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Iterative HOMER with uncertainties

We present iHOMER, an iterative version of the HOMER method to extract Lund fragmentation functions from experimental data. Through iterations, we address the information gap between latent and observable phase spaces and systematically remove bias. To quantify uncertainties on the inferred weights, we use a combination of Bayesian neural networks and uncertainty-aware regression. We find that the combination of iterations and uncertainty quantification produces well-calibrated weights that accurately reproduce the data distribution. A parametric closure test shows that the iteratively learned fragmentation function is compatible with the true fragmentation function.

Butter, Anja [Heidelberg Univ. (Germany); Sorbonne↗

First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole

We present measurements of the properties of the central radio source in M87 using Event Horizon Telescope data obtained during the 2017 campaign. We develop and fit geometric crescent models (asymmetric rings with interior brightness depressions) using two independent sampling algorithms that consider distinct representations of the visibility data. We show that the crescent family of models is statistically preferred over other comparably complex geometric models that we explore. We calibrate the geometric model parameters using general relativistic magnetohydrodynamic (GRMHD) models of the emission region and estimate physical properties of the source. We further fit images generated from GRMHD models directly to the data. We compare the derived emission region and black hole parameters from these analyses with those recovered from reconstructed images. There is a remarkable consistency among all methods and data sets. We find that >50% of the total flux at arcsecond scales comes from near the horizon, and that the emission is dramatically suppressed interior to this region by a factor >10, providing direct evidence of the predicted shadow of a black hole. Across all methods, we measure a crescent diameter of 42 ± 3 μas and constrain its fractional width to be <0.5. Associating the crescent feature with the emission surrounding the black hole shadow, we infer an angular gravitational radius of GM/Dc2 = 3.8 ± 0.4 μas. Folding in a distance measurement of ${16.8}_{-0.7}^{+0.8}\,\mathrm{Mpc}$ gives a black hole mass of $M=6.5\pm 0.2{| }_{\mathrm{stat}}\pm 0.7{| }_{\mathrm{sys}}\times {10}^{9}\hspace{2pt}{M}_{\odot }$. This measurement from lensed emission near the event horizon is consistent with the presence of a central Kerr black hole, as predicted by the general theory of relativity.

79 ASTRONOMY AND ASTROPHYSICS↗

A technique for interpretation of auroral bremsstrahlung X-ray spectra

A procedure is described for determining the average energy and the energy flux of precipitating electrons from the resulting bremsstrahlung X-rays based on the assumption that the precipitating electrons have an exponential electron distribution. It is shown that, for nonexponential electron distributions, this method yields an exponential electron distribution that produces an altitude profile of ionization very similar to that expected from the actual spectrum. The Hall and Pedersen electrical conductances calculated from the inferred ionization profile were also found to be close to the true values.

Robinson, Robert M.↗

Physics-based, Bayesian sequential detection method and system for radioactive contraband

A distributed sequential method and system for detecting and identifying radioactive contraband from highly uncertain (noisy) low-count, radionuclide measurements, i.e. an event mode sequence (EMS), using a statistical approach based on Bayesian inference and physics-model-based signal processing based on the representation of a radionuclide as a monoenergetic decomposition of monoenergetic sources. For a given photon event of the EMS, the appropriate monoenergy processing channel is determined using a confidence interval condition-based discriminator for the energy amplitude and interarrival time and parameter estimates are used to update a measured probability density function estimate for a target radionuclide. A sequential likelihood ratio test is then used to determine one of two threshold conditions signifying that the EMS is either identified as the target radionuclide or not, and if not, then repeating the process for the next sequential photon event of the EMS until one of the two threshold conditions is satisfied.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

Initial Assessment of Ruthenium Removal Systems for Tritium Pretreatment Off-Gas

The processing of used nuclear fuel releases volatile and semivolatile radionuclides into the off-gas streams associated with the various processing steps. Ongoing efforts to develop methods to control the release of gaseous radionuclides to the environment have identified ruthenium (specifically 106 Ru) as one of the semivolatile nuclides requiring a high degree of abatement. The tritium pretreatment process will release a fraction of the ruthenium as volatile ruthenium tetroxide (RuO 4 ). Scoping tests were conducted to characterize the adsorption/deposition of RuO 4 from dry gas streams by two different substrates: steel wool and silica gel. A simple test system was assembled to produce RuO 4 through oxidation of powdered ruthenium metal, followed by a temperature-controlled sorption column. Tests were conducted with no sorbent present in the column to obtain baseline data on the evolution of RuO 4 and deposition of ruthenium on the internal surfaces of the column. Silica gel and steel wool were tested as adsorbent media. Gaseous effluent from the column was scrubbed with caustic solution to remove any remaining ruthenium. Ultraviolet/visible spectroscopy (UV/Vis) and inductively coupled plasma–mass spectroscopy analyses were used to measure the amount of ruthenium in the scrub solutions. The UV/Vis analysis proved to be problematic because of precipitation of ruthenates, ruthenium oxides, and hydrated ruthenium oxides that occurred over time. An effective stabilization agent that does not adversely affect the desired spectra needs to be identified if this method is to be used in the future. There were indications that oxidation of powdered ruthenium metal did not generate a stream with steady concentration of RuO 4 , so improved techniques are needed for follow-on experiments designed to infer deposition rate. Both steel wool and silica gel are able to decrease the effluent Ru concentration by a factor of 10–100 during the 5-hour tests conducted. These tests were conducted using a dilute RuO 4 -bearing dry gas stream and the short test duration did not result in sorbent saturation or in an estimation of Ru breakthrough time. RuO 4 desorbs from silica gel at room temperature. As interactions between RuO 4 and silica gel are not irreversible, it cannot function as a disposal media without pretreatment to reduce RuO 4 to a less mobile species. Silica gel may be useful as a RuO 4 sorbent or delay media in some applications. Based on visual observations, steel wool was an effective deposition substrate with no evidence of RuO 4 release. The deposition form should be identified and methods to assay the ruthenium content need to be developed.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

Multimodal representation learning for predicting molecule–disease relations

Motivation: Predicting molecule–disease indications and side effects is important for drug development and pharmacovigilance. Comprehensively mining molecule–molecule, molecule–disease and disease–disease semantic dependencies can potentially improve prediction performance. Methods: We introduce a Multi-Modal REpresentation Mapping Approach to Predicting molecular-disease relations (M2REMAP) by incorporating clinical semantics learned from electronic health records (EHR) of 12.6 million patients. Specifically, M2REMAP first learns a multimodal molecule representation that synthesizes chemical property and clinical semantic information by mapping molecule chemicals via a deep neural network onto the clinical semantic embedding space shared by drugs, diseases and other common clinical concepts. To infer molecule–disease relations, M2REMAP combines multimodal molecule representation and disease semantic embedding to jointly infer indications and side effects. Results: We extensively evaluate M2REMAP on molecule indications, side effects and interactions. Results show that incorporating EHR embeddings improves performance significantly, for example, attaining an improvement over the baseline models by 23.6% in PRC-AUC on indications and 23.9% on side effects. Further, M2REMAP overcomes the limitation of existing methods and effectively predicts drugs for novel diseases and emerging pathogens. Availability and implementation: The code is available at https://github.com/celehs/M2REMAP, and prediction results are provided at https://shiny.parse-health.org/drugs-diseases-dev/.

59 BASIC BIOLOGICAL SCIENCES↗

First-principles study of electronic and elastic properties of LuAl{sub 3}

A systematic theoretical study of electronic structure of rare earth intermetallic LuAl{sub 3} has been carried out using full potential linearized augmented plane wave (FP-LAPW) method based on density functional theory (DFT) within the generalized gradient approximation(GGA) for exchange and correlation potential. The ground state properties such as lattice constant (a{sub o}), bulk modulus (B) and pressure derivative of bulk modulus (B′) were evaluated. LuAl{sub 3} has the cubic AuCu{sub 3} type crystal structure. The electronic properties of this compound have been analyzed quantatively from band structure and DOS. It is clear from band structure that this compound is metallic in nature. The calculated elastic constants infer that this compound is mechanically stable.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Tracking the PACE of household energy usage: Energy usage impacts of projects financed through Property Assessed Clean Energy programs in California

This report examines the household-level energy impacts of residential property assessed clean energy (R-PACE) projects using normalized metered energy consumption methods. Our analysis covers projects that occurred through an R-PACE program between 2009 and 2017 and includes more than 25,000 electricity meters and 15,000 gas meters. We employ a comparison group, drawn from other R-PACE households with similar locational and usage characteristics whose projects were implemented at different times, to control for some non-project and non-weather factors that may impact energy use. We find that projects consisting of energy efficiency technologies save, on average, about 3% of household electricity usage and 3.5% of household gas usage. R-PACE financing, however, can be used to install central heating or air conditioning equipment for the first time. These projects would be expected to increase energy consumption. Since our data do not directly indicate which projects are new installations, we develop a simple algorithm for identifying them. Removing these inferred installation projects yields average savings of about 5% for electricity and 6% for gas for those households that remain in the sample. Given the mild California climate and the results of another study of similar California projects using similar methods, these results are in line with expectations. Solar PV projects yield large reductions in grid electricity use, averaging 69% of household consumption. We estimate that, collectively, all R-PACE projects installed in California through 2019 would generate annual reductions in grid-tied electricity consumption of 506 GWh (mostly due to solar PV) and gas consumption reductions of 2 million therms in a normal weather year. These impacts are equivalent to the electricity consumption of about 74,000 California households (including both efficiency and PV generation) and the gas consumption of about 4700 California households.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

Robust cosmological inference from non-linear scales with k -th nearest neighbour statistics

ABSTRACT We present the methodology for deriving accurate and reliable cosmological constraints from non-linear scales ($\lt 50\, h^{-1}$ Mpc) with k-th nearest neighbour (kNN) statistics. We detail our methods for choosing robust minimum scale cuts and validating galaxy–halo connection models. Using cross-validation, we identify the galaxy–halo model that ensures both good fits and unbiased predictions across diverse summary statistics. We demonstrate that we can model kNNs effectively down to transverse scales of $r_{\rm p}\sim 3\, h^{-1}$ Mpc and achieve precise and unbiased constraints on the matter density and clustering amplitude, leading to a 2 per cent constraint on σ8. Our simulation-based model pipeline is resilient to varied model systematics, spanning simulation codes, halo finding, and cosmology priors. We demonstrate the effectiveness of this approach through an application to the Beyond-2p mock challenge. We propose further explorations to test more complex galaxy–halo connection models and tackle potential observational systematics.

79 ASTRONOMY AND ASTROPHYSICS↗

SU-E-J-128: Two-Stage Atlas Selection in Multi-Atlas-Based Image Segmentation

Purpose: In the new era of big data, multi-atlas-based image segmentation is challenged by heterogeneous atlas quality and high computation burden from extensive atlas collection, demanding efficient identification of the most relevant atlases. This study aims to develop a two-stage atlas selection scheme to achieve computational economy with performance guarantee. Methods: We develop a low-cost fusion set selection scheme by introducing a preliminary selection to trim full atlas collection into an augmented subset, alleviating the need for extensive full-fledged registrations. More specifically, fusion set selection is performed in two successive steps: preliminary selection and refinement. An augmented subset is first roughly selected from the whole atlas collection with a simple registration scheme and the corresponding preliminary relevance metric; the augmented subset is further refined into the desired fusion set size, using full-fledged registration and the associated relevance metric. The main novelty of this work is the introduction of an inference model to relate the preliminary and refined relevance metrics, based on which the augmented subset size is rigorously derived to ensure the desired atlases survive the preliminary selection with high probability. Results: The performance and complexity of the proposed two-stage atlas selection method were assessed using a collection of 30 prostate MR images. It achieved comparable segmentation accuracy as the conventional one-stage method with full-fledged registration, but significantly reduced computation time to 1/3 (from 30.82 to 11.04 min per segmentation). Compared with alternative one-stage cost-saving approach, the proposed scheme yielded superior performance with mean and medium DSC of (0.83, 0.85) compared to (0.74, 0.78). Conclusion: This work has developed a model-guided two-stage atlas selection scheme to achieve significant cost reduction while guaranteeing high segmentation accuracy. The benefit in both complexity and performance is expected to be most pronounced with large-scale heterogeneous data.

60 APPLIED LIFE SCIENCES↗

A deep learning-based direct forecasting of CO 2 plume migration

Accurate and timely forecasts of CO 2 plume evolution in geological reservoirs are crucial for CO 2 migration detection, leakage risk assessment, and operation decision support. Conventional forecasting usually adopts a two-step strategy, first calibrating reservoir model parameters against observations using iterative inverse modeling (or history matching) and then applying the calibrated model for predictions. This method impedes real-time forecasts due to the heavy computational demand in inverse modeling and may suffer from poor prediction accuracy because of the limited observation data. In this work, we propose a deep learning-based latent space mapping framework to forecast CO 2 plume migration directly by avoiding the inverse modeling. We first use the convolutional autoencoder to map the high-dimensional complex plume extents onto low-dimensional latent space. Next, we use neural networks to learn the relationship between the observation variables and the prediction latent variables. And then for given observation data, we infer the prediction values directly. This one-step direct forecasting is computationally efficient which requires a few number of parallelizable reservoir simulations and it can provide accurate predictions with limited observations by learning the observation-prediction relationship in the reduced dimension. Therefore, our proposed method enables an in-time forecast of dynamic CO 2 plume distributions. In this work, we demonstrate the effectiveness and accuracy of our method in predicting the CO 2 plume migration using four metrics such as plume area, centroid movement distance, and plume spreading in the primary and secondary directions. And the spatio-temporal evolution patterns of plume migration under diverse geological complexities are also accurately quantified.

15 GEOTHERMAL ENERGY↗

Bayesian inference of structured latent spaces from neural population activity with the orthogonal stochastic linear mixing model

The brain produces diverse functions, from perceiving sounds to producing arm reaches, through the collective activity of populations of many neurons. Determining if and how the features of these exogenous variables (e.g., sound frequency, reach angle) are reflected in population neural activity is important for understanding how the brain operates. Often, high-dimensional neural population activity is confined to low-dimensional latent spaces. However, many current methods fail to extract latent spaces that are clearly structured by exogenous variables. This has contributed to a debate about whether or not brains should be thought of as dynamical systems or representational systems. Here, we developed a new latent process Bayesian regression framework, the orthogonal stochastic linear mixing model (OSLMM) which introduces an orthogonality constraint amongst time-varying mixture coefficients, and provide Markov chain Monte Carlo inference procedures. We demonstrate superior performance of OSLMM on latent trajectory recovery in synthetic experiments and show superior computational efficiency and prediction performance on several real-world benchmark data sets. We primarily focus on demonstrating the utility of OSLMM in two neural data sets: μ ECoG recordings from rat auditory cortex during presentation of pure tones and multi-single unit recordings form monkey motor cortex during complex arm reaching. We show that OSLMM achieves superior or comparable predictive accuracy of neural data and decoding of external variables (e.g., reach velocity). Most importantly, in both experimental contexts, we demonstrate that OSLMM latent trajectories directly reflect features of the sounds and reaches, demonstrating that neural dynamics are structured by neural representations. Together, these results demonstrate that OSLMM will be useful for the analysis of diverse, large-scale biological time-series datasets.

59 BASIC BIOLOGICAL SCIENCES↗

Order-by-order uncertainties of nucleon-nucleon Wolfenstein amplitudes in chiral effective field theory

Quantum mechanical invariance principles dictate the most general operator structure that can be present in the nucleon-nucleon (NN) interaction. Five independent operators appear in the on-shell NN amplitude together with five corresponding coefficient functions. The usual choice for these coefficient functions is known as the NN Wolfenstein amplitudes. We analyze the order-by-order convergence of each of the five NN Wolfenstein amplitudes predicted by a semilocal coordinate space potential implementation of chiral effective field theory (𝜒⁢EFT). We do this at laboratory kinetic energies between 25 and 200 MeV for both neutron-proton and proton-proton scattering. Our analysis uses the Gaussian-process methods developed by the BUQEYE Collaboration to describe the contributions of each 𝜒⁢EFT order, and so yields truncation uncertainties for each Wolfenstein amplitude that are correlated across scattering angles. We combine information on the size of different orders in the EFT to infer the 𝜒⁢EFT breakdown scale for each amplitude, finding, on average, Λ 𝑏 between 750 and 800 MeV. Furthermore, with this choice of Λ 𝑏 , the EFT truncation uncertainties cover both higher-order results and empirical Wolfenstein amplitudes well for all orders other than the leading order.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Hierarchical Bayesian Modeling for Cosmology: Can NPE reliably replace MCMC?

Hierarchical neural posterior estimation has its place Hierarchical Bayesian Modeling (HBM) combined with MCMC algorithms has been shown to provide more robust and accurate inference for real-world phenomena in which nature takes a nested form. However, MCMC-based inference can be computationally expensive, and its performance often suffers for complex posterior geometries. These costs are especially pertinent for HBM. Studies have recently demonstrated the potential for a flexible, expressive, and amortized hierarchical neural posterior estimator (HNPE) built on Normalizing Flows. These studies have mostly been performed on simple datasets, or they focus on a single parameter from each level of the hierarchy. A systematic study analyzing how both hierarchical methods compare for more complex and realistic datasets is necessary before applying HNPE for scientific measurements. Here, we re-explore the theory behind HNPE and conduct comparative numerical experiments of HNPE and MCMC-based HBM methods on real and synthetic data, including strong gravitational lensing simulations. In particular, we use a suite of diagnostics to show trade-offs in terms of accuracy, precision, time to train or sample, reproducibility, and the need for expert domain knowledge. Especially for higher dimensional and complex posteriors, HNPE is expected to drastically improve on time for inference, accuracy, and precision with an upfront training time cost.

Hur, Rachel [Chicago U.] (ORCID:000900089890445X)↗

Two-stage atlas subset selection in multi-atlas based image segmentation

Purpose: Fast growing access to large databases and cloud stored data presents a unique opportunity for multi-atlas based image segmentation and also presents challenges in heterogeneous atlas quality and computation burden. This work aims to develop a novel two-stage method tailored to the special needs in the face of large atlas collection with varied quality, so that high-accuracy segmentation can be achieved with low computational cost. Methods: An atlas subset selection scheme is proposed to substitute a significant portion of the computationally expensive full-fledged registration in the conventional scheme with a low-cost alternative. More specifically, the authors introduce a two-stage atlas subset selection method. In the first stage, an augmented subset is obtained based on a low-cost registration configuration and a preliminary relevance metric; in the second stage, the subset is further narrowed down to a fusion set of desired size, based on full-fledged registration and a refined relevance metric. An inference model is developed to characterize the relationship between the preliminary and refined relevance metrics, and a proper augmented subset size is derived to ensure that the desired atlases survive the preliminary selection with high probability. Results: The performance of the proposed scheme has been assessed with cross validation based on two clinical datasets consisting of manually segmented prostate and brain magnetic resonance images, respectively. The proposed scheme demonstrates comparable end-to-end segmentation performance as the conventional single-stage selection method, but with significant computation reduction. Compared with the alternative computation reduction method, their scheme improves the mean and medium Dice similarity coefficient value from (0.74, 0.78) to (0.83, 0.85) and from (0.82, 0.84) to (0.95, 0.95) for prostate and corpus callosum segmentation, respectively, with statistical significance. Conclusions: The authors have developed a novel two-stage atlas subset selection scheme for multi-atlas based segmentation. It achieves good segmentation accuracy with significantly reduced computation cost, making it a suitable configuration in the presence of extensive heterogeneous atlases.

60 APPLIED LIFE SCIENCES↗

Using SIMS to decode noisy stratigraphic δ 13 C variations in Ediacaran carbonates

Carbonate carbon isotope (δ 13 C carb ) chemostratigraphy is a valuable tool in Precambrian stratigraphic correlation. The effectiveness of this tool rests on the assumption that δ 13 C carb data record global seawater signals. However, in some cases δ 13 C carb data exhibit rapid and noisy stratigraphic variations that may have been influenced by authigenic or diagenetic carbonate minerals. To assess the contribution of non-primary minerals towards bulk carbonate carbon isotope values, we acquired SIMS (secondary ion mass spectrometry) δ 13 C carb data, electron microscopic data, and CL (cathodoluminescence) microscopic data from two Ediacaran successions—the lower Member II of the Doushantuo Formation in South China and the Mooifontein Member of the Zaris Formation in southern Namibia. The Doushantuo samples came from a stratigraphic interval with noisy meter-scale δ 13 C carb variations of up to 10‰, whereas the Mooifontein Member is characterized by consistent δ13Ccarb values with limited meter-scale variations of less than ~4‰. Our data show that the meter-scale stratigraphic variations in δ 13 C carb are also mirrored in the SIMS data at μm-mm scales in both Doushantuo and Mooifontein samples. In the Doushantuo samples, SIMS δ 13 C carb values of authigenic calcite vary by up to 10‰ over μm-mm scales and can be either higher or lower than those of the co-existing dolomite matrix, which also appears to be affected by authigenic carbonate on the basis of petrographic observation. Bulk-sample δ 13 C carb values measured on powders microdrilled from the same SIMS specimens are within the range of SIMS δ 13 C carb values. Thus, we infer that bulk-sample δ 13 C carb values of Doushantuo samples represent mixtures of different carbonate components. The Doushantuo and Mooifontein SIMS δ 13 C carb data validate the traditional method of using chemostratigraphic consistency to evaluate authigenic/diagenetic alteration, but given that secondary phases may be 13 C-enriched relative to matrix carbonate, caution is warranted when taking maximum values to approximate primary chemostratigraphic trends.

58 GEOSCIENCES↗

The SRG/eROSITA All-Sky Survey: Optical identification and properties of galaxy clusters and groups in the western galactic hemisphere

The first SRG/eROSITA All-Sky Survey (eRASS1) provides the largest intracluster medium-selected galaxy cluster and group catalog covering the western Galactic hemisphere. Compared to samples selected purely on X-ray extent, the sample purity can be enhanced by identifying cluster candidates using optical and near-infrared data from the DESI Legacy Imaging Surveys. Using the red-sequence-based cluster findereROMaPPer, we measured individual photometric properties (redshiftz λ , richnessλ, optical center, and BCG position) for 12000 eRASS1 clusters over a sky area of 13 116 deg 2 , augmented by 247 cases identified by matching the candidates with known clusters from the literature. The median redshift of the identified eRASS1 sample isz= 0.31, with 10% of the clusters atz> 0.72. The photometric redshifts have an accuracy ofδz/(1 +z) ≲ 0.005 for 0.05 specand velocity dispersionσ) were measured a posteriori for a subsample of 3210 and 1499 eRASS1 clusters, respectively, using an extensive compilation of spectroscopic redshifts of galaxies from the literature. We infer that the primary eRASS1 sample has a purity of 86% and optical completeness >95% forz> 0.05. For these and further quality assessments of the eRASS1 identified catalog, we applied our identification method to a collection of galaxy cluster catalogs in the literature, as well as blindly on the full Legacy Surveys covering 24069 deg 2 . Using a combination of these cluster samples, we investigated the velocity dispersion-richness relation, finding that it scales with richness as log(λ norm ) = 2.401 × log(σ) − 5.074 with an intrinsic scatter ofδ in = 0.10 ± 0.01 dex. The primary product of our work is the identified eRASS1 cluster catalog with high purity and a well-defined X-ray selection process, opening the path for precise cosmological analyses presented in companion papers.

Astronomy & Astrophysics↗

Simultaneous inference of equation of state parameters and unknown data errors with uncertainty quantification via hierarchical Bayesian posterior maximization

Equations of state (EOSs) are a key component in running hydrodynamic simulations as they relate the thermodynamic states for the material. The Davis reactants EOS is commonly used for modeling high explosives (HEs), and the EOS model parameters are calibrated using material specific data. The calibrations are often performed with uncertainty quantification via Bayesian inference to account for uncertainty in the data and generate ensembles of likely parameters. However, there are relatively few HE data sets to use for calibration and many are historical and lack error information. In this work, we simultaneously calibrate the Davis reactants EOS model parameters and unknown data error terms for the high explosive PBX 9501. To quantify the uncertainty in the models and the data, we use a Bayesian framework for the calibration and compute the hierarchical Bayesian posterior distribution with both a posteriori maximization approach and Markov Chain Monte Carlo. In general, we find that, given our assumptions, the two approaches result in similar calibrated parameters, posterior covariance matrices, and insights about the parameters but that the posterior maximization requires far less computational resources.

97 MATHEMATICS AND COMPUTING↗