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Bayesian Monte Carlo is a tool to address imperfect data & models, non-linear models, non-normal PDFs, and integration of differential and integral data. Simple propagation to criticality shows effects of new $\mathcal{L}$($β|z,γ$).
Conventional nuclear data evaluation methods using generalized linear least squares make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate to minimize the cost function (even for nonlinear models); the model (e.g., of neutron cross section) and experimental data (including covariance data) are without defect and prior PDFs of parameters and measured data are known perfectly. Neglect of covariance between model parameters and measured data in conventional evaluations contributes to imperfections. These assumptions are inherent to the generalized linear least squares minimization method commonly used for resolved resonance region neutron cross section evaluations but are often not justified due to the presence of non-normal PDFs, nonlinear models (e.g., R-matrix formalism), and inherent imperfections in data and models (e.g., imperfect covariance data). Here, these assumptions are removed in a mathematical framework of Bayes’ theorem, which is implemented using the Metropolis-Hastings Monte Carlo method. Most importantly, new parameters are introduced to parameterize discrepancies between the theoretical model and measured data to quantify judgement about discrepancies or imperfections in a reproducible manner. An evaluation of 233U in the eV region using the ENDF-B/VIII.0 library and transmission data (Guber et al.) is presented, and posterior parameters are compared to those obtained by conventional evaluation methods. This example illustrates the effects of removing the most harmful assumption: that of model-data perfection.
We propose a novel Bayesian Monte Carlo Integration (BMCI) technique to retrieve the profiles of temperature, water vapor, and cloud liquid/ice water content from microwave cloudy measurements in the rainbands of tropical cyclones (TC). These retrievals then can either be directly used by meteorologists to analyze the structure of TCs or be assimilated into numerical models to provide accurate initial conditions for the NWP models. The BMCI technique is applied to the data from the Advanced Technology Microwave Sounder (ATMS) onboard Suomi National Polar-orbiting Partnership (NPP) and Global Precipitation Measurement (GPM) Microwave Imager (GMI).
Bayesian evaluation of resolved resonance region (RRR) nuclear data has historically been carried out using the generalized least squares (GLS) formalism, as implemented in, e.g., SAMMY. We have recently developed a prototype of Bayesian Monte Carlo (BMC) evaluation framework, implemented using a Markov Chain Monte Carlo (MCMC) method with a Metropolis-Hastings (MH) acceptance criterion. This was done in order to remove the approximations underlying the conventional GLS evaluations, namely, the linear approximation, and the approximation that all probability density functions (PDFs) are of the normal kind. Recent works by others have used similar stochastic approaches to quantify cross section uncertainties from ENDF evaluated co-variances, and/or, from integral benchmark data, but those have not been conceived as an evaluation framework like the one presented here.
Part I presented a Monte Carlo Bayesian method for constraining a complex statistical model of GCM sub-gridcolumn moisture variability using high-resolution MODIS cloud data, thereby permitting large-scale model parameter estimation and cloud data assimilation. This part performs some basic testing of this new approach, verifying that it does indeed significantly reduce mean and standard deviation biases with respect to the assimilated MODIS cloud optical depth, brightness temperature and cloud top pressure, and that it also improves the simulated rotational-Ramman scattering cloud optical centroid pressure (OCP) against independent (non-assimilated) retrievals from the OMI instrument. Of particular interest, the Monte Carlo method does show skill in the especially difficult case where the background state is clear but cloudy observations exist. In traditional linearized data assimilation methods, a subsaturated background cannot produce clouds via any infinitesimal equilibrium perturbation, but the Monte Carlo approach allows finite jumps into regions of non-zero cloud probability. In the example provided, the method is able to restore marine stratocumulus near the Californian coast where the background state has a clear swath. This paper also examines a number of algorithmic and physical sensitivities of the new method and provides guidance for its cost-effective implementation. One obvious difficulty for the method, and other cloud data assimilation methods as well, is the lack of information content in the cloud observables on cloud vertical structure, beyond cloud top pressure and optical thickness, thus necessitating strong dependence on the background vertical moisture structure. It is found that a simple flow-dependent correlation modification due to Riishojgaard (1998) provides some help in this respect, by better honoring inversion structures in the background state.
Conventional training methods for neural networks involve starting al a random location in the solution space of the network weights, navigating an error hyper surface to reach a minimum, and sometime stochastic based techniques (e.g., genetic algorithms) to avoid entrapment in a local minimum. It is further typically necessary to preprocess the data (e.g., normalization) to keep the training algorithm on course. Conversely, Bayesian based learning is an epistemological approach concerned with formally updating the plausibility of competing candidate hypotheses thereby obtaining a posterior distribution for the network weights conditioned on the available data and a prior distribution. In this paper, we developed a powerful methodology for estimating the full residual uncertainty in network weights and therefore network predictions by using a modified Jeffery's prior combined with a Metropolis Markov Chain Monte Carlo method.
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Despite the importance of clouds and their influence on atmospheric water and energy balance, Numerical Weather Prediction (NWP) centers systematically exclude cloud information from the assimilation process and only assimilate clear-sky radiances (Janiskov´a et al. 2012). In order to ensure that only clear sky radiances are assimilated, strict cloud detection thresholds are applied before radiances are fed into data assimilation (DA) systems. This process not only excludes a large portion of satellite radiances, but causes loss of information in the regions that are of high interest to meteorologists and are most challenging for weather forecasts (Errico et al. 2007; Haddad et al. 2015). Although, in recent years there has been great advances in the operational weather forecasting, the prediction of tropical cyclones (TC), especially the intensity of TCs, remains challenging. According to Aksoy et al. (2013), in addition to the model deficiencies, another important factor that contributes to this challenge includes lack of observations in the peripheral environment (rain- bands) of TCs mainly because of the selective assimilation of existing observations. Satellite observations provide more than 90 % of the input data for the initialization of NWP models but more than 75 % of satellite observations are discarded due to the cloud contamination as well as land, snow, and ice emissivity issues (Bauer et al. 2010).
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Nuclear data evaluation methods conventionally make the following assumptions: prior and posterior probability distribution functions (PDFs) of all model parameters and data are normal (Gaussian); the linear approximation is sufficiently accurate for minimization of a cost function (even for non-linear models); and that both the model (of, e.g., neutron cross section) and experimental data (including their covariance data) are perfect. These assumptions are inherent to the well-known generalized linear least squares (GLLS) minimization method commonly used for evaluations of resolved resonance region (RRR) neutron cross sections. However, these assumptions are often not justified due to the presence of non-normal PDFs, non-linear models (e.g. R -matrix formalism), and inherent imperfections in data and models (e.g. discrepant data sets, discrepancies between the previous evaluation and newly measured data, or imperfect covariance data). We remove the said assumptions in a mathematical framework of Bayes’ theorem, and implement it using the Metropolis-Hastings Monte Carlo method. Parameters of a new kind are introduced to parameterize inherent imperfections, e.g. , any discrepancies between the theoretical model and measured data. These new parameters enable evaluators to quantify their expert judgement about any discrepancies or imperfections in a reproducible manner. We demonstrate the framework with an ongoing evaluation of 233 U in the eV region using the ENDF-B/VIII library and transmission data measured by Guber, et al. , and compare the posterior parameters to those obtained by conventional evaluation methods.