Distribution Of Model Parameters Research Articles

A Mechanistic Bayesian Inferential Workflow for Estimation of In Vivo Skin Permeation from In Vitro Measurements

Computational models can play an integral role in the chemical risk assessment of dermatological products. However, a limitation on the ability of mathematical models to extrapolate from in vitro measurements to in human predictions arises from context-dependence: modeling assumptions made in one setting may not carry over to another scenario. Mechanistic models of dermal absorption relate the skin penetration kinetics of permeants to their partitioning and diffusion across elementary sub-compartments of the skin. This endows them with a flexibility through which specific model components can be adjusted to better reflect dermal absorption in contexts that differ from the in vitro setting, while keeping fixed any context-invariant parameters that remain unchanged in the two scenarios. This paper presents a workflow for predicting in vivo dermal absorption by integrating a mechanistic model of skin penetration with in vitro permeation test (IVPT) measurements. A Bayesian approach is adopted to infer a joint posterior distribution of context-invariant model parameters. By populating the model with samples of context-invariant parameters from this distribution and adjusting context-dependent parameters to suit the in vivo setting, simulations of the model yield estimates of the likely range of in vivo dermal absorption given the IVPT data. This workflow is applied to five compounds previously tested in vivo. In each case, the range of in vivo predictions encompassed the range observed experimentally. These studies demonstrate that the proposed workflow enables the derivation of mechanistically derived upper bounds on dermal absorption for the purposes of chemical risk assessment.

Cosmology with the submillimetre galaxies magnification bias

Context.High-z submillimetre galaxies can be used as a background sample for gravitational lensing studies thanks to their magnification bias. In particular, the magnification bias can be exploited in order to constrain the free parameters of a halo occupation distribution (HOD) model and some of the main cosmological parameters. A pseudo-tomographic analysis shows that the tomographic approach should improve the parameter estimation.Aims.In this work the magnification bias has been evaluated as cosmological tool in a tomographic set-up. The cross-correlation function (CCF) data have been used to jointly constrain the astrophysical parametersMmin,M1, andαin each of the selected redshift bins as well as the cosmological parameters ΩM,σ8, andH0for the lambda cold dark matter (ΛCDM) model. Moreover, we explore the possible time evolution of the dark energy density by also introducing theω0, ωaparameters in the joint analysis (ω0CDM andω0ωaCDM).Methods.The CCF was measured between a foreground spectroscopic sample of Galaxy And Mass Assembly galaxies and a background sample ofHerschelAstrophysical Terahertz Large Area Survey (H-ATLAS) galaxies. The foreground sample was divided into four redshift bins (0.1–0.2, 0.2–0.3, 0.3–0.5, and 0.5–0.8) and the sample of H-ATLAS galaxies has photometric redshifts > 1.2. The CCF was modelled using a halo model description that depends on HOD and cosmological parameters. Then a Markov chain Monte Carlo method was used to estimate the parameters for different cases.Results.For the ΛCDM model the analysis yields a maximum posterior value at 0.26 with [0.17, 0.41] 68% C.I. for ΩMand at 0.87 with [0.75, 1] 68% C.I. forσ8. With our current resultsH0is not yet constrained. With a more generalω0CDM model, the constraints on ΩMandσ8are similar, but we found a maximum posterior value forω0at −1 with [ − 1.56, −0.47] 68% C.I. In theω0ωaCDM model, the results are −1.09 with [ − 1.72, −0.66] 68% C.I. forω0and −0.19 with [ − 1.88, 1.48] 68% C.I. forωa.Conclusions.The results onMminshow a trend towards higher values at higher redshift confirming recent findings. The tomographic analysis presented in this work improves the constraints in theσ8 − ΩMplane with respect to previous findings exploiting the magnification bias and it confirms that magnification bias results do not show the degeneracy found with cosmic shear measurements. Moreover, related to dark energy, we found a trend of higherω0values for lowerH0values.

An Improved Tandem Neural Network Architecture for Inverse Modeling of Multicomponent Reactive Transport in Porous Media

AbstractParameter estimation for reactive transport models (RTMs) is important in improving their predictive capacity for accurately simulating subsurface hydrogeochemical processes. This paper introduces a deep learning approach called the tandem neural network architecture (TNNA), which consists of a forward network and a reverse network to estimate input parameters for RTMs. The TNNA approach has a limitation in that the approximation error from the forward network often results in biased inversion results. To solve this problem, we proposed to enhance TNNA using an adaptive updating strategy (AUS), which locally reduces the approximation error of the forward network. The developed framework updates the forward network by iteratively using local sampling and transfer learning. The TNNA‐AUS was verified by a cation exchange example. The results show that TNNA‐AUS successfully reduces the inversion bias and improves the computational efficiency and inversion accuracy, compared with the global improvement strategy of adding training samples according to the prior distribution of model parameters. After verification, the TNNA‐AUS was applied to a real‐world and well‐documented RTM problem of the Aquia aquifer, Maryland, USA. The inversion results demonstrate that the developed TNNA‐AUS algorithm is an excellent tool for us to understand the complex subsurface hydrogeochemical processes and estimate the associated reaction parameters.

A Novel Method for Generation of In Silico Subjects with Type 2 Diabetes.

A type 2 diabetes (T2D) simulator has been recently proposed for supporting drug development and treatment optimization. This tool consists of a physiological model of glucose/insulin/C-peptide dynamics and a virtual cohort of T2D subjects (i.e., random extractions of model parameterizations from a joint parameter distribution) well describing both average and variability realistic T2D dynamics . However, the state-of-art procedure to get a reliable virtual population requires some post-processing after subject extraction, in order to discard implausible behaviors. We propose an improved method for virtual subjects' generation to overcome this burdensome task. To do so, we first assessed a refined joint parameter distribution, from which extracting a number of subjects, greater than the target population size. Then, target-size subsets are undersampled from the large cohort. The final virtual population is selected among the subsets as the one maximizing the similarity with T2D data and model parameter distribution, by means of measurement' outcome metrics and Euclidian distance (Δ), respectively. In the final population, almost all the outcome metrics are statistically identical to the clinical counterparts (p-value>0.05) and model parameters' distribution differs by ~5-10% from that derived from data. The methodology described here is flexible, thus resulting suitable for different T2D stages and type 1 diabetes, as well.Clinical Relevance- A straightforward subjects' generation would ease the availability of tailored in silico trials for testing diabetes treatment in a specific population.

The Alpha Power Transformed Dagum Distribution: Properties and Applications

In this study, we propose a new extension of the Dagum distribution called the alpha power transformed Dagum distribution. Basic statistical properties of the new distribution such as; quantile function, raw and incomplete moments, moment generating function, order statistics, Rényi entropy, stochastic ordering and stress strength model are investigated. The characterizations of the new model is investigated. The method of maximum likelihood is used to estimate the model parameters of the new distribution and the observed information matrix is also obtained. A Monte Carlo simulation is presented to examine the behavior of the parameter estimates. The applicability of the new model is demonstrated by means of three applications.

Bayesian consensus clustering for multivariate longitudinal data.

In clinical and epidemiological studies, there is a growing interest in studying the heterogeneity among patients based on longitudinal characteristics to identify subtypes of the study population. Compared to clustering a single longitudinal marker, simultaneously clustering multiple longitudinal markers allow additional information to be incorporated into the clustering process, which reveals co-existing longitudinal patterns and generates deeper biological insight. In the current study, we propose a Bayesian consensus clustering (BCC) model for multivariate longitudinal data. Instead of arriving at a single overall clustering, the proposed model allows each marker to follow marker-specific local clustering and these local clusterings are aggregated to find a global (consensus) clustering. To estimate the posterior distribution of model parameters, a Gibbs sampling algorithm is proposed. We apply our proposed model to the primary biliary cirrhosis study to identify patient subtypes that may be associated with their prognosis. We also perform simulation studies to compare the clustering performance between the proposed model and existing models under several scenarios. The results demonstrate that the proposed BCC model serves as a useful tool for clustering multivariate longitudinal data.

Bayesian uncertainty analysis of SA turbulence model for supersonic jet interaction simulations

The Reynolds Averaged Navier-Stokes (RANS) models are still the workhorse in current engineering applications due to its high efficiency and robustness. However, the closure coefficients of RANS turbulence models are determined by model builders according to some simple fundamental flows, and the suggested values may not be applicable to complex flows, especially supersonic jet interaction flow. In this work, the Bayesian method is employed to recalibrate the closure coefficients of Spalart-Allmaras (SA) turbulence model to improve its performance in supersonic jet interaction problem and quantify the uncertainty of wall pressure and separation length. The embedded model error approach is applied to the Bayesian uncertainty analysis. Firstly, the total Sobol index is calculated by non-intrusive polynomial chaos method to represent the sensitivity of wall pressure and separation length to model parameters. Then, the pressure data and the separation length are respectively served as calibration data to get the posterior uncertainty of model parameters and Quantities of Interests (QoIs). The results show that the relative error of the wall pressure predicted by the SA turbulence model can be reduced from 14.99% to 2.95% through effective Bayesian parameter estimation. Besides, the calibration effects of four likelihood functions are systematically evaluated. The posterior uncertainties of wall pressure and separation length estimated by different likelihood functions are significantly discrepant, and the Maximum a Posteriori (MAP) values of parameters inferred by all functions show better performance than the nominal values. Finally, the closure coefficients are also estimated at different jet total pressures. The similar posterior distributions of model parameters are obtained in different cases, and the MAP values of parameters calibrated in one case are also applicable to other cases.

High-Order Deterministic Sensitivity Analysis and Uncertainty Quantification: Review and New Developments

This work reviews the state-of-the-art methodologies for the deterministic sensitivity analysis of nonlinear systems and deterministic quantification of uncertainties induced in model responses by uncertainties in the model parameters. The need for computing high-order sensitivities is underscored by presenting an analytically solvable model of neutron scattering in a hydrogenous medium, for which all of the response’s relative sensitivities have the same absolute value of unity. It is shown that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis. This work also presents new mathematical expressions that extend to the sixth-order of the current state-of-the-art fourth-order formulas for computing fourth-order correlations among computed model response and model parameters. Another novelty presented in this work is the mathematical framework of the 3rd-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (3rd-CASAM-N), which enables the most efficient computation of the exact expressions of the 1st-, 2nd- and 3rd-order functional derivatives (“sensitivities”) of a model’s response to the underlying model parameters, including imprecisely known initial, boundary and/or interface conditions. The 2nd- and 3rd-level adjoint functions are computed using the same forward and adjoint computer solvers as used for solving the original forward and adjoint systems. Comparisons between the CPU times are also presented for an OECD/NEA reactor physics benchmark, highlighting the fact that finite-difference schemes would not only provide approximate values for the respective sensitivities (in contradistinction to the 3rd-CASAM-N, which provides exact expressions for the sensitivities) but would simply be unfeasible for computing sensitivities of an order higher than first-order. Ongoing work will generalize the 3rd-CASAM-N to a higher order while aiming to overcome the curse of dimensionality.

Joint seismic and petrophysical nonlinear inversion with Gaussian mixture-based adaptive regularization

The estimation of reservoir parameters from seismic observations is one of the main objectives in reservoir characterization. However, the forward model relating the petrophysical properties of rocks to observed seismic data is highly nonlinear, and solving the relevant inverse problem is a challenging task. We have developed a novel inversion method for jointly estimating the elastic and petrophysical parameters of rocks from prestack seismic data. We combine a full rock-physics model and the exact Zoeppritz equation as the forward model. To overcome the ill conditioning of the inverse problem and address the complex prior distribution of model parameters given lithofacies variations, we have developed a regularization term based on the prior Gaussian mixture model under a Bayesian framework. The objective function is optimized by the fast simulated annealing algorithm, during which the Gaussian mixture-based regularization terms are adaptively and iteratively adjusted by the maximum likelihood estimator, allowing the posterior distribution to be more consistent with the observed seismic data. The adaptive regularization method improves the accuracy of petrophysical parameters compared with sequential inversion and nonadaptive regularization methods, and the inversion result can be used for indicating gas-saturated areas when applied to field data.

Forecasting Development of COVID-19 Epidemic in European Union Using Entropy-Randomized Approach

The paper is devoted to the forecasting of the COVID-19 epidemic by the novel method of randomized machine learning. This method is based on the idea of estimation of probability distributions of model parameters and noises on real data. Entropy-optimal distributions correspond to the state of maximum uncertainty which allows the resulting forecasts to be used as forecasts of the most "negative" scenario of the process under study. The resulting estimates of parameters and noises, which are probability distributions, must be generated, thus obtaining an ensemble of trajectories that considered to be analyzed by statistical methods. In this work, for the purposes of such an analysis, the mean and median trajectories over the ensemble are calculated, as well as the trajectory corresponding to the mean over distribution values of the model parameters. The proposed approach is used to predict the total number of infected people using a three-parameter logistic growth model. The conducted experiment is based on real COVID-19 epidemic data in several countries of the European Union. The main goal of the experiment is to demonstrate an entropy-randomized approach for predicting the epidemic process based on real data near the peak. The significant uncertainty contained in the available real data is modeled by an additive noise within 30%, which is used both at the training and predicting stages. To tune the hyperparameters of the model, the scheme is used to configure them according to a testing dataset with subsequent retraining of the model. It is shown that with the same datasets, the proposed approach makes it possible to predict the development of the epidemic more efficiently in comparison with the standard approach based on the least-squares method.

A Bayesian approach to dynamical modeling of eye-movement control in reading of normal, mirrored, and scrambled texts.

In eye-movement control during reading, advanced process-oriented models have been developed to reproduce behavioral data. So far, model complexity and large numbers of model parameters prevented rigorous statistical inference and modeling of interindividual differences. Here we propose a Bayesian approach to both problems for one representative computational model of sentence reading (SWIFT; Engbert et al., Psychological Review, 112, 2005, pp. 777-813). We used experimental data from 36 subjects who read the text in a normal and one of four manipulated text layouts (e.g., mirrored and scrambled letters). The SWIFT model was fitted to subjects and experimental conditions individually to investigate between-subject variability. Based on posterior distributions of model parameters, fixation probabilities and durations are reliably recovered from simulated data and reproduced for withheld empirical data, at both the experimental condition and subject levels. A subsequent statistical analysis of model parameters across reading conditions generates model-driven explanations for observable effects between conditions. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Polynomial surrogates for Bayesian traveltime tomography

This paper tackles the issue of the computational load encountered in seismic imaging by Bayesian traveltime inversion. In Bayesian inference, the exploration of the posterior distribution of the velocity model parameters requires extensive sampling. The computational cost of this sampling step can be prohibitive when the first arrival traveltime prediction involves the resolution of an expensive number of forward models based on the eikonal equation. We propose to rely on polynomial chaos surrogates of the traveltimes between sources and receivers to alleviate the computational burden of solving the eikonal equation during the sampling stage. In an offline stage, the approach builds a functional approximation of the traveltimes from a set of solutions of the eikonal equation corresponding to a few values of the velocity model parameters selected in their prior range. These surrogates then substitute the eikonal-based predictions in the posterior evaluation, enabling very efficient extensive sampling of Bayesian posterior, for instance, by a Markov Chain Monte Carlo algorithm. We demonstrate the potential of the approach using numerical experiments on the inference of two-dimensional domains with layered velocity models and different acquisition geometries (microseismic and seismic refraction contexts). The results show that, in our experiments, the number of eikonal model evaluations required to construct accurate surrogates of the traveltimes is low. Further, an accurate and complete characterization of the posterior distribution of the velocity model is possible, thanks to the generation of large sample sets at a low cost. Finally, we discuss the extension of the current approach to more realistic velocity models and operational situations.

Quantifying Distributions of Parameters for Cardiac Action Potential Models Using the Hamiltonian Monte Carlo Method.

Cardiac action potential (AP) models are typically given with a single set of parameter values; however, this approach does not consider variability and uncertainty across individuals and experimental conditions. As an alternative to single-value parameter fitting, we sought to use a Bayesian approach, the Hamiltonian Monte Carlo (HMC) algorithm, to find distributions of physiological parameter values for cardiac AP models across a range of cycle lengths (CLs) and dynamics. To assess HMC's accuracy for cardiac data, we applied it to synthetic APs from the Mitchell-Shaeffer (MS) and Fenton-Karma (FK) models with added noise over a range of physiological CLs, some of which included alternans. To show the applicability of HMC to experimental data, we calculated parameter distributions for both models using micro-electrode recordings of zebrafish APs from a range of CLs. For synthetic APs generated from three CLs using the MS (FK) models, HMC produced unimodal quasi-symmetric distributions for all five (13) parameters. APs generated by setting all parameters in the MS (FK) model to the modes of their corresponding marginal distributions yielded errors in voltage traces below 5.0% (0.6%). We also obtained distributions for the MS (FK) model parameters using zebrafish data to construct the first minimal model of the zebrafish AP, with voltage trace errors below 4.8% (3.4%). We have shown that HMC can identify not only a single set of parameter values but also viable distributions for cardiac AP model parameters using synthetic and experimental data. Because HMC generates samples from the parameter distributions based on input data, it can produce families of parameterizations that can be used in population-based modeling approaches without the need for rejecting a large number of randomly generated candidate parameterizations. HMC also has the potential to provide quantitative measures of spatial/individual variability and uncertainty.

Autoregressive model of an underwater acoustic channel in the frequency domain

The underwater acoustic (UWA) channel model based on the multipath delay–amplitude or the angle of departure (AoD)–angle of arrival (AoA) require many parameters to describe a broadband underwater acoustic (UWA) channel. In this paper, an autoregressive (AR) model is developed to describe the channel in the frequency domain with few parameters. The AR model used here is with an order P, where P is also related to the number of main multipath clusters. Hence, only P + 1 parameters, which consist of the variance of the complex Gaussian white noise (CGWN) and the P pole values of the AR model, are required to describe UWA channel. Each pole value is corresponded to a multipath cluster in the channel, where its phase is correlated with the average multipath delay, and its absolute value is related to the multipath energy. A comprehensive analysis was performed to characterize the statistical distribution of each pole value and the CGWN variance. The results showed that the absolute value and phase of each pole satisfied the Weibull and gamma distributions, respectively, and the variance obeyed a lognormal distribution. Simulation channels were generated with different distributions of the AR model parameters, and the root mean square (RMS) delay obeyed a similar cumulative distribution function as the measured channel.

Parameter Estimation for Reynolds-Averaged Navier–Stokes Models Using Approximate Bayesian Computation

Approximate Bayesian computation (ABC) is used to estimate unknown parameter values, as well as their uncertainties, in Reynolds-averaged Navier–Stokes (RANS) simulations of turbulent flows. The ABC method approximates posterior distributions of model parameters, but does not require the direct computation, or estimation, of a likelihood function. This method thus enables relatively simple and flexible parameter estimation for complex models and a wide range of reference data. This paper describes the ABC approach, including the use of a calibration step, adaptive proposal, and Markov chain Monte Carlo (MCMC) technique to accelerate the parameter estimation, resulting in an ABC approach with improved MCMC, denoted ABC-IMCMC. As a test of the classic ABC rejection algorithm and ABC-IMCMC, parameters are estimated in a nonequilibrium RANS model using reference data from direct numerical simulations of periodically sheared homogeneous turbulence. The use of ABC-IMCMC is then demonstrated to estimate parameters in the Menter shear-stress-transport (SST) model using experimental reference data for an axisymmetric transonic bump. This paper shows that the accuracy of the SST model for this test case can be improved using ABC-IMCMC, indicating that ABC-IMCMC is a promising method for the calibration of RANS models using a wide range of reference data.

Bayesian model updating and class selection of a wing-engine structure with nonlinear connections using nonlinear normal modes

This paper presents a Bayesian model updating and model class selection approach based on nonlinear normal modes (NNMs). The performance of the proposed approach is demonstrated on a conceptually simple wing-engine structure. Control-based continuation is exploited to measure experimentally the NNMs of the structure by tracking the phase quadrature condition between the structural response and single input excitation. A two-phase Bayesian model updating framework is implemented to estimate the joint posterior distribution of unknown model parameters: (1) at phase I, the effective Young’s modulus of a detailed linear finite element model and its estimation uncertainty are inferred from the data; (2) at phase II, a reduced-order model is obtained from the updated linear model using Craig-Bampton method, and coefficient parameters of structural nonlinearities are updated using the measured NNMs. Five different model classes representing different nonlinear functions are investigated, and their Bayesian evidence are compared to reveal the most plausible model. The obtained model is used to predict NNMs by propagating uncertainties of parameters and error function. Good agreement is observed between model-predicted and experimentally identified NNMs, which verifies the effectiveness of the proposed approach for nonlinear model updating and model class selection.

Continuous updating of individual headache forecasting models using Bayesian methods.

To illustrate the benefits of deploying individual headache forecasting models using continuous updating with Bayesian methods. The ability to reliably forecast headache attacks within an individual over time would enhance the study of attacks and allow preemptive treatment. However, deploying a suitable forecasting model in a clinical setting will likely involve several unique challenges related to heterogeneity in the predictor weights, limited or sparse data, and the need for a quick "warm-up." The use of Bayesian methods offers solutions to each of these specific challenges. This was a post hoc analysis of a cohort study of individuals with episodic migraine attacks. Individuals completed daily diaries that allowed the estimation of several forecasting models, each using different types of ancillary information incorporated into formal prior probability distributions. An in silico analysis was conducted that mimicked the deployment of these models in a clinical-like setting where the parameters of the models were continuously updated and evaluated each day using root mean square error (RMSE). Individuals (N=95) were followed for 50days and contributed 3359days of nonmissing diary data. During the observation period, there were 1293/3359 (38.5%) days with a headache attack. Self-reported baseline headache frequency was associated with the corresponding predicted probability of future attacks, r=0.15-0.39. At Day 25, the correlation between baseline information and predicted attack likelihood was r=0.29 (95% CI: 0.09-0.47). Additionally, the use of prior probability distributions for model parameters improved the model fit, especially early in the deployment of the models (e.g., Day 5 RMSE 0.45 vs. 0.43). Models using informative prior probability distributions outperformed the models estimated without this information during the first 42days of observation, although performance became more similar as more data were collected. This analysis demonstrates the value of Bayesian methods in using additional available information to improve forecasting model performance, especially early in the deployment of a forecasting model. To obtain the full value of such models or to apply any model in clinical settings, a model with adequate discrimination and calibration will be needed.

Nonlinear random vibration analysis: A Bayesian nonparametric approach

Random vibration analysis aims to estimate the response statistics of dynamical systems subject to stochastic excitations. Stochastic differential equations (SDEs) that govern the response of general nonlinear systems are often complicated, and their analytical solutions are scarce. Thus, a range of approximate methods and simulation techniques have been developed. This paper develops a hybrid approach that approximates the governing SDE of nonlinear systems using a small number of response simulations and information available a priori. The main idea is to identify a set of surrogate linear systems such that their response probability distributions collectively estimate the response probability distribution of the original nonlinear system. To identify the surrogate linear systems, the proposed method integrates the simulated responses of the original nonlinear system with information available a priori about the number and parameters of the surrogate linear systems. There will be epistemic uncertainty in the number and parameters of the surrogate linear systems because of the limited data. This paper proposes a Bayesian nonparametric approach, called a Dirichlet Process Mixture Model, to capture these uncertainties. The Dirichlet process models the uncertainty over an infinite-dimensional parameter space, representing an infinite number of potential surrogate linear systems. Specifically, the proposed method allows the number of surrogate linear systems to grow indefinitely as the nonlinear system observed dynamic unveil new patterns. The quantified uncertainty in the estimates of the unknown model parameters propagates into the response probability distribution. The paper then shows that, under some mild conditions, the estimated probability distribution approaches, as close as desired, to the original nonlinear system’s response probability distribution. As a measure of model accuracy, the paper provides the convergence rate of the response probability distribution. Because the posterior distribution of the unknown model parameters is often not analytically tractable, a Gibbs sampling algorithm is presented to draw samples from the posterior distribution. Variational Bayesian inference is also introduced to derive an approximate closed-form expression for the posterior distribution. The paper illustrates the proposed method through the random vibration analysis of a nonlinear elastic and a nonlinear hysteretic system.

Upper mantle radial anisotropy under the Indian Ocean from higher mode surface waves and a hierarchical transdismensional approach

SUMMARYWe investigated the likelihood of radial anisotropy in the shallow and deep upper mantle, including the mantle transition zone (MTZ) under the Indian Ocean. Seismic anisotropy can be an indicator of mantle deformation through lattice preferred orientation of anisotropic crystals in the mantle. It has thus the potential to illuminate Earth’s dynamic interior, but previous seismic tomography studies have not achieved consensus on the existence of radial anisotropy below ∼250 km depth. We developed a fully nonlinear transdimensional hierarchical Bayesian Markov Chain Monte Carlo approach to invert fundamental and higher mode surface wave dispersion data and applied it to a subset of a global Love and Rayleigh wave data set. We obtained posterior model parameter distributions for shear wave velocity (VS) and radial anisotropy ξ under the Indian Ocean. These posterior model distributions were used to calculate the probability of having radial anisotropy at different depths. We demonstrated that separate inversions of Love and Rayleigh waves yield models compatible with the results of joint inversions within uncertainties. The obtained pattern of VS anomalies agrees with most previous studies. They display negative anomalies along ridges in the uppermost mantle, but those are stronger than for regularized inversions. The Central Indian Ridge and the Southeastern Indian Ridge present velocity anomalies that extend to ∼200 km depth, whereas the Southwestern Indian Ridge seems to have a shallower origin. Weaker, laterally variable velocity perturbations were found at larger depths. The anisotropy models differ more strongly from regularized inversion results and their uncertainties were rather large. We found that anisotropy models from regularized inversions also depend on the chosen parametrization, which is consistent with the existence of a large model null-space. Apart from a fast horizontally polarized shear wave signal in the top 100 km, likely reflecting the horizontal plate motion due to asthenospheric deformation, no clear relation to surface geology was detected in the anisotropy models. We found that, although the anisotropy model uncertainties are rather large, and lateral variations are present, the data generally prefer at least 1 per cent anisotropy in the MTZ with fast vertically polarized shear waves, within errors. Incorporating group velocity data did not help better constrain deep structure by reducing parameter trade-offs. We also tested the effect of prior constraints on the 410- and 660-km topography and found that the undulations of these discontinuities had little effect on the resulting models in our study region.

A multivariate statistical approach to predict COVID-19 count data with epidemiological interpretation and uncertainty quantification.

For the analysis of COVID‐19 pandemic data, we propose Bayesian multinomial and Dirichlet‐multinomial autoregressive models for time‐series of counts of patients in mutually exclusive and exhaustive observational categories, defined according to the severity of the patient status and the required treatment. Categories include hospitalized in regular wards (H) and in intensive care units (ICU), together with deceased (D) and recovered (R). These models explicitly formulate assumptions on the transition probabilities between these categories across time, thanks to a flexible formulation based on parameters that a priori follow normal distributions, possibly truncated to incorporate specific hypotheses having an epidemiological interpretation. The posterior distribution of model parameters and the transition matrices are estimated by a Markov chain Monte Carlo algorithm that also provides predictions and allows us to compute the reproduction number Rt. All estimates and predictions are endowed with an accuracy measure obtained thanks to the Bayesian approach. We present results concerning data collected during the first wave of the pandemic in Italy and Lombardy and study the effect of nonpharmaceutical interventions. Suitable discrepancy measures defined to check and compare models show that the Dirichlet‐multinomial model has an adequate fit and provides good predictive performance in particular for H and ICU patients.