An Adaptive Metropolis Algorithm

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

Grapham: Graphical models with adaptive random walk Metropolis algorithms

Abstract. Calibration of terrestrial ecosystem models is important but challenging. Bayesian inference implemented by Markov chain Monte Carlo (MCMC) sampling provides a comprehensive framework to estimate model parameters and associated uncertainties using their posterior distributions. The effectiveness and efficiency of the method strongly depend on the MCMC algorithm used. In this work, a differential evolution adaptive Metropolis (DREAM) algorithm is used to estimate posterior distributions of 21 parameters for the data assimilation linked ecosystem carbon (DALEC) model using 14 years of daily net ecosystem exchange data collected at the Harvard Forest Environmental Measurement Site eddy-flux tower. The calibration of DREAM results in a better model fit and predictive performance compared to the popular adaptive Metropolis (AM) scheme. Moreover, DREAM indicates that two parameters controlling autumn phenology have multiple modes in their posterior distributions while AM only identifies one mode. The application suggests that DREAM is very suitable to calibrate complex terrestrial ecosystem models, where the uncertain parameter size is usually large and existence of local optima is always a concern. In addition, this effort justifies the assumptions of the error model used in Bayesian calibration according to the residual analysis. The result indicates that a heteroscedastic, correlated, Gaussian error model is appropriate for the problem, and the consequent constructed likelihood function can alleviate the underestimation of parameter uncertainty that is usually caused by using uncorrelated error models.

In this paper Bayesian method was used to estimate the parameters of Pearson III distribution, the posterior distribution was calculated by the Markov Chain Monte Carlo (MCMC) approach which avoiding the difficulty of boresome integration. Metropolis-Hasting (MH) algorithm, Delayed Rejection (DR) algorithm and Adaptive Metropolis (AM) algorithm are employed for parameter sampling at the site of interest, respectively, then EM (Ensemble Mean) and ES (Ensemble Spread) were applied to evaluate convergence of the parameter sampling. The results indicate that the AM sampling is much better than the other two algorithms in convergence rate.

Recurrent event data, which represent the occurrence of repeated incidences, are common in observational studies. Furthermore, collecting possible spatial correlations in health and environmental data is likely to provide more information for risk prediction. This article proposes a comprehensive proportional intensity model considering spatial random effects for recurrent event data using a Bayesian approach. The spatial information for areal data (where the spatial location is known up to a geographic unit such as a county) and georeferenced data (where the location is exactly observed) is examined. A traditional constant baseline intensity function, as well as a flexible piecewise constant baseline intensity function, are both under consideration. To estimate the parameters, a Markov chain Monte Carlo method with the Metropolis-Hastings algorithm and the adaptive Metropolis algorithm are applied. To assess the performance of model fitting, the deviance information criterion and log pseudo marginal likelihood are proposed. Overall, simulation studies demonstrate that the proposed model is significantly better than models that do not consider spatial effects if spatial correlations exist. Finally, our approach is implemented using a dataset related to the recurrence of cardiovascular diseases, which incorporates spatial information.

On the stability and ergodicity of adaptive scaling Metropolis algorithms

Stochastic parameter estimation and inversion have become increasingly popular in recent years. Nowadays, it is computationally reasonable and regular to solve complex inverse problems within the Bayesian framework. Applications of Bayesian inferences for inverse problems require investigation of the posterior distribution, which usually has a complex landscape and is highly dimensional. In these cases, Markov chain Monte Carlo methods (MCMC) are often used. This paper discusses a Bayesian approach for identifying adsorption and desorption rates in combination with a pore-scale reactive flow. Markov chain Monte Carlo sampling is used to estimate adsorption and desorption rates. The reactive transport in porous media is governed by incompressible Stokes equations, coupled with convection–diffusion equation for species’ transport. Adsorption and desorption are accounted via Robin boundary conditions. The Henry isotherm is considered for describing the reaction terms. The measured concentration at the outlet boundary is provided as additional information for the identification procedure. Metropolis–Hastings and Adaptive Metropolis algorithms are implemented. Credible intervals have been plotted from sampled posterior distributions for both algorithms. The impact of the noise in the measurements and influence of several measurements for Bayesian identification procedure is studied. Sample analysis using the autocorrelation function and acceptance rate is performed to estimate mixing of the Markov chain. As result, we conclude that MCMC sampling algorithm within the Bayesian framework is good enough to determine an admissible set of parameters via credible intervals.

The Markov chain Monte Carlo （MCMC） method is used to estimate the unknown parameters of air-sea oscillator system, which is a model of Eastern Pacific sea surface temperature （SST）. Firstly，the posterior probability density function for unknown parameters of air-sea oscillator system is deduced with the Bayesian formula. Secondly， the Adaptive Metropolis algorithm is used to construct the Markov Chains of unknown parameters. And the converged samples are used to calculate the mathematic expectation. The results of numerical experiments show that parameters estimated by the new method have high precision and the noise is filtered effectively from the observations.

For high precision in source reconstruction of magnetoencephalography (MEG) or electroencephalography data, high accuracy of the coregistration of sources and sensors is mandatory. Usually, the source space is derived from magnetic resonance imaging (MRI). In most cases, however, no quality assessment is reported for sensor-to-MRI coregistrations. If any, typically root mean squares (RMS) of point residuals are provided. It has been shown, however, that RMS of residuals do not correlate with coregistration errors. We suggest using target registration error (TRE) as criterion for the quality of sensor-to-MRI coregistrations. TRE measures the effect of uncertainty in coregistrations at all points of interest. In total, 5544 data sets with sensor-to-head and 128 head-to-MRI coregistrations, from a single MEG laboratory, were analyzed. An adaptive Metropolis algorithm was used to estimate the optimal coregistration and to sample the coregistration parameters (rotation and translation). We found an average TRE between 1.3 and 2.3 mm at the head surface. Further, we observed a mean absolute difference in coregistration parameters between the Metropolis and iterative closest point algorithm of and m. A paired sample t-test indicated a significant improvement in goal function minimization by using the Metropolis algorithm. The sampled parameters allowed computation of TRE on the entire grid of the MRI volume. Hence, we recommend the Metropolis algorithm for head-to-MRI coregistrations.

<p>The actual temperature profile is a determinant of ice rheology, which controls ice deformation and flow, and sliding over the underlying bedrock. Importantly, the ice flow in turn affects its temperature profile through strain heating, which makes observed temperature profiles a powerful input for ice sheet model validation.</p><p><span>Up to now temperature profile was available in few boreholes or from glaciological models. Recently, </span><span>Macelloni et al. (2016)</span><span> opened up new opportunities for probing ice </span><span>temperature</span><span> from space with the low-frequency passive sensors. </span><span>Indeed</span><span>, at L-band frequency, the very low absorption of ice and the low scattering by particles (grain size, bubbles in ice) allow a large penetration in the dry snow and ice (several hundreds of meters). Macelloni et al. (2019) performed the first retrieval of the ice sheet temperature in Antarctica by using the European Space Agency (ESA)’s Soil Moisture and Ocean Salinity (SMOS) L-band observations. The</span><span>y used </span><span>the</span><span> minimization of the difference between SMOS brightness temperature and microwave emission model simulations that include</span><span>s</span><span> a glaciological model.</span></p><p><span>Here, in the framework of the ESA 4D-Antarctica project, we propose a new method based on a Bayesian approach in order to improve the accuracy of the retrieved ice temperature and to provide an uncertainty estimation along the profiles. As a first step, a one-dimensional ice temperature profile model (Robin 1955) is used, which limits the retrieval to the Antarctic Plateau. Then, the new temperature emulator based on the three-dimensional glaciological GRISLI (Quiquet et al., 2018) will be used to enable retrievals over the entire continent (cf. Ritz’s presentation in this session for the GRISLI emulator </span><span>description</span><span>).</span></p><p><span>The Bayesian inference takes as free parameters: </span><span>ice thickness, </span><span>surface ice temperature, snow accumulation and geothermal heat flux (GHF). Their prior probability distribution is defined as normal, centered around a priori values taken from literature, and truncated to stay in a realistic range. The observed brightness temperature distribution is normal and a normal likelihood function is used to quantify the matching between the observed and simulated brightness temperature. The parameter space investigation is achieved through a Markov Chain Monte Carlo (MCMC) method. Here, the differential evolution adaptive Metropolis (DREAM) algorithm is used, which runs multiple different Markov chains in parallel and uses a discrete proposal distribution to evolve the sampler to the posterior distribution </span><span>(</span><span>Laloy and Vrugt, 2012).</span></p><p>For each SMOS brightness temperature observation, 1000 iterations are run on 5 parallel chains. The 2500 first iterations are discarded (aka. burn-in) and only the last 2500 are used for the final ice temperature profile estimation. The posterior probability distribution captures the most likely parameter set (i.e. a surface temperature, snow accumulation and GHF combination), and so, the most likely ice temperature profiles associated to this SMOS observation. It also provides the standard deviation which is an accurate estimate of the temperature uncertainty along the depth obtained with the method.</p>

Laser ablation (LA) is a minimally invasive cancer therapy that uses laser energy to selectively heat and destroy cancerous tissues while minimizing damage to surrounding healthy tissue. Accurate temperature predictions derived from mathematical models are instrumental in optimizing treatment outcomes. These models assist surgeons during both pre-operative planning and intraoperative guidance. However, their accuracy can be compromised by inherent model limitations, measurement errors, or parameter uncertainties. To address this challenge, we combined the Delayed Rejection Adaptive Metropolis (DRAM) algorithm, an advanced version of the Markov Chain Monte Carlo (MCMC) method, with bioheat equations to tune and quantify key parameters influencing temperature distribution. Our findings showed that laser standard distribution, tissue absorption coefficient, and thermal conductivity impact the temperature profile, with correlation coefficients of -0.64, -0.36, and 0.15, respectively. To validate the model, temperature data were obtained from an experimental LA setup using ex vivo porcine liver, monitored by fiber Bragg grating sensors. After tuning the model parameters, the simulation accurately predicted temperature distributions within 1.0±0.5°C. Additionally, the parameter distributions were available at each time point during ablation, offering valuable insights for real-time decision-making, particularly regarding laser energy delivery.

Fast transdimensional Bayesian transient electromagnetic imaging for urban underground space detection

Estimation of low or high conditional quantiles is called for in many applications, but commonly encountered data sparsity at the tails of distributions makes this a challenging task. We develop a Bayesian joint-quantile regression method to borrow information across tail quantiles through a linear approximation of quantile coefficients. Motivated by a working likelihood linked to the asymmetric Laplace distributions, we propose a new Bayesian estimator for high quantiles by using a delayed rejection and adaptive Metropolis and Gibbs algorithm. We demonstrate through numerical studies that the proposed estimator is generally more stable and efficient than conventional methods for estimating tail quantiles, especially at small and modest sample sizes.

A Bayesian approach for evaluation of the effect of water quality model parameter uncertainty on TMDLs: A case study of Miyun Reservoir

Simple Monte Carlo and the Metropolis algorithm