertainties in governing physical parameters limit the accuracy of traditional statistical and machinelearning imputation frameworks. To address these challenges, we develop a Bayesian Physics-Informed Spatio-Temporal Network (BPI-STNet) that jointly captures spatial and temporal dependencies while enforcing hydrologic consistency through embedded physical constraints. The framework integrates a GraphSAGE-LSTM architecture to model spatial connectivity across gauges and temporal flow dynamics, coupled with a Bayesian update mechanism to estimate uncertain parameters in a simplified water-balance framework. Unlike conventional physics-informed networks that rely on sampling-based posterior estimation, BPI-STNet derives an analytic solution to the inverse problem, allowing closed-form Bayesian updates of uncertain parameters Λ={α,β,k} using Gaussian priors and likelihoods. Applied to daily observations from the Susquehanna River Basin (1980-2022), BPI-STNet achieves substantial improvements over a purely data-driven RGNN baseline, which reduced RMSE by 23 % and MAE by 9 %, and achieving an average NSE values up to 0.96. The results demonstrate that coupling Bayesian inference with physics-informed learning yields physically consistent, uncertainty-aware reconstructions that preserve the temporal persistence and statistical distribution of observed flows. The proposed framework establishes a generalizable paradigm for data-sparse hydrologic systems where both data fidelity and physical interpretability are essential.
Krishnan Kutty Ambika, Anukesh [ORNL] (ORCID:00000↗
Detecting outbreaks using a spatial latent field
In this paper, we present a method for estimating the infection-rate of a disease as a spatial-temporal field. Our data comprises time-series case-counts of symptomatic patients in various areal units of a region. We extend an epidemiological model, originally designed for a single areal unit, to accommodate multiple units. The field estimation is framed within a Bayesian context, utilizing a parameterized Gaussian random field as a spatial prior. We apply an adaptive Markov chain Monte Carlo method to sample the posterior distribution of the model parameters condition on COVID-19 case-count data from three adjacent counties in New Mexico, USA. Our results suggest that the correlation between epidemiological dynamics in neighboring regions helps regularize estimations in areas with high variance (i.e., poor quality) data. Using the calibrated epidemic model, we forecast the infection-rate over each areal unit and develop a simple anomaly detector to signal new epidemic waves. Our findings show that anomaly detector based on estimated infection-rates outperforms a conventional algorithm that relies solely on case-counts.
Safta, Cosmin [Sandia National Laboratories (SNL-C↗
A D-term Modeling Code (DMC) for Simultaneous Calibration and Full-Stokes Imaging of Very Long Baseline Interferometric Data
In this paper we present DMC, a model and associated tool for polarimetric imaging of very long baseline interferometry data sets that simultaneously reconstructs the full-Stokes emission structure along with the station-based gain and leakage calibration terms. DMC formulates the imaging problem in terms of posterior exploration, which is achieved using Hamiltonian Monte Carlo sampling. The resulting posterior distribution provides a natural quantification of uncertainty in both the image structure and the data calibration. We run DMC on both synthetic and real data sets, the results of which demonstrate its ability to accurately recover both the image structure and calibration quantities, as well as to assess their corresponding uncertainties. The framework underpinning DMC is flexible, and its specific implementation is under continued development.
47 OTHER INSTRUMENTATION↗
State of Stress Uncertainty Quantification and Geomechanical Risk Analysis for Subsurface Engineering
All subsurface stress measurement methods available for deep subsurface applications are indirect and involve a significant amount of uncertainty. Quantifying these uncertainties is critical for applications where the state of stress is a component of a risk analysis with significant safety, environmental, and economic considerations. A publicly available and open source tool called the State of Stress Analysis Tool (SOSAT) has been developed to aid in performing a Bayesian uncertainty quantification and geomechanical risk analysis using many types of commonly available data. A new feature of this tool is using data from the FutureGen 2.0 site. The new feature of the tool was uses information about the absence or presence of borehole breakouts to constrain the state of stress. This new feature allows the user to specify probability distributions for the relevant parameters such as the drilling mud pressure, temperature, and the formation strength properties. The tool computes the posterior joint probability distribution of the horizontal principal stresses. Using random samples from this posterior distribution the tool then computes the probability of activating a critically oriented fault at a range of pore pressures. This latter feature can be used to aid in the assessment of the risk of induced seismicity and leakage in geologic carbon sequestration and enhanced geothermal systems.
Burghardt, Jeffrey A.↗
Personalized and uncertainty-aware coronary hemodynamics simulations: From Bayesian estimation to improved multi-fidelity uncertainty quantification
Non-invasive simulations of coronary hemodynamics have improved clinical risk stratification and treatment outcomes for coronary artery disease, compared to relying on anatomical imaging alone. However, simulations typically use empirical approaches to distribute total coronary flow amongst the arteries in the coronary tree, which ignores patient variability, the presence of disease, and other clinical factors. Further, uncertainty in the clinical data often remains unaccounted for in the modeling pipeline. We present an end-to-end uncertainty-aware pipeline to (1) personalize coronary flow simulations by incorporating vessel-specific coronary flows as well as cardiac function; and (2) predict clinical and biomechanical quantities of interest with improved precision, while accounting for uncertainty in the clinical data. We assimilate patient-specific measurements of myocardial blood flow from clinical CT myocardial perfusion imaging to estimate branch-specific coronary artery flows. Simulated noise in the clinical data is used to estimate the joint posterior distributions of the model parameters using adaptive Markov Chain Monte Carlo sampling. Additionally, the posterior predictive distribution for the relevant quantities of interest is determined using a new approach combining multi-fidelity Monte Carlo estimation with non-linear, data-driven dimensionality reduction. This leads to improved correlations between high- and low-fidelity model outputs. Our framework accurately recapitulates clinically measured cardiac function as well as branch-specific coronary flows under measurement noise uncertainty. We observe substantial reductions in confidence intervals for estimated quantities of interest compared to single-fidelity Monte Carlo estimation and state-of-the-art multi-fidelity Monte Carlo methods. This holds especially true for quantities of interest that showed limited correlation between the low- and high-fidelity model predictions. In addition, the proposed multi-fidelity Monte Carlo estimators are significantly cheaper to compute than traditional estimators, under a specified confidence level or variance. The proposed pipeline for personalized and uncertainty-aware predictions of coronary hemodynamics is based on routine clinical measurements and recently developed techniques for CT myocardial perfusion imaging. The proposed pipeline offers significant improvements in precision and reduction in computational cost.
Bayesian parameter estimation↗
Microsimulation model calibration using incremental mixture approximate Bayesian computation
Microsimulation models (MSMs) are used to inform policy by predicting population-level outcomes under different scenarios. MSMs simulate individual-level event histories that mark the disease process (such as the development of cancer) and the effect of policy actions (such as screening) on these events. MSMs often have many unknown parameters; calibration is the process of searching the parameter space to select parameters that result in accurate MSM prediction of a wide range of targets. We develop Incremental Mixture Approximate Bayesian Computation (IMABC) for MSM calibration which results in a simulated sample from the posterior distribution of model parameters given calibration targets. IMABC begins with a rejection-based ABC step, drawing a sample of points from the prior distribution of model parameters and accepting points that result in simulated targets that are near observed targets. Next, the sample is iteratively updated by drawing additional points from a mixture of multivariate normal distributions and accepting points that result in accurate predictions. Posterior estimates are obtained by weighting the final set of accepted points to account for the adaptive sampling scheme. We demonstrate IMABC by calibrating CRC-SPIN 2.0, an updated version of a MSM for colorectal cancer (CRC) that has been used to inform national CRC screening guidelines.
97 MATHEMATICS AND COMPUTING↗
An adaptive Hessian approximated stochastic gradient MCMC method
Bayesian approaches have been successfully integrated into training deep neural networks. One popular family is stochastic gradient Markov chain Monte Carlo methods (SG-MCMC), which have gained increasing interest due to their ability to handle large datasets and the potential to avoid overfitting. Although standard SG-MCMC methods have shown great performance in a variety of problems, they may be inefficient when the random variables in the target posterior densities have scale differences or are highly correlated. Here, we present an adaptive Hessian approximated stochastic gradient MCMC method to incorporate local geometric information while sampling from the posterior. The idea is to apply stochastic approximation (SA) to sequentially update a preconditioning matrix at each iteration. The preconditioner possesses second-order information and can guide the random walk of a sampler efficiently. Instead of computing and saving the full Hessian of the log posterior, we use limited memory of the samples and their stochastic gradients to approximate the inverse Hessian-vector multiplication in the updating formula. Moreover, by smoothly optimizing the preconditioning matrix via SA, our proposed algorithm can asymptotically converge to the target distribution with a controllable bias under mild conditions. To reduce the training and testing computational burden, we adopt a magnitude-based weight pruning method to enforce the sparsity of the network. Our method is user-friendly and demonstrates better learning results compared to standard SG-MCMC updating rules. The approximation of inverse Hessian alleviates storage and computational complexities for large dimensional models. Numerical experiments are performed on several problems, including sampling from 2D correlated distribution, synthetic regression problems, and learning the numerical solutions of heterogeneous elliptic PDE. The numerical results demonstrate great improvement in both the convergence rate and accuracy.
97 MATHEMATICS AND COMPUTING↗
Probabilistic Calibration of Expensive Models using Efficiently Trained Surrogates
Calibration of computational models in the presence of uncertainty is often cast as a Bayesian inference problem and solved via sampling methods, e.g., Markov chain Monte Carlo. When the computational model is expensive, this task becomes intractable due to the large number of samples required to accurately estimate the posterior distribution of the calibration parameters. A popular solution to this problem is to use machine learning to develop a faster-to-evaluate, lower-fidelity substitute for the original model to serve as a surrogate while solving the inference problem. Although considered an offline cost, generating training data to construct this surrogate model can still be an expensive task in practice. An active learning algorithm is presented that focuses training on improving surrogate accuracy specifically in and around the bulk of the posterior distribution, as this is where the model is exercised during calibration. Candidate samples are drawn from families of distributions related to an approximation of the posterior. The sample maximizing predictive variance is then selected for evaluation by the original computational model, yielding a label for the training point. Iterating this approach increases efficiency relative to space filling designs (e.g., Latin hypercube sampling) by avoiding low probability points. Practical considerations are discussed, including the benefits of using a sequential Monte Carlo sampling approach, convergence heuristics, and the importance of both exploration and exploitation given that the true posterior is unknown a priori.
uncertainty quantification↗
Dimensional Reduction for Sampled Priors and Application to Photometric Redshift Distributions
A typical Bayesian inference on the values of some parameters of interest q from some data D involves running a Markov Chain (MC) to sample from the posterior $p$($q$,$n$|$D$) $\propto$ $\mathcal{L}$($D$|$q$,$n$)$p$(q)$p$($n$), where n are some nuisance parameters with a separable prior. In some cases, the nuisance parameters are high-dimensional, and their prior p(n) is itself defined only by a set of samples that have been drawn from some other MC. The MC for the posterior will typically require evaluation of p(n) at arbitrary values of n, i.e., one needs to provide a density estimator over the full n space from the provided samples. But the high dimensionality of n hinders both the density estimation and the efficiency of the MC for the posterior. We describe a solution to this problem: a linear compression of the n space into a much lower-dimensional space u, which projects away directions in n space that cannot appreciably alter $\mathcal{L}$. The algorithm for doing so is a slight modification to principal components analysis, and is less restrictive on p(n) than other proposed solutions to this issue. We demonstrate this “mode projection” technique using the analysis of 2-point correlation functions of weak lensing fields and galaxy density in the Dark Energy Survey, where n is a binned representation of the redshift distribution n(z) of the galaxies.
79 ASTRONOMY AND ASTROPHYSICS↗
Bayesian operator inference for data-driven reduced-order modeling
This work proposes a Bayesian inference method for the reduced-order modeling of time-dependent systems. Informed by the structure of the governing equations, the task of learning a reduced-order model from data is posed as a Bayesian inverse problem with Gaussian prior and likelihood. The resulting posterior distribution characterizes the operators defining the reduced-order model, hence the predictions subsequently issued by the reduced-order model are endowed with uncertainty. The statistical moments of these predictions are estimated via a Monte Carlo sampling of the posterior distribution. Since the reduced models are fast to solve, this sampling is computationally efficient. Furthermore, the proposed Bayesian framework provides a statistical interpretation of the regularization term that is present in the deterministic operator inference problem, and the empirical Bayes approach of maximum marginal likelihood suggests a selection algorithm for the regularization hyperparameters. The proposed method is demonstrated on two examples: the compressible Euler equations with noise-corrupted observations, and a single-injector combustion process.
97 MATHEMATICS AND COMPUTING↗