Posterior Distribution Of Parameters Research Articles

Parameter estimation for allometric trophic network models: A variational Bayesian inverse problem approach

Abstract Differential equation models are powerful tools for predicting biological systems, capable of projecting far into the future and incorporating data recorded at arbitrary times. However, estimating these models' parameters from observations can be challenging because numerical methods are often required to approximate their solution. An example of such a model is the allometric trophic network model, for which studies considering its inverse problem are limited, particularly in the Bayesian framework. Here we develop a variational Bayesian method for parameter inference of the allometric trophic network model and explore how accurately we can recover its parameter values. We represent the model as a Bayesian neural network, which combines an artificial neural network with Bayesian inference, using a surrogate for the posterior distribution of model parameters, and train this model by evolutionary optimization to avoid potentially costly computation of the gradient with respect to the model parameters. Using synthetic data, we compare the accuracy of this variational inference to ordinary least squares estimation. To reduce the number of estimated parameters, we focus on the inference of functional response parameters. Our variational Bayesian method yields parameter estimates that are comparable to the ordinary least squares results in terms of accuracy. The method provides a promising approach for including uncertainty quantification in parameter estimation, which the simple ordinary least squares approach as it is does not address. Regardless of the method, potential multimodality of the inference problem is nonetheless important to keep in mind. The present study suggests a technique for parameter inference of ordinary differential equation models in the Bayesian context. We propose the method especially for validation of the allometric trophic network model against empirical data.

Inferring stellar parameters and their uncertainties from high-resolution spectroscopy using invertible neural networks

New spectroscopic surveys will increase the number of astronomical objects in need of characterisation by more than an order of magnitude. Machine learning tools are required to address this data deluge in a fast and accurate fashion. Most machine learning algorithms cannot directly estimate error, making them unsuitable for reliable science. We aim to train a supervised deep-learning algorithm tailored for high-resolution observational stellar spectra. This algorithm accurately infers precise estimates while providing coherent estimates of uncertainties by leveraging information from both the neural network and the spectra. We trained a conditional invertible neural network (cINN) on observational spectroscopic data obtained from the GIRAFFE spectrograph (HR10 and HR21 setups) within the Gaia-ESO survey. A key feature of cINN is its ability to produce the Bayesian posterior distribution of parameters for each spectrum. By analysing this distribution, we inferred stellar parameters and their corresponding uncertainties. We carried out several tests to investigate how parameters are inferred and errors are estimated. We achieved an accuracy of 28K in $T_ eff $, 0.06 dex in $ g$, 0.03 dex in Fe/H $, and between 0.05 dex and 0.17 dex for the other abundances for high-quality spectra. Accuracy remains stable with low signal-to-noise ratio (between 5 and 25) spectra, with an accuracy of 39K in $T_ eff $, 0.08 dex in $ g$, and 0.05 dex in Fe/H $. The uncertainties obtained are well within the same order of magnitude. The network accurately reproduces astrophysical relationships both on the scale of the Milky Way and within smaller star clusters. We created a table containing the new parameters generated by our cINN. This neural network represents a compelling proposition for future astronomical surveys. These derived uncertainties are coherent and can therefore be reused in future works as Bayesian priors.

Stochastic Nonlinear Model Updating in Structural Dynamics Using a Novel Likelihood Function within the Bayesian-MCMC Framework

The study presents a novel approach for stochastic nonlinear model updating in structural dynamics, employing a Bayesian framework integrated with Markov Chain Monte Carlo (MCMC) sampling for parameter estimation by using an approximated likelihood function. The proposed methodology is applied to both numerical and experimental cases. The paper commences by introducing Bayesian inference and its constituents: the likelihood function, prior distribution, and posterior distribution. The resonant decay method is employed to extract backbone curves, which capture the non-linear behaviour of the system. A mathematical model based on a single degree of freedom (SDOF) system is formulated, and backbone curves are obtained from time response data. Subsequently, MCMC sampling is employed to estimate the parameters using both numerical and experimental data. The obtained results demonstrate the convergence of the Markov chain, present parameter trace plots, and provide estimates of posterior distributions of updated parameters along with their uncertainties. Experimental validation is performed on a cantilever beam system equipped with permanent magnets and electromagnets. The proposed methodology demonstrates promising results in estimating parameters of stochastic non-linear dynamical systems. Through the use of the proposed likelihood functions using backbone curves, the probability distributions of both linear and non-linear parameters are simultaneously identified. Based on this view, the necessity to segregate stochastic linear and non-linear model updating is eliminated.

Bayesian estimation of stochastic volatility jump diffusion model parameters using S&P 500 and VIX data

The Stochastic Volatility Jump Diffusion (SVJD) model is a common tool used for option pricing. Existing literature has employed many techniques in order to estimate the parameters of this model, including both Bayesian and frequentist approaches. In 2010, Duan and Yeh [Jump and volatility risk premiums implied by VIX. J Econ Dyn Control. 2010;34(11):2232–2244] used a frequentist approach with maximum likelihood estimation that included the VIX, the Chicago Board Options Exchange (CBOE) Volatility Index, which carries real time S&P 500 option price information in addition to the S&P 500 data. In this paper, we combine the previous Bayesian approach in Cape et al. [Estimating Heston's and Bates' models parameters using Markov Chain Monte Carlo simulation. J Stat Comput Simul. 2015;85(11):2295–2314] with the inclusion of VIX by Duan and Yeh. We provide the derivations of the conditional posterior distributions of the SVJD model's parameters and a guide to a Markov Chain Monte Carlo (MCMC) algorithm used to estimate the parameters. We apply the algorithm to the historical S&P 500 and VIX data and provide the subsequent parameter estimates. We offer these tools for others performing similar research with our R code available upon request.

Euclid preparation. LIII. LensMC, weak lensing cosmic shear measurement with forward modelling and Markov Chain Monte Carlo sampling

is a weak lensing shear measurement method developed for and Stage-IV surveys. It is based on forward modelling in order to deal with convolution by a point spread function (PSF) with comparable size to many galaxies, sampling the posterior distribution of galaxy parameters via Markov chain Monte Carlo, and marginalisation over nuisance parameters for each of the 1.5 billion galaxies observed by We quantified the scientific performance through high-fidelity images based on the Flagship simulations and emulation of the VIS images, realistic clustering with a mean surface number density of $ arcmin^ for galaxies, and $ arcmin^ for stars, and a diffraction-limited chromatic PSF with a full width at half maximum of $ $ and spatial variation across the field of view. measured objects with a density of $ arcmin^ in $4500 deg^2 $. The total shear bias was broken down into measurement (our main focus here) and selection effects (which will be addressed in future work). We found measurement multiplicative and additive biases of $m_1=(-3.6 $, $m_2=(-4.3 $, $c_1=(-1.78 $, and $c_2=(0.09 $; a large detection bias with a multiplicative component of $1.2 $ and an additive component of $-3 $; and a measurement PSF leakage of $ $ and $ $. When model bias is suppressed, the obtained measurement biases are close to requirement and largely dominated by undetected faint galaxies ($-5 $). Although significant, model bias will be straightforward to calibrate given its weak sensitivity on galaxy morphology parameters. is publicly available at

Hematocrits of red blood cell units increase during storage due to changes in mean corpuscular volume, impacting outcomes of red cell exchange procedures

Abstract Introduction Hematocrits of red blood cell (RBC) units are generally not measured by blood centers or transfusion services. Red blood cell exchange (RCE) is commonly used to treat sickle cell disease, and a key parameter needed in RCE is the average hematocrit of RBC units. However, there is no consensus on the average hematocrit to use or how it is affected by storage. The goal of our study is to determine the average hematocrit of RBC units by measuring hematologic parameters RBC units during storage and performing statistical modeling of hematocrit changes over time. Methods We sampled aliquots of leukoreduced RBC units in AS-1 (n=10) using the Terumo TSCD-II Sterile Tubing Welder to sterilely collect ~15mL of RBCs in transfer-pack containers. Aliquots were collected weekly from Day 7 to Day 42 of storage. Complete blood counts were performed using the Sysmex XN-3100 Hematology Analyzer. We statistically modeled changes in RBC units during storage by utilizing Bayesian multilevel linear regression models, allowing for capturing effects at the individual RBC-unit level. Posterior distributions of model parameters were estimated and processed using Stan and the rethinking and rstan packages in R. Posterior distributions are summarized with posterior means and 95%-credible intervals in parentheses. Model selection was done by calculating Akaike weights using the widely applicable information criterion (WAIC) to signify statistical support for each model. Results Multilevel linear regression models of hematologic parameters as a function of storage time were developed. Hematocrits increased with a posterior mean of 1.4% (0.98–1.75%) per week of storage, with posterior mean hematocrits of: 61.9% (60.2–63.6%) on Day 7, 63.9% (61.5–66.4%) on Day 21, and 66.0% (62.8–69.7%) on Day 42. These results indicate that a single average cannot be used for RBC units since hematocrits dramatically increase during storage. Mean corpuscular volume (MCV) increased with a posterior mean of 2.31 fL (1.4–3.71) per week whereas hemoglobin remained stable with a posterior mean increase of 0.008 g/dL (-0.016–0.031) per week. Using these results, we subsequently developed a multilevel regression model of hematocrit as a function of MCV. The model using MCV (WAIC=100.2, weight=100%) was favored over the model using time (WAIC=120.0, weight=0%), suggesting that changes in MCV are driving the increasing hematocrits. Conclusion Our study is the first to leverage Bayesian statistics to carefully quantify the average hematocrits of RBC units during storage time and to demonstrate that increasing hematocrits are caused by changes in MCV. These results indicate that transfusion protocols must consider the age of RBC units for RCE calculations, and provisions (e.g., requiring use of fresh units for RCE) may help minimize the variability in post-exchange testing. Moreover, our findings suggest that transfusion services should regularly measure hematocrits of RBC units in their inventory.

How uncertainty in calibration data affects the modeling of non-point source pollutant loads in baseflow

Baseflow is a major transport pathway for non-point source (NPS) pollutants. Watershed water quality (WWQ) models calibrated by low-quality data may produce misleading predictions of baseflow NPS pollutant loads, resulting in poor management decisions. We evaluated how models of the baseflow nitrate loads in the Huron River basin, southwest of Lake Erie, were affected by uncertainty in the calibration data. Based on a five-year time series of daily streamflow, nitrate concentration, and specific conductance, two sets of “observed” baseflow nitrate load data that include uncertainty were estimated using various tracer-based and non-tracer-based hydrograph separation methods, in conjunction with assumptions regarding baseflow nitrate concentrations. We calibrated the Soil and Water Assessment Tool plus (SWAT+) model with the two “observed” data sets and used the Generalized Likelihood Uncertainty Estimation (GLUE) approach to quantify parameter and predictive uncertainties. The results showed that baseflow accounted for 26 %–34 % of the mean annual total streamflow (11.8 m3/s) and 8 %–37 % of the mean annual total nitrate load (14.3 kg·ha−1·year−1) in the Huron River basin. The baseflow and nitrate load estimates from the non-tracer-based methods resembled those from the tracer-based method but had greater uncertainty. The posterior parameter distributions, as well as the weighted means and 90 % prediction intervals of the simulated baseflow nitrate loads, exhibited minimal variation when different calibration data sets for SWAT+ and different threshold likelihood values for GLUE were used. Our analysis emphasizes the necessity of calibrating WWQ models with baseflow pollutant loads/concentrations when addressing water quality issues related to baseflow. It also demonstrates the feasibility of utilizing multiple non-tracer-based hydrograph separation methods to estimate baseflow NPS pollutant loads. These non-tracer-based methods offer a simplicity and broader applicability compared to tracer-based methods. This study has provided insights into how calibration data uncertainty impacts the modeling of NPS pollution in baseflow and highlights the practical value of non-tracer-based hydrograph separation methods.

Calibration of stochastic, agent-based neuron growth models with approximate Bayesian computation.

Understanding how genetically encoded rules drive and guide complex neuronal growth processes is essential to comprehending the brain's architecture, and agent-based models (ABMs) offer a powerful simulation approach to further develop this understanding. However, accurately calibrating these models remains a challenge. Here, we present a novel application of Approximate Bayesian Computation (ABC) to address this issue. ABMs are based on parametrized stochastic rules that describe the time evolution of small components-the so-called agents-discretizing the system, leading to stochastic simulations that require appropriate treatment. Mathematically, the calibration defines a stochastic inverse problem. We propose to address it in a Bayesian setting using ABC. We facilitate the repeated comparison between data and simulations by quantifying the morphological information of single neurons with so-called morphometrics and resort to statistical distances to measure discrepancies between populations thereof. We conduct experiments on synthetic as well as experimental data. We find that ABC utilizing Sequential Monte Carlo sampling and the Wasserstein distance finds accurate posterior parameter distributions for representative ABMs. We further demonstrate that these ABMs capture specific features of pyramidal cells of the hippocampus (CA1). Overall, this work establishes a robust framework for calibrating agent-based neuronal growth models and opens the door for future investigations using Bayesian techniques for model building, verification, and adequacy assessment.

Bayesian inference of coupled groundwater flow and radiogenic helium-4 production and transport at the catchment scale

Hydrogeological numerical models are essential for assessing radioactive waste disposal by understanding groundwater flow systems. These models typically rely on hydraulic head data, with other state variables often underutilized in model inversions. In Flanders' Neogene aquifer, where safety studies for Boom Clay are ongoing, existing models face uncertainties due to dependence on hydraulic heads alone. This study aims to: i) develop a conceptual model of radiogenic Helium-4 (4Herad) production and transport to reproduce observed concentrations; ii) evaluate the usefulness of 4Herad observations as an additional state variable for inverse conditioning; and iii) assess how including 4Herad affects model calibration and uncertainty reduction. We address these objectives by incorporating radiogenic Helium-4 (4Herad) as an unconventional state variable in model inversion to tackle parameter non-uniqueness. Utilizing a Bayesian approach, we employ two 3D models: a groundwater flow (GWF) model conditioned by hydraulic heads and a coupled GWF model with 4Herad production and transport, conditioned by both hydraulic heads and 4Herad concentrations. We hypothesize that integrating 4Herad will improve model accuracy and reduce uncertainties compared to using hydraulic head data alone. Findings indicate that major 4Herad sources include crustal fluxes (~57 %) and in situ Boom Clay production (~25 %). The coupled inference narrows the posterior distributions of parameters, especially for hydraulic conductivity (e.g., Mol, Diest, Berchem formations, and the Rauw Fault) and recharge, showing that 4Herad observations effectively reduce uncertainty beyond hydraulic head data. Including 4Herad improves model reliability and prediction accuracy, highlighting its importance for refining groundwater flow parameters and derived fluxes. To the best of our knowledge, this study marks the first-ever attempt at harnessing 4Herad quantitative insights into flow and transport parameters following a Bayesian approach at a catchment scale, setting a precedent for future research and emphasizing the innovative nature of this work. This study showcases a state-of-the-art approach in hydrogeological modelling, demonstrating the effectiveness of integrating 4Herad with hydraulic head observations to produce more reliable model predictions. By reducing uncertainties in crucial groundwater fluxes, this approach offers significant potential for improving geological disposal safety studies. This pioneering work advocates for the continued use of 4Herad in hydrogeological modelling, emphasizing its innovative contribution to future research and safety studies. Plain language summaryUnderstanding how groundwater flows is crucial for safely storing radioactive waste. Traditionally, models used to study groundwater rely mostly on measurements of water pressure, while other valuable data often go unused. In the Neogene aquifer in Flanders, where researchers are studying the Boom Clay formation for potential waste storage, relying only on water pressure has created uncertainties in the models. This study explores a new approach by using a different type of measurement: radiogenic Helium-4 (4Herad). The goals of the study are threefold: first, to build a model that accurately represents how 4Herad is produced and moves through the groundwater; second, to test whether including 4Herad as an additional measurement improves the model compared to using only water pressure data; and third, to see how using 4Herad affects the accuracy and reliability of the model. The study uses two types of models: one that relies only on water pressure data and another that includes both water pressure and 4Herad measurements. The idea is that 4Herad could provide additional insights and help reduce uncertainties in the model. The findings show that 4Herad comes mainly from two sources: the surrounding crust and the Boom Clay itself. Including 4Herad in the model helps narrow down uncertainties, particularly regarding how water moves through different layers and how much water is being replenished. This research is the first to use 4Herad in this way, using a formal statistical method to improve groundwater models. The results suggest that 4Herad is a valuable addition for making more accurate predictions and improving safety assessments for radioactive waste storage.

Fault diagnosis based on incomplete sensor variables with a hierarchical semi-supervised Gaussian mixture classifier

We propose a hierarchical semi-supervised variational semi-Bayesian Gaussian mixture classifier based on the partially incomplete and unlabeled samples for the fault diagnosis of mechanical and electrical systems. These systems are typically complex in structure and are monitored by multiple sensors simultaneously. Some sensor variables in the collected data may be incomplete due to sensor malfunctions or transmission errors. Additionally, labeling the data can be a time-consuming and labor-intensive task, resulting in many unlabeled samples. To address these challenges, the missing sensor variables and the unavailable labels are treated as hidden values and handled within the framework of the Expectation Maximization algorithm. We employ a semi-Bayesian technique with variational inference to estimate the model parameters. Specifically, we introduce prior distribution to the mean and covariance matrix to address the possible singularity of the empirical covariance matrix. The weighting coefficients are left without putting prior distribution so that their values can decay to zero, effectively removing redundant components of the mixture model. The factorized distribution is utilized to approximate the posterior distribution of the model parameters, as well as the missing labels and other latent variables. Numerical and real case studies are carried out to verify the effectiveness of the proposed technique.

Efficient Bayesian updating for deformation prediction of high rock slopes induced by excavation with monitoring data

This study develops an efficient Bayesian updating method with monitoring data for predicting the deformation of high rock slopes induced by excavation. The importance ranking based on random forest is introduced to identify the key rock parameters as random variables in the Bayesian updating. The surrogate models using support vector machine are constructed to approximate the physical numerical models using FLAC3D for evaluating slope deformation. A practical example involving deformation prediction of the excavated left-bank rock slope for the well-known Baihetan hydropower station in southwest China is presented. The results indicate that the developed Bayesian updating method can efficiently and accurately update the posterior distributions of rock parameters and predict the deformation of high rock slopes induced by excavation. Incorporating the monitoring data of displacement into the Bayesian updating can effectively reduce the uncertainty of rock parameters and displacement prediction. As a result, the displacement predictions made by the Bayesian updating are closer to the monitoring data than the prior displacement predictions. In addition, incorporating more monitoring data of displacement from the previous excavation stages produces more accurate displacement predictions for subsequent excavation stages.

Can the young water fraction reduce predictive uncertainty in water transit time estimations?

Transit time distributions (TTDs) of streamflow are informative descriptors of catchment hydrological functioning and solute transport mechanisms. Conventional methods for estimating TTDs generally require model calibration against extensive tracer data time series, which are often limited to well-studied experimental catchments. We challenge this limitation and propose an alternative approach that uses the young water fraction (Fywobs), an increasingly used water age metric which represents the proportion of streamflow with a transit time younger than 2-3 months, and that can be robustly estimated with sparsely measured tracer data. To this end, we conducted a proof of concept study by modelling TTDs using StorAge Selection (SAS) functions with oxygen isotopes (δ18O) measurements for 23 diverse catchments in Germany. In a Monte-Carlo approach, we computed the (averaged) marginal TTDs of a prior parameter distribution and derived a model-based Fyw (Fywsim). We compared Fywsim with Fywobs, obtained from δ18O measurements, and constrained the prior SAS parameters distribution. Subsequently, we derived a posterior distribution of parameters and resulting model simulations. Our findings showed that using Fywobs to constrain the model effectively reduced parameter equifinality and simulation uncertainty. However, the value of Fywobs on reducing model uncertainty varied across sites, with larger values (Fywobs≥0.10) leading to simulations with a narrower uncertainty band and higher model efficiency, whilst smaller values (Fywobs≤0.05) had limited influence on reducing model output uncertainty. We discussed the potential and limitations of combining SAS functions with Fywobs, and considered broader implications of this approach for enhancing our understanding of catchment functioning and water quality status.

EXPRESS: Stratified Distributional Analysis - a Novel Perspective on RT Distributions.

Response times and their distributions serve as a powerful lens into cognitive processes. We present a novel statistical methodology called Stratified Distributional Analysis (SDA) to quantitatively assess how key determinants of response times (word frequency and length) shape their distributions. Taking advantage of the availability of millions of lexical decision response times in the English Lexicon Project and the British Lexicon Project, we made important advances into the theoretical issue of linking response times and word frequency by analysing RT distributions as a function of word frequency and word length. We tested these distributions against the lognormal, Wald, and Gamma distributions and three measures of word occurrence (word form frequencies obtained from subtitles and contextual diversity as operationalized as discourse contextual diversity and user contextual diversity). We found that the RT distributions were best described by a lnorm distribution across both megastudies when word occurrence was quantified by a contextual diversity measure. The link between the lnorm distribution and its generative process highlights the power of SDA in elucidating mechanisms that govern the generation of RTs through the fitting of probability distributions. Using a hierarchical Bayesian framework, SDA yielded posterior distributions for the distributional parameters at the single-participant level, enabling probabilistic predictions of response times as a function of word frequency and word length, which has the potential to serve as a diagnostic tool to uncover idiosyncratic features of word processing. Crucially, while we applied our parsimonious methodology to lexical decision response times, it is applicable to a variety of tasks such as word-naming and eye-tracking data.

Kalman tracking and parameter estimation of continuous gravitational waves with a pulsar timing array

Abstract Continuous nanohertz gravitational waves from individual supermassive black hole binaries may be detectable with pulsar timing arrays. A novel search strategy is developed, wherein intrinsic achromatic spin wandering is tracked simultaneously with the modulation induced by a single gravitational wave source in the pulse times of arrival. A two-step inference procedure is applied within a state-space framework, such that the modulation is tracked with a Kalman filter, which then provides a likelihood for nested sampling. The procedure estimates the static parameters in the problem, such as the sky position of the source, without fitting for ensemble-averaged statistics such as the power spectral density of the timing noise, and therefore complements traditional parameter estimation methods. It also returns the Bayes factor relating a model with a single gravitational wave source to one without, complementing traditional detection methods. It is shown via astrophysically representative software injections in Gaussian measurement noise that the procedure distinguishes a gravitational wave from pure noise down to a characteristic wave strain of h0 ≈ 2 × 10−15. Full posterior distributions of model parameters are recovered and tested for accuracy. There is a bias of ≈0.3 rad in the marginalised one-dimensional posterior for the orbital inclination ι, introduced by dropping the so-called ‘pulsar terms’. Smaller biases $\lesssim 10~{{\%}}$ are also observed in other static parameters.

A Comparative Study of Single-Chain and Multi-Chain MCMC Algorithms for Bayesian Model Updating-Based Structural Damage Detection

Bayesian model updating has received considerable attention and has been extensively used in structural damage detection. It provides a rigorous statistical framework for realizing structural system identification and characterizing uncertainties associated with modeling and measurements. The Markov Chain Monte Carlo (MCMC) is a promising tool for inferring the posterior distribution of model parameters to avoid the intractable evaluation of multi-dimensional integration. However, the efficacy of most MCMC techniques suffers from the curse of parameter dimension, which restricts the application of Bayesian model updating to the damage detection of large-scale systems. In addition, there are several MCMC techniques that require users to properly choose application-specific models, based on the understanding of algorithm mechanisms and limitations. As seen in the literature, there is a lack of comprehensive work that investigates the performances of various MCMC algorithms in their application of structural damage detection. In this study, the Differential Evolutionary Adaptive Metropolis (DREAM), a multi-chain MCMC, is explored and adapted to Bayesian model updating. This paper illustrates how DREAM is used for model updating with many uncertainty parameters (i.e., 40 parameters). Furthermore, the study provides a tutorial to users who may be less experienced with Bayesian model updating and MCMC. Two advanced single-chain MCMC algorithms, namely, the Delayed Rejection Adaptive Metropolis (DRAM) and Transitional Markov Chain Monte Carlo (TMCMC), and DREAM are elaborately introduced to allow practitioners to understand better the concepts and practical implementations. Their performances in model updating and damage detection are compared through three different engineering applications with increased complexity, e.g., a forty-story shear building, a two-span continuous steel beam, and a large-scale steel pedestrian bridge.

Copula Approximate Bayesian Computation Using Distribution Random Forests

Ongoing modern computational advancements continue to make it easier to collect increasingly large and complex datasets, which can often only be realistically analyzed using models defined by intractable likelihood functions. This Stats invited feature article introduces and provides an extensive simulation study of a new approximate Bayesian computation (ABC) framework for estimating the posterior distribution and the maximum likelihood estimate (MLE) of the parameters of models defined by intractable likelihoods, that unifies and extends previous ABC methods proposed separately. This framework, copulaABCdrf, aims to accurately estimate and describe the possibly skewed and high-dimensional posterior distribution by a novel multivariate copula-based meta-t distribution based on univariate marginal posterior distributions that can be accurately estimated by distribution random forests (drf), while performing automatic summary statistics (covariates) selection, based on robustly estimated copula dependence parameters. The copulaABCdrf framework also provides a novel multivariate mode estimator to perform MLE and posterior mode estimation and an optional step to perform model selection from a given set of models using posterior probabilities estimated by drf. The posterior distribution estimation accuracy of the ABC framework is illustrated and compared with previous standard ABC methods through several simulation studies involving low- and high-dimensional models with computable posterior distributions, which are either unimodal, skewed, or multimodal; and exponential random graph and mechanistic network models, each defined by an intractable likelihood from which it is costly to simulate large network datasets. This paper also proposes and studies a new solution to the simulation cost problem in ABC involving the posterior estimation of parameters from datasets simulated from the given model that are smaller compared to the potentially large size of the dataset being analyzed. This proposal is motivated by the fact that, for many models defined by intractable likelihoods, such as the network models when they are applied to analyze massive networks, the repeated simulation of large datasets (networks) for posterior-based parameter estimation can be too computationally costly and vastly slow down or prohibit the use of standard ABC methods. The copulaABCdrf framework and standard ABC methods are further illustrated through analyses of large real-life networks of sizes ranging between 28,000 and 65.6 million nodes (between 3 million and 1.8 billion edges), including a large multilayer network with weighted directed edges. The results of the simulation studies show that, in settings where the true posterior distribution is not highly multimodal, copulaABCdrf usually produced similar point estimates from the posterior distribution for low-dimensional parametric models as previous ABC methods, but the copula-based method can produce more accurate estimates from the posterior distribution for high-dimensional models, and, in both dimensionality cases, usually produced more accurate estimates of univariate marginal posterior distributions of parameters. Also, posterior estimation accuracy was usually improved when pre-selecting the important summary statistics using drf compared to ABC employing no pre-selection of the subset of important summaries. For all ABC methods studied, accurate estimation of a highly multimodal posterior distribution was challenging. In light of the results of all the simulation studies, this article concludes by discussing how the copulaABCdrf framework can be improved for future research.

Ensemble surrogate modeling of advective-dispersive transport with intraparticle diffusion model for column-leaching test

Column-leaching tests are a common approach for assessing the leaching behavior and resulting environmental risks of contaminated soils and waste materials, which are frequently reused for various construction purposes. The observed breakthrough curves of the contaminants are influenced by the complex dynamics of solute transport and kinetic inter-phase mass transfer. Disentangling these interactions necessitates numerical models. However, inverse modeling and sensitivity analysis can be time-consuming, especially when sorption kinetics are explicitly described by intraparticle diffusion, which requires discretizing the domain both in the flow direction along the column axis and inside the grains. To circumvent the need for such computationally intensive models, we have developed two different ensemble surrogate models. These models employ two separate ensemble methods: random forest stacking and inverse-distance weighted interpolation. Each method is applied to base surrogate models that cover different parts of the parameter space. The base surrogate models use the method of Extremely randomized Trees (ExtraTrees). The defined parameter range is based on the German standard for column-leaching tests. To optimize the base surrogate models, we utilized adaptive-sampling methods based on three distinct infill criteria: maximizing the expected improvement, staying within a certain Mahalanobis distance to the best estimate (both for exploitation), and maximizing the standard deviation (for exploration). The ensemble surrogate model demonstrates excellent performance in emulating the behavior of the original numerical model, with a relative root mean squared error of 0.09. We applied our proposed ensemble surrogate model to estimate the complete posterior parameter distribution using Simulation-Based Inference, specifically Neural Posterior Estimation, to determine the full parameter distribution conditioned on copper-leaching data from two different soils. Samples drawn from the posterior distribution align perfectly with the observed data for both the surrogate and original models.

Bayesian modeling and optimization for split-plot experiments with multiple responses

In many industrial processes, cost or time constraints make some input variables harder to change or control than others. An appropriate experimental design method is restricted randomization, which results in split-plot experiments. Empirical models that connect multiple quality characteristics with input variables play a crucial role in robust parameter design for split-plot experiments. At present, many modeling methods typically adopt the single response model for analyzing industrial processes in the split-plot experiments without considering correlation among multiple responses, correlation among whole plots, and uncertainty of model parameters. However, ignoring these issues can lead to poor product or process design. To solve these issues, this paper suggests a novel Bayesian modeling and optimization approach. We first construct a Bayesian multi-response linear mixed-effects model and obtain the posterior distribution for model parameters by employing Bayesian theorem. Then, the Gibbs sampling procedure is employed for the estimation of model parameters. Finally, the overall weighted desirability optimization function meeting the specification is developed to avoid acquiring ideal input settings with outliers. A simulation and an engineering case study demonstrate the validity of the proposed method. In comparison to existing methods, the optimization results given the proposed method are more robust and reliable.

Sequential and adaptive probabilistic integration for Bayesian model updating

The present study attempts to introduce a Bayesian perspective to marginal likelihood estimation. For this purpose, the Bayesian marginal likelihood inference (BMLI) framework is first developed, where the posterior statistics of the marginal likelihood are estimated using efficient random process sampling. The sequential and adaptive probabilistic integration (SAPI) method is then proposed within the BMLI framework to efficiently infer the posterior distribution of the model parameters as a result of the marginal likelihood estimation. In this method, an adaptive tempering of the likelihood function is presented to draw samples from a series of intermediate distributions to perform probabilistic integration sequentially, and the samples are finally converged to the target posterior domain. An active learning process is also designed to accelerate the sequential integration with a minimum number of likelihood function observations. Four numerical examples are investigated and the superiority of the SAPI method over other sequential sampling methods is demonstrated.

Integrating Bayesian inference and neural ODEs for microgrids dynamics parameters estimation

The integration of solar and wind energy sources in microgrids has witnessed significant growth, giving rise to distinct challenges due to their intermittent nature when it comes to achieving efficient microgrid control. However, estimating the parameters of the dynamic microgrid components facilitates capturing the complex and time-varying characteristics of renewable energy generation. This requires an accurate estimation of the parameters from the dynamic differential equations for effective modeling and control. In this research paper, we presented a novel methodology based on the integration of Bayesian inference and Neural ODEs. The Bayesian inference quantifies the uncertainty, and the Neural ODEs model the dynamic systems. By combining the strengths of both methods, we aimed to achieve a precise and robust parameter estimation of the dynamic microgrid components. The methodology is validated on a simulated microgrid that consists of a diesel generator, Solar PV array, double-fed induction generator, and a battery energy storage system. The results showed promised inferences estimation obtained from the parameter posterior distribution even in the presence of uncertainty. This can enhance our understanding of the dynamics of renewable energy systems and can contribute to the advancement of decision-making microgrid control strategies.