Posterior Posterior Probability Density Function Research Articles

Bayesian Estimation of Past Astronomical Frequencies, Lunar Distance, and Length of Day From Sediment Cycles

AbstractAstronomical cycles recorded in stratigraphic sequences offer a powerful data source to estimate Earth's axial precession frequency k, as well as the frequency of rotation of the planetary perihelia (gi) and of the ascending nodes of their orbital planes (si). Together, these frequencies control the insolation cycles (eccentricity, obliquity and climatic precession) that affect climate and sedimentation, providing a geologic record of ancient Solar system behavior spanning billions of years. Here, we introduce two Bayesian methods that harness stratigraphic data to quantitatively estimate ancient astronomical frequencies and their uncertainties. The first method (TimeOptB) calculates the posterior probability density function (PDF) of the axial precession frequency k and of the sedimentation rate u for a given cyclostratigraphic data set, while setting the Solar system frequencies gi and si to fixed values. The second method (TimeOptBMCMC) applies an adaptive Markov chain Monte Carlo algorithm to efficiently sample the posterior PDF of all the parameters that affect astronomical cycles recorded in stratigraphy: five gi, five si, k, and u. We also include an approach to assess the significance of detecting astronomical cycles in cyclostratigraphic records. The methods provide an extension of current approaches that is computationally efficient and well suited to recover the history of astronomical cycles, Earth‐Moon history, and the evolution of the Solar system from geological records. As case studies, data from the Xiamaling Formation (N. China, 1.4 Ga) and ODP Site 1262 (S. Atlantic, 55 Ma) are evaluated, providing updated estimates of astronomical frequencies, Earth‐Moon history, and secular resonance terms.

Gaussian process metamodel and Markov chain Monte Carlo-based Bayesian inference framework for stochastic nonlinear model updating with uncertainties

The estimation of the posterior probability density function (PDF) of unknown parameters remains a challenge in stochastic nonlinear model updating with uncertainties; thus, a novel Bayesian inference framework based on the Gaussian process metamodel (GPM) and the advanced Markov chain Monte Carlo (MCMC) method is proposed in this paper. The instantaneous characteristics (ICs) of the decomposed measurement response, calculated using the Hilbert transform and the discrete analytical mode decomposition methodology, are extracted as nonlinear indices and further used to construct the likelihood function. Then, the posterior PDFs of structural nonlinear model parameters are derived based on the Bayesian theorem. To precisely calculate the posterior PDF, an advanced MCMC approach, i.e., delayed rejection adaptive Metropolis-Hastings (DRAM) algorithm, is adopted with the advantages of a high acceptance ratio and strong robustness. However, as a common shortage in most MCMC methods, the resampling technology is still applied, and numerous iterations of nonlinear simulations are conducted to ensure accuracy, thus directly reducing the computational efficiency of the DRAM. To address the abovementioned issue, a mathematical regression metamodel of the GPM with a polynomial kernel function is adopted in this paper instead of the traditional finite element model (FEM) to simulate a nonlinear response for the reduction of computational cost, and the hyperparameters are further estimated using the conjugate gradient optimization methodology. Finally, numerical simulations concerning a Giuffré–Menegotto–Pinto modeled steel-frame structure and a seven-storey base-isolated structure are conducted. Furthermore, a shake-table experimental test of a nonlinear steel framework is investigated to validate the accuracy of the Bayesian inference method. Both simulations and experiment demonstrate that the proposed GPM and DRAM-based Bayesian method effectively estimates the posterior PDF of unknown parameters and is appropriate for stochastic nonlinear model updating even with multisource uncertainties.

Gaussian Bayesian network model of healthcare, food and energy sectors in the pandemic: Türkiye case

Healthcare, food, and energy are the basic needs of life in the globalizing world. Humankind's quest to maintain a healthy life and find ways to meet its needs has continued since its existence. The causal relations between the healthcare, food, and energy sectors are explored using data from the pandemic when COVID-19 was a global risk, and human health sustainability underwent a complicated process. It aims to examine the interaction between the healthcare, food, and energy sectors and model the causal relationship within the framework of probabilistic dependencies. For this purpose, the relationships between these sectors during the pandemic are modeled via Bayesian Networks (BNs). This highly successful inference method makes the complex structure of causal relationships graphically understandable. The data consists of stock returns at the end of the business day between March 11, 2020, when the pandemic was declared, and December 26, 2022. Data on 13 stocks actively traded on the Istanbul Stock Exchange (BIST) during the 700 days were obtained from tr.investing.com. Causal modeling uses Gaussian Bayesian Networks (GBNs) for continuous variables. To make the inferences drawn from the data more successful and minimize the loss of information, the GBN model is built with continuous variables. The posterior Probability Density Functions (PDFs) of the stocks in the network are constructed over the structure of the Directed Acyclic Graphs (DAG) of the BNs, and inferences are made by querying possible cases (tips). Markov Chain Monte Carlo (MCMC) simulations are performed with the posterior PDFs, and measures of the central tendency of the stocks are calculated. GBNs are used to generate daily return estimates for ULKER with the lowest MSE (1.06e-03) and RMSE (3.22e-02) values and ULUUN with the highest MSE (3.43e-03) and RMSE (5.83e-02) values.

Bayesian modal identification of non-classically damped systems using time-domain data

In this work, we present a Bayesian time–domain method for identifying the complex modal parameters of a linear dynamic system from ambient vibration data. Complex modal parameter identification is valid for both classically and non-classically damped systems, in contrast to real modal parameter identification, which is exclusively applicable to classically damped systems. A Bayesian method is adopted here which makes it possible to determine the joint posterior Probability Density Function (PDF) of the modal parameters for given measured data and modeling assumptions. For sufficient data, it is seen that the modal parameters are globally identifiable, and the posterior PDF is approximated by a Gaussian distribution with mean at the optimal parameters that maximize the posterior PDF. The uncertainty in the identified modal parameters is given by the covariance matrix of the posterior PDF. Analytical expressions are derived for computing the gradient for efficient optimization and the Hessian for determining the covariance matrix. Further, it is shown that the assumption of classical damping during analysis results in poor prediction of damping ratios in the case of non-classically damped systems. The application of the proposed approach is illustrated by means of simulated examples and an experimental study.

On the Bayesian model updating based on model reduction using complex modal data for damage detection

An approach for the Bayesian model updating of a linear dynamic system using complex modal data identified from dynamic test data is proposed in this paper. Very few works have utilized complex modal data for Bayesian model updating, and these works consider only the Most Probable Value (MPV) of the modal parameters as the modal data. On the other hand, the present work considers the posterior uncertainties of the modal parameters along with the MPVs as the modal data. Additionally, dynamic reduction has been applied to downsize the full system model to a reduced model with master Degrees of Freedom (DOFs) of the reduced model confined to the observed DOFs. Dynamic reduction facilitates updating of structural parameters without the requirement of mode matching. Additional uncertain parameters in the form of system modal frequencies, damping ratios, and partial mode shapes are introduced to establish a link between the structural model parameters and the modal data through the eigenvalue equation. Detailed formulation leading to the development of the posterior Probability Density Function (PDF) is presented, and a new Metropolis-within-Gibbs (MWG) sampler is proposed to simulate samples from the posterior PDF. Furthermore, a formulation is presented for evaluating the probability of damage based on the posterior samples obtained using the proposed approach from the structure's undamaged and possibly-damaged state. The proposed approach is validated and demonstrated comprehensively through simulated examples and an experimental study. Lastly, the performance of the proposed approach is compared with an alternative approach which is developed by integrating out the partial system mode shape from the posterior PDF.

CA mortar void identification for slab track utilizing time-domain Markov chain Monte Carlo-based Bayesian approach

This paper investigates the feasibility and practicability study on the use of Markov chain Monte Carlo (MCMC)-based Bayesian approach for identifying the cement-emulsified asphalt (CA) void of the slab track system utilizing the measured vibration data. A newly developed model class identification algorithm was extended and integrated with the MCMC-based Bayesian approach for the first time to identify the CA mortar void that may partly extend to a neighborhood region. Not only the most probable values of the scaling factors to the mortar stiffness can be calculated, but also the damage probability of model parameters using the posterior probability density function (PDF) can be estimated, and the void can be clearly identified by the MCMC-based Bayesian approach. The proposed methodology was experimentally verified and positive outcomes were obtained. The detection results illustrate that the proposed method not only can successfully assess the void location of the CA mortar but also provide the information of the damage severity, and the posterior PDFs of model parameters can be also calculated by using kernel density estimation to quantitatively describe the uncertainty of the model.

Adaptive genetic algorithm-based particle herding scheme for mitigating particle impoverishment

Particle filters (PFs) estimate the true parameter state from samples of states, called particles, that are drawn to construct an approximated posterior probability density function (PDF). Each drawn particle has its own probability of being the true state called importance weight. Both particle degeneracy and impoverishment are important problems found in generic PFs that cause particles to become trapped at local maxima. Genetic algorithms (GAs) were applied to herd the drawn particles to discover new high-likelihood state values in order to reshape the approximated posterior PDF and to enhance the accuracy of state estimation, while the particle diversity is maintained. We calculated the threshold weight for classifying particles as low-weight particles and high-weight particles. Each parent pair consists of a high-weight particle and a low-weight particle. High-weight particles are always kept unchanged. After each new offspring particle is found using GA operations, we must check its weight to ensure that it is actually superior to its low-weight parent. Otherwise, this offspring must not be allowed to replace its low-weight parent and must be destroyed. To ensure that the number of high-weight particles is adequate, some offspring particles evolve to be the new high-weight parent particles to produce high-weight offspring with further promoted diversity. Results showed that our approach significantly improves posterior PDF approximation and reduces errors in state estimation.

Bayesian updating of reliability by cross entropy-based importance sampling

Bayesian analysis enables a consistent updating of the failure probability of engineering systems when new data is available. To this end, we introduce an adaptive importance sampling (IS) method based on the principle of cross entropy (CE) minimization. The key contribution is a novel IS density associated with the posterior probability density function (PDF) of the uncertain parameters, that facilitates efficient sampling from the important region of the failure domain, especially when the failure event is rare. The IS density is designed via a two-step procedure. The first step involves construction of a sample-based approximation of the posterior, which we build using the CE method. Here the aim is to determine the parameters of a chosen parametric distribution family that minimize its Kullback–Leibler divergence from the posterior PDF. The second step of the proposed method constructs the desired IS density for sampling the failure domain through a second round of CE minimization, starting from the approximate posterior obtained in the first step. An adaptive, multi-level approach is employed to solve the two CE optimization problems. The IS densities deduced in the two steps are then applied to construct an efficient estimator for the posterior probability of failure. Through numerical studies, we investigate and demonstrate the efficacy of the method in accurately estimating the reliability of engineering systems with rare failure events.

Bayesian updating of model parameters using adaptive Gaussian process regression and particle filter

Bayesian model updating provides a powerful framework for updating and uncertainty quantification of models by making use of observations, following probability rules in the treatment of uncertainty. Particle filter (PF) and Bayesian Updating with Structural Reliability method (BUS) have been developed by researchers as promising computational tools for this purpose. However, reducing computational cost in the updating process, especially for complex models, remains one of the key challenges. Surrogate model approach achieves this by appropriately replacing, possibly adaptively, the evaluation of the original computationally costly models with approximate ones that are much less costly. This study proposes an efficient method to estimate the posterior probability density function (PDF) of model parameters by using a surrogate model constructed using adaptive Gaussian Process Regression and PF. Of critical importance is the development of ‘learning function’, which finds the location of large values of posterior PDF and avoids those that have been visited. The proposed methodology is illustrated using a single-variable example and compared with PF and BUS. Its application is illustrated through an example of structural dynamics and another one on settlement prediction by soil–water coupled FEM with Cam-clay model.

Bayesian estimation of the low-energy constants up to fourth order in the nucleon-nucleon sector of chiral effective field theory

We use Bayesian methods and Hamiltonian Monte Carlo (HMC) sampling to infer the posterior probability density function (PDF) for the low-energy constants (LECs) up to next-to-next-to-next-to-leading order (N3LO) in a chiral effective field theory ($\ensuremath{\chi}\mathrm{EFT}$) description of the nucleon-nucleon interaction. In a first step, we condition the inference on neutron-proton and proton-proton scattering data and account for uncorrelated $\ensuremath{\chi}\mathrm{EFT}$ truncation errors. We demonstrate how to successfully sample the 31-dimensional space of LECs at N3LO using a revised HMC inference protocol. In a second step we extend the analysis by means of importance sampling and an empirical determination of the neutron-neutron scattering length to infer the posterior PDF for the leading charge-dependent contact LEC in the $^{1}S_{0}$ neutron-neutron interaction channel. While doing so we account for the $\ensuremath{\chi}\mathrm{EFT}$ truncation error via a conjugate prior. We use the resulting posterior PDF to sample the posterior predictive distributions for the effective range parameters in the $^{1}S_{0}$ wave as well as the strengths of charge-symmetry breaking and charge-independence breaking. We conclude that empirical point-estimate results of isospin breaking in the $^{1}S_{0}$ channel are consistent with the PDFs obtained in our Bayesian analysis and that, when accounting for $\ensuremath{\chi}\mathrm{EFT}$ truncation errors, one must go to next-to-next-to-leading order to confidently detect isospin breaking effects.

Bayesian updating and marginal likelihood estimation by cross entropy based importance sampling

We introduce a novel simulation-based method for Bayesian analysis to learn model parameters based on data. The method employs importance sampling (IS) to construct a sample-based approximation of the posterior probability density function (PDF) and estimate the marginal likelihood. We propose to build the IS density through an adaptive sampling approach based on the cross entropy (CE) method. The aim is to identify the parameters of a chosen parametric distribution family that minimize its Kullback-Leibler divergence from the target posterior PDF. An adaptive multi-level approach, based on tempering of the likelihood function, is proposed to efficiently solve this CE optimization problem. We investigate the appropriate choice of the parametric distribution, depending on the number of uncertain model parameters and nature of the posterior density. Numerical studies demonstrate the performance of the proposed method.

Development of a two-phase adaptive MCMC method for efficient Bayesian model updating of complex dynamic systems

The fundamental problem of Bayesian model updating is identifying the posterior probability density function (PDF) of uncertain model parameters. Markov chain Monte Carlo (MCMC) has been recognized to be potential for this problem. However, the main difficulty is that many samples need to be generated first to locate the regions of high probability. This could be time consuming for complex dynamic systems. A novel two-phase adaptive MCMC method is developed in this work to tackle this difficulty. A parameter-space search algorithm is used to do a systematic search to find output-equivalent points, which are in the neighborhood of the regions of high probability. To generate samples of uncertain model parameters in these regions for describing the shape of the posterior PDF, a Markov chain is constructed with these output-equivalent points as seeds. This is done by the proposed weighted MCMC algorithm based on the modified Metropolis-Hasting (MH) algorithm. To enhance the efficiency of the algorithm, a new proposal PDF is developed using the seeds. The covariance matrix of the proposal PDF is adaptively updated with the latest set of generated samples. Furthermore, the optimal prediction-error variance as a function of model parameters is analytical derived and its values at the seeds are adaptively used to estimate the acceptance ratio in the modified MH algorithm. The proposed weighted MCMC algorithm generates samples in the region of high probability adaptively without going through multiple levels, which are computationally demanding. The posterior uncertainties in model updating can be rigorously considered by the generated samples with their weights proportional to their posterior PDF values. To verify the proposed method, numerical simulation of a two-story shear building with the challenging locally identifiable and unidentifiable cases was conducted. Further verification was also conducted by updating a finite element model of a full-scaled coupled building system utilizing measured vibration data from field tests.

Hydrogeological multiple-point statistics inversion by adaptive sequential Monte Carlo

For strongly non-linear and high-dimensional inverse problems, Markov chain Monte Carlo (MCMC) methods may fail to properly explore the posterior probability density function (PDF) given a realistic computational budget and are generally poorly amenable to parallelization. Particle methods approximate the posterior PDF using the states and weights of a population of evolving particles and they are very well suited to parallelization. We focus on adaptive sequential Monte Carlo (ASMC), an extension of annealed importance sampling (AIS). In AIS and ASMC, importance sampling is performed over a sequence of intermediate distributions, known as power posteriors, linking the prior to the posterior PDF. The AIS and ASMC algorithms also provide estimates of the evidence (marginal likelihood) as needed for Bayesian model selection, at basically no additional cost. ASMC performs better than AIS as it adaptively tunes the tempering schedule and performs resampling of particles when the variance of the particle weights becomes too large. We consider a challenging synthetic groundwater transport inverse problem with a categorical channelized 2D hydraulic conductivity field defined such that the posterior facies distribution includes two distinct modes. The model proposals are obtained by iteratively re-simulating a fraction of the current model using conditional multiple-point statistics (MPS) simulations. We examine how ASMC explores the posterior PDF and compare with results obtained with parallel tempering (PT), a state-of-the-art MCMC inversion approach that runs multiple interacting chains targeting different power posteriors. For a similar computational budget, ASMC outperforms PT as the ASMC-derived models fit the data better and recover the reference likelihood. Moreover, we show that ASMC partly retrieves both posterior modes, while none of them is recovered by PT. Lastly, we demonstrate how the power posteriors obtained by ASMC can be used to assess the influence of the assumed data errors on the posterior means and variances, as well as on the evidence. We suggest that ASMC can advantageously replace MCMC for solving many challenging inverse problems arising in the field of water resources.

Practical Bayesian Inversions for Rock Composition and Petrophysical Endpoints in Multimineral Analysis

Summary Rock composition can be related to conventional well logs through theoretical equations and petrophysical endpoints. Multimineral analysis is a formation evaluation tool that uses inversions to quantify rock composition from well logs. However, because of data errors and the multivariate selection of petrophysical endpoints, solutions from the multimineral analysis are nonunique. Many plausible realizations exhibit comparable data misfits. Therefore, the uncertainties in rock composition and petrophysical endpoints must be quantified but cannot be fulfilled by deterministic solvers. Stochastic Bayesian methods have been applied to assess the uncertainties, but the high run time, tedious parameter tuning, and need for specific prior information hinder their practical use. We implement Markov chain Monte Carlo with ensemble samplers (MCMCES) to assess the uncertainties of rock composition or petrophysical endpoints in the Bayesian framework. The resultant posterior probability density functions (PDFs) quantify the uncertainties. Our method has fewer tuning parameters and is more efficient in convergence than the conventional random walk Markov chain Monte Carlo (MCMC) methods in high-dimensional problems. We present two independent applications of MCMCES in multimineral analysis. We first apply MCMCES to assess the uncertainties in volume fractions with a suite of well logs and petrophysical endpoints. However, defining the petrophysical endpoints can be challenging in complex geological settings because the values of standard endpoints may not be optimal. Next, we use MCMCES to estimate petrophysical endpoints’ posterior PDFs when the endpoints are uncertain. Our methods provide posterior volume-fraction or petrophysical-endpoint realizations for interpreters to evaluate multimineral solutions. We demonstrate our approach with synthetic and field examples. Reproducible results are supplemented with the paper.

Bayesian Inference of Seismogenic Stress for the 2016 Mw 7.8 Kaikōura, New Zealand, Earthquake

ABSTRACT We use a Bayesian Markov chain Monte Carlo estimation strategy to estimate the state of pre-earthquake stress from eight published coseismic slip models (CSMs) of the 2016 Mw 7.8 Kaikōura, New Zealand, earthquake. Our estimation relies on the Wallace–Bott assumption, which equates the direction of slip and the orientation of maximum shear stress along a fault plane, and is analogous to methods that infer stress from focal mechanisms. We infer the orientations and relative magnitudes of the principal components of tensorial stress preceding the 2016 Mw 7.8 Kaikōura, New Zealand, earthquake as stress posterior probability density functions (PDFs). We find that the orientations of the principal stresses are well resolved, with weaker to no resolution of the relative magnitudes of the principal stresses. We form a nonmutually exclusive composite posterior PDF as an aggregate of the individual posterior PDFs estimated from each CSM, which allows for the true coseismic slip to be included in any or all of the CSMs. We find that the Kaikōura earthquake can be described by a homogeneous pre-earthquake tensorial stress, despite the complex pattern of slip on multiple fault segments. This state of crustal stress is best represented by an Andersonian thrust regime with a west-northwest–east-southeast trending, horizontal most compressive stress. In addition to describing the state of pre-earthquake stress, the nonmutually exclusive composite posterior PDF allows for the evaluation of the mechanical consistency of the features within CSMs, both individually and as an ensemble. The estimated stress posteriors allow for testing of the CSMs for consistency with known slip mechanisms on the Hope fault, as well as arbitration between differences in inferred slip presented by the CSMs.

Strategies to Enhance the Performance of Gaussian Mixture Model Fitting for Uncertainty Quantification

Summary When formulating history matching within the Bayesian framework, we may quantify the uncertainty of model parameters and production forecasts using conditional realizations sampled from the posterior probability density function (PDF). It is quite challenging to sample such a posterior PDF. Some methods [e.g., Markov chain Monte Carlo (MCMC)] are very expensive, whereas other methods are cheaper but may generate biased samples. In this paper, we propose an unconstrained Gaussian mixture model (GMM) fitting method to approximate the posterior PDF and investigate new strategies to further enhance its performance. To reduce the central processing unit (CPU) time of handling bound constraints, we reformulate the GMM fitting formulation such that an unconstrained optimization algorithm can be applied to find the optimal solution of unknown GMM parameters. To obtain a sufficiently accurate GMM approximation with the lowest number of Gaussian components, we generate random initial guesses, remove components with very small or very large mixture weights after each GMM fitting iteration, and prevent their reappearance using a dedicated filter. To prevent overfitting, we add a new Gaussian component only if the quality of the GMM approximation on a (large) set of blind-test data sufficiently improves. The unconstrained GMM fitting method with the new strategies proposed in this paper is validated using nonlinear toy problems and then applied to a synthetic history-matching example. It can construct a GMM approximation of the posterior PDF that is comparable to the MCMC method, and it is significantly more efficient than the constrained GMM fitting formulation (e.g., reducing the CPU time by a factor of 800 to 7,300 for problems we tested), which makes it quite attractive for large-scale history-matching problems. NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

Novel Solutions to the Three-Anchor ToA-Based Three-Dimensional Positioning Problem.

At least four non-coplanar anchor nodes (ANs) are required for the time-of-arrival (ToA)-based three-dimensional (3D) positioning to enable unique position estimation. Direct method (DM) and particle filter (PF) algorithms were developed to address the three-anchor ToA-based 3D positioning problem. The proposed DM reduces this problem to the solution of a quadratic equation, exploiting the knowledge about the workspace, to first estimate the x- or z-coordinate, and then the remaining two coordinates. The implemented PF uses 1000 particles to represent the posterior probability density function (PDF) of the AN’s 3D position. The prediction step generates new particles by a resampling procedure. The ToA measurements determine the importance of these particles to enable updating the posterior PDF and estimating the 3D position of the AN. Simulation results corroborate the viability of the developed DM and PF algorithms, in terms of accuracy and computational cost, in the pursuit and circumnavigation scenarios, and even with a horizontally coplanar arrangement of the three ANs. Therefore, it is possible to enable applications requiring real-time positioning, such as unmanned aerial vehicle (UAV) autonomous docking and circling a stationary (or moving) position, without the need for an excessive number of ANs.

Source estimation through sensitivity-based mode decomposition of sequential data

Characterizing explosions from infrasound is usually performed using posterior probability density functions (PDFs), which involves a large number of propagation model runs. These propagation models require atmospheric specifications and a model for representing small-scale atmospheric fluctuations, such as gravity waves. While defining a proper probabilistic model for the input parameters of these models may become a burden, in real-world cases, only a limited number of input parameters happens to influence the acoustic waveforms significantly. These parameters can be selected using quantitative importance measures or sensitivity indices that allow one to rank the sources of uncertainty. In this work, we introduce a method which is able to extract information from these sensitivity indices in time-frequency space, using waveforms generated by a full-wave propagation model. The extracted modes can be used to determine the relevant patterns that contribute most to the sensitivity captured in the data sequence or to project the perturbed model onto a system of fewer degrees of freedom. The restriction of the likelihood to subdomains where the sensitivity to gravity waves is weak allows a better estimate of the posterior PDF for source parameters. Demonstrations of the method are presented using waveforms recorded at thousands of kilometers of explosions.

Adaptive sequential Monte Carlo for posterior inference and model selection among complex geological priors

SUMMARY Bayesian model selection enables comparison and ranking of conceptual subsurface models described by spatial prior models, according to the support provided by available geophysical data. Deep generative neural networks can efficiently encode such complex spatial priors, thereby, allowing for a strong model dimensionality reduction that comes at the price of enhanced non-linearity. In this setting, we explore a recent adaptive sequential Monte Carlo (ASMC) approach that builds on annealed importance sampling (AIS); a method that provides both the posterior probability density function (PDF) and the evidence (a central quantity for Bayesian model selection) through a particle approximation. Both techniques are well suited to parallel computation and rely on importance sampling over a sequence of intermediate distributions, linking the prior and the posterior PDF. Each subsequent distribution is approximated by updating the particle weights and states, compared with the previous approximation, using a small pre-defined number of Markov chain Monte Carlo (MCMC) proposal steps. Compared with AIS, the ASMC method adaptively tunes the tempering between neighboring distributions and performs resampling of particles when the variance of the particle weights becomes too large. We evaluate ASMC using two different conceptual models and associated synthetic cross-hole ground penetrating radar tomography data. For the most challenging test case, we find that the ASMC method is faster and more reliable in locating the posterior PDF than state-of-the-art adaptive MCMC. The evidence estimates are found to be robust with respect to the choice of ASMC algorithmic variables and much less sensitive to the model proposal type than MCMC. The variance of the evidence estimates are best estimated by replication of ASMC runs, while approximations based on single runs provide comparable estimates when using a sufficient number of proposal steps in approximating each intermediate distribution.

Large-scale Gravitational Lens Modeling with Bayesian Neural Networks for Accurate and Precise Inference of the Hubble Constant

Abstract We investigate the use of approximate Bayesian neural networks (BNNs) in modeling hundreds of time delay gravitational lenses for Hubble constant (H 0) determination. Our BNN was trained on synthetic Hubble Space Telescope quality images of strongly lensed active galactic nuclei with lens galaxy light included. The BNN can accurately characterize the posterior probability density functions (PDFs) of model parameters governing the elliptical power-law mass profile in an external shear field. We then propagate the BNN-inferred posterior PDFs into an ensemble H 0 inference, using simulated time delay measurements from a plausible dedicated monitoring campaign. Assuming well-measured time delays and a reasonable set of priors on the environment of the lens, we achieve a median precision of 9.3% per lens in the inferred H 0. A simple combination of a set of 200 test lenses results in a precision of 0.5 km s−1 Mpc−1 (0.7%), with no detectable bias in this H 0 recovery test. The computation time for the entire pipeline—including the generation of the training set, BNN training and H 0 inference—translates to 9 minutes per lens on average for 200 lenses and converges to 6 minutes per lens as the sample size is increased. Being fully automated and efficient, our pipeline is a promising tool for exploring ensemble-level systematics in lens modeling for H 0 inference.