Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Bayesian method”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Intelligent Process Visualization through Nuclear Operation Process Modeling, Reasoning, and Object Detection from Field Videos (Final Report)

This report is a deliverable for the “Final Report” task of DOE NEET Project 19-16790, "Context-Aware Safety Information Display for Nuclear Field Workers." This project's overall goal is to test the hypothesis that integrating computer vision and process reasoning methods will enable proactive visualization of the safe operation and maintenance processes of Nuclear Power Plants (NPP) for field workers. Augmented Reality (AR) glasses adopting such proactive safety information visualization techniques can significantly increase personnel safety and reduce the NPP’s operating costs. The current practice of monitoring NPPs requires workers to switch between digital models, data, and physical workspaces in identifying relevant but potentially occluded objects and in assessing the risks of operation and maintenance processes. On the other hand, frequently changed field conditions require field workers to report to supervisors for real-time guidance. Such guidance is essential to ensure that changing conditions will not invalidate or endanger the work order and other ongoing processes that may jeopardize NPP operations. Additionally, incorrect recognition of equipment objects can result in communication errors and safety problems. AR techniques can assist engineers in viewing the physical workspaces with objects labeled with detailed operation procedures and safety reminders during field operations. The project team developed an “Intelligent Context-Aware Safety Information Display” (ICAD) for supporting Nuclear Power Plant (NPP) field workers in achieving safe and efficient execution of a series of operational tasks in uncertain and changing workspaces of an NPP. Before designing the ICAD-AR prototype, the project team synthesized NPP operational knowledge models through literature review studies, surveys, interviews with domain experts, and knowledge modeling. The project team conducted an extensive study of the operational procedures of various NPPs, and digital technologies that can support the safe and efficient execution of those procedures in different NPP operational contexts. This literature review helped the project team conduct surveys and interviews with nuclear engineers and field workers to identify three categories of information. The NPP knowledge modeling efforts reveal that the three categories of information identified have different levels of importance in a typical procedure of carrying out a series of tasks to achieve a specific NPP operation goal (e.g., shutdown, mode changes). These three categories of information include 1) Workspace dynamics – the changing spatial arrangements of workspaces, tools, protection equipment, and supporting materials, 2) Workflow prognostics – the dynamic dependencies between different parts of an NPP that functionally support and influence each other in terms of safety and efficiency, and 3) Hazards – objects and spaces that contain hazardous materials or physical conditions that can pose risks to workers or mechanical systems. The project team has profiled the importance levels of these categories of information into a knowledge model. This knowledge model specifies what types of information are more critical for a given task in a given workspace so that computers can automatically identify critical objects and sensors in a scene for delivering context-ware safety information to field workers through AR devices. Significant research development of this project results in technical research outcomes and a prototyping system that illustrates the technical feasibility of establishing an ICAD-AR system supporting the proactive safety information display for nuclear field workers. This final report summarizes the project team’s technological achievements in the past three years. Overall, the project team completed the development and integration of five techniques into a prototype ICAD Augmented Reality (ICAD-AR) system and demonstrated the developed system’s real-time execution in a mechanical room. The project team completed the analysis of using this prototype in other types of workspaces based on 3D image data and digital design models collected from two additional workspaces (a water treatment plant and a flow loop training facility). The integrated techniques include 1) Natural Language Processing (NLP) algorithms supporting the generation and updates of nuclear fieldwork process models based on text analysis of work packages and operation manuals; 2) sensor log analysis for predicting control actions in given sensor reading contexts; 3) computer vision algorithms for automatic localization and navigation of workers; 4) object detection algorithms for identifying task-related objects and correlated sensors for safety checking; 5) AR technique as a platform for supporting the integration. The testing results of these five techniques have shown that 1) the sensor log analysis model can predict the next control action with an accuracy of 0.883; 2) the trained natural language processing model can extract more than 80% of the critical information from paper-based procedures (PBPs); 3) the navigation algorithm with the integration of Visual Inertial Odometry (VIO) and Non-Recursive Bayesian Filter methods make operator’s trajectory estimation resilient to drift error; 4) the computer vision algorithm can detect task-specific and safety-critical objects with an average accuracy of 95.3%. The project team used work procedures collected from a flow loop training facility and two datasets collected from two mechanical rooms simulating the workspaces of NPPs to demonstrate the technical capabilities of the developed ICAD-AR prototype. The demonstration validated the technical feasibility of establishing the ICAD-AR system for nuclear field workers and identified the challenges in 1) automatic text analysis of work packages; 2) use of limited samples of sensor logs for predicting the proper timings of control actions; 3) reliably tracking workers and their task progress in mechanical rooms with many similar objects.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Effect of thermal conditioning on the initiation threshold of secondary high‐explosives

While most performance metrics of high–explosive (HE) based devices like detonation velocity, detonation pressure, and energy output are expected to degrade over time, the evolution of initiation threshold appears less clear, with claims of both increasing and decreasing trends in threshold having been made in the literature. This work analyzes D–optimally designed sequential binary test data for a few thermally conditioned porous–powder and polymer–bonded HE initiator systems using a Bayesian likelihood method employing the probit regression model. Here we find that in most cases the initiation threshold decreases (i.e., sensitivity increases) upon accelerated thermal conditioning. However, such results are nuanced and influenced by factors like the contact area of initiating stimulus, HE characteristics like density and specific surface area, as well as possible thermally induced changes to other materials and interfaces involved.

36 MATERIALS SCIENCE↗
DOE OSTI2022

B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD

Here, the Deep Operator Network (DeepONet) is a neural network architecture used to approximate operators, including the solution operator of parametric PDEs. DeepONets have shown remarkable approximation ability. However, the performance of DeepONets deteriorates when the training data is polluted with noise, a scenario that occurs in practice. To handle noisy data, we propose a Bayesian DeepONet based on replica exchange Langevin diffusion (reLD). Replica exchange uses two particles. The first particle trains a DeepONet to exploit the loss landscape and make predictions. The other particle trains a different DeepONet to explore the loss landscape and escape local minima via swapping. Compared to DeepONets trained with state-of-the-art gradient-based algorithms (e.g., Adam), the proposed Bayesian DeepONet greatly improves the training convergence for noisy scenarios and accurately estimates the uncertainty. To further reduce the high computational cost of the reLD training of DeepONets, we propose (1) an accelerated training framework that exploits the DeepONet's architecture to reduce its computational cost up to 25% without compromising performance and (2) a transfer learning strategy that accelerates training DeepONets for PDEs with different parameter values. Finally, we illustrate the effectiveness of the proposed Bayesian DeepONet using four parametric PDE problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Optimizing long-term monitoring of radiation air-dose rates after the Fukushima Daiichi Nuclear Power Plant

Radiation air dose rates near the Fukushima Daiichi Nuclear Power Plant (FDNPP) have been steadily decreasing over the past eight years since the release of radioactive elements in March 2011. Currently, the radiation monitoring program is expected to transition to long-term monitoring after most of the remediation activities are completed. The main long-term monitoring objectives are to (1) confirm the continuing reduction of contaminant and hazard levels, (2) provide assurance for the public, (3) accumulate the basic datasets for scientific knowledge and future preparation, and (4) detect changes or anomalies in contaminant mobility (if they occur), or any unexpected processes or events. In this work, we have developed a methodology for optimizing the monitoring locations of radiation air dose-rate monitoring. Our approach consists of three steps in order to determine monitoring locations in a systematic manner: (1) prioritizing the critical locations, such as schools or regulatory requirement locations, (2) diversifying locations that cover the key environmental controls that are known to influence contaminant mobility and distributions, and (3) capturing the heterogeneity of radiation air-dose rates across the domain. Therefore, for the second step, we use a Gaussian mixture model to identify the representative locations among multiple environmental variables, such as elevation and land-cover types. For the third step, we use a Gaussian process model to capture and estimate the heterogeneity of air-dose rates across the domain. Employing an integrated dose-rate map derived from Bayesian geostatistical methods as a reference map, we distribute the monitoring locations in such a way as to capture the heterogeneity of the reference map. Our results have shown that this approach allows us to select monitoring locations in a systematic manner such that the heterogeneity of air dose rates is captured by the minimal number of monitoring locations.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2022

Estimation of process steady state with autoregressive models and Bayesian inference

To improve efficiency, separations engineers will typically design process circuits containing recirculating streams, which mix one or more of the process outputs with the feed material. Doing so can improve efficiency, but will cause a delay in the system reaching steady state conditions until the recirculating load mass flows stabilize. In testing separation circuits, engineers will often test a variety of factors and complete an analysis from sample results. Knowledge of if a process is at steady state, as well as the steady state conditions of a process, is essential for a valid techno-economic analysis. However, the definition of process steady state is often poorly defined, or does not include uncertainty quantification. If the performance of a process operating under two different sets of conditions are compared, an engineer who does not test for steady state or quantify steady state conditions risks producing a faulty analysis. In this work, a Bayesian statistical method for testing if all streams are at steady state is further motivated and then derived. Then after testing for steady state, the same model is used with a prior distribution that enforces a steady state assumption to estimate steady state conditions. Further, these methods were validated in a solvent extraction pilot plant where steady state conditions for all outflows were inferred with uncertainty quantification. Analysis is completed with functions available to the reader as part of the BayesMassBal (V 1.1.0) software package written in R.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2025

Bayesian model-data comparison incorporating theoretical uncertainties

Accurate comparisons between theoretical models and experimental data are critical for scientific progress. However, inferred physical model parameters can vary significantly with the chosen physics model, highlighting the importance of properly accounting for theoretical uncertainties. In this Letter, we present a Bayesian framework that explicitly quantifies these uncertainties by statistically modeling theory errors, guided by qualitative knowledge of a theory’s varying reliability across the input domain. We demonstrate the effectiveness of this approach using two systems: a simple ball drop experiment and multi-stage heavy-ion simulations. In both cases incorporating model discrepancy leads to improved parameter estimates, with systematic improvements observed as additional experimental observables are integrated.

Bayesian methods↗
DOE OSTI2020

An assessment of Earth's climate sensitivity using multiple lines of evidence

In this work, we assess evidence relevant to Earth's equilibrium climate sensitivity per doubling of atmospheric CO 2 , characterized by an effective sensitivity S . This evidence includes feedback process understanding, the historical climate record, and the paleoclimate record. An S value lower than 2 K is difficult to reconcile with any of the three lines of evidence. The amount of cooling during the Last Glacial Maximum provides strong evidence against values of S greater than 4.5 K. Other lines of evidence in combination also show that this is relatively unlikely. We use a Bayesian approach to produce a probability density (PDF) for S given all the evidence, including tests of robustness to difficult-to-quantify uncertainties and different priors. The 66% range is 2.6-3.9 K for our Baseline calculation, and remains within 2.3-4.5 K under the robustness tests; corresponding 5-95% ranges are 2.3-4.7 K, bounded by 2.0-5.7 K (although such high-confidence ranges should be regarded more cautiously). This indicates a stronger constraint on S than reported in past assessments, by lifting the low end of the range. This narrowing occurs because the three lines of evidence agree and are judged to be largely independent, and because of greater confidence in understanding feedback processes and in combining evidence. We identify promising avenues for further narrowing the range in S , in particular using comprehensive models and process understanding to address limitations in the traditional forcing-feedback paradigm for interpreting past changes.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Neglecting Model Parametric Uncertainty Can Drastically Underestimate Flood Risks

Abstract Floods drive dynamic and deeply uncertain risks for people and infrastructures. Uncertainty characterization is a crucial step in improving the predictive understanding of multi‐sector dynamics and the design of risk‐management strategies. Current approaches to estimate flood hazards often sample only a relatively small subset of the known unknowns, for example, the uncertainties surrounding the model parameters. This approach neglects the impacts of key uncertainties on hazards and system dynamics. Here we mainstream a recently developed method for Bayesian inference to calibrate a computationally expensive distributed hydrologic model. We compare three different calibration approaches: (a) stepwise line search, (b) precalibration or screening, and (c) the Fast Model Calibrations (FaMoS) approach. FaMoS deploys a particle‐based approach that takes advantage of the massive parallelization afforded by modern high‐performance computing systems. We quantify how neglecting parametric uncertainty and data discrepancy can drastically underestimate extreme flood events and risks. Precalibration improves prediction skill score over a stepwise line search. The Bayesian calibration improves the uncertainty characterization of model parameters and flood risk projections.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

On-the-fly active learning of interpretable Bayesian force fields for atomistic rare events

Abstract Machine learned force fields typically require manual construction of training sets consisting of thousands of first principles calculations, which can result in low training efficiency and unpredictable errors when applied to structures not represented in the training set of the model. This severely limits the practical application of these models in systems with dynamics governed by important rare events, such as chemical reactions and diffusion. We present an adaptive Bayesian inference method for automating the training of interpretable, low-dimensional, and multi-element interatomic force fields using structures drawn on the fly from molecular dynamics simulations. Within an active learning framework, the internal uncertainty of a Gaussian process regression model is used to decide whether to accept the model prediction or to perform a first principles calculation to augment the training set of the model. The method is applied to a range of single- and multi-element systems and shown to achieve a favorable balance of accuracy and computational efficiency, while requiring a minimal amount of ab initio training data. We provide a fully open-source implementation of our method, as well as a procedure to map trained models to computationally efficient tabulated force fields.

36 MATERIALS SCIENCE↗
DOE OSTI2021

A probabilistic graphical model foundation for enabling predictive digital twins at scale

A unifying mathematical formulation is needed to move from one-off digital twins built through custom implementations to robust digital twin implementations at scale. This work proposes a probabilistic graphical model as a formal mathematical representation of a digital twin and its associated physical asset. We create an abstraction of the asset–twin system as a set of coupled dynamical systems, evolving over time through their respective state spaces and interacting via observed data and control inputs. The formal definition of this coupled system as a probabilistic graphical model enables us to draw upon well-established theory and methods from Bayesian statistics, dynamical systems and control theory. The declarative and general nature of the proposed digital twin model make it rigorous yet flexible, enabling its application at scale in a diverse range of application areas. Here, we demonstrate how the model is instantiated to enable a structural digital twin of an unmanned aerial vehicle (UAV). The digital twin is calibrated using experimental data from a physical UAV asset. Its use in dynamic decision-making is then illustrated in a synthetic example where the UAV undergoes an in-flight damage event and the digital twin is dynamically updated using sensor data. The graphical model foundation ensures that the digital twin calibration and updating process is principled, unified and able to scale to an entire fleet of digital twins.

42 ENGINEERING↗
DOE OSTI2020

Dependence of the impurity transport on the dominant turbulent regime in ELM-y H-mode discharges on the DIII-D tokamak

Laser blow-off injections of aluminum and tungsten have been performed on the DIII-D tokamak to investigate the variation of impurity transport in a set of dedicated ion and electron heating scans with a fixed value of the external torque. The particle transport is quantified via the Bayesian inference method, which, constrained by a combination of a charge exchange recombination spectroscopy, soft X-ray measurements, and VUV spectroscopy provides a detailed uncertainty quantification of transport coefficients. Contrasting discharge phases with a dominant electron and ion heating reveal 3-fold drop in the impurity confinement time and order of magnitude increase in midradius impurity diffusion, when additional electron heating is applied. Further, the calculated stationary aluminum density profiles reverse from peaked in electron heated to hollow in the ion heated case, following a similar trend as electron and carbon density. Comparable values of a core diffusion have been observed for W and Al ions, while differences in the propagation dynamics of these impurities are attributed to pedestal and edge transport. Modeling of the core transport with non-linear gyrokinetics code CGYRO, significantly underpredicts the magnitude of the variation in Al transport. Diffusion increases 3-times steeper with additional electron heat flux and 10-times lower diffusion is observed in ion heated case than predicted by the modeling. Here, the CGYRO model quantitatively matches the increase of the Al diffusion when approaching the linear threshold for the transition from the ion temperature gradient (ITG) to trapped electron mode (TEM).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Analysis of differential scanning calorimetry data for aged plutonium

Differential scanning calorimetry data for samples of a 52 year old plutonium alloy with 3.3 at. % Ga that were heated beyond the melting point is analyzed using transition state theory to find activation energies for the δ to ε and ε to liquid phase transitions. A Bayesian statistical method involving a Gaussian process model is used to find mean values and confidence intervals for the activation energies. The activation energy for the δ to ε phase transition increases by 3.3 ± 3.8% per decade, relative to the case when all age related plutonium lattice point defects have been removed through annealing. The corresponding increase in activation energy for the ε to liquid transition is shown to be 7.1 ± 1.8% per decade. It is postulated that the change in activation energy with age for both phase transitions is caused, in part, by the accumulation of the same type of lattice point defects associated with the observed increase in elastic bulk modulus over time.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

Sequential Selection for Minimizing the Variance with Application to Crystallization Experiments

For many crystal-based products (e.g., pharmaceuticals, energy storage), the size uniformity is not only a key quality attribute, but sometimes also an indicator of other attributes such as solid purity. This article proposes a sequential selection approach to find a proper experimental setting that leads to high uniformity, or equivalently, small variance for crystal sizes, from the advanced slug flow reaction crystallization process of a model crystal, called manganese oxalate hydrate. The proposed sequential selection approach contains a Bayesian adaptive method to incorporate new uniformity measurements in each step and two design acquisition functions to improve the selection of the most promising experimental setting in terms of minimizing the variance. We study the performance of the proposed approach through multiple synthetic numerical studies, as well as a case study based on data from slug flow crystallization experiments. Throughout these studies, the proposed approach shows competitive performance in identifying the best experimental setting.

Expected improvement↗
DOE OSTI2020

S -factor and scattering-parameter extractions from ${}^{3}\mathrm{He}+{}^{4}\mathrm{He}{ \rightarrow }^{7}\mathrm{Be}+\gamma $

Previous studies of the reaction ${}^{3}\mathrm{He}+{}^{4}\mathrm{He}\to {}^{7}\mathrm{Be}+\gamma $ have mainly focused on providing the best central value and error bar for the S factor at solar energies. Experimental measurements of this capture reaction at higher energies, the ${}^{3}\mathrm{He}$–${}^{4}\mathrm{He}$ scattering phase shifts, as well as properties of ${}^{7}\mathrm{Be}$ and its excited state, have been used to constrain the theoretical models employed for this purpose. Here we show that much more information than was previously appreciated can be extracted from angle-integrated capture data alone. We use the next-to-leading-order (NLO) amplitude in an effective field theory (EFT) for ${}^{3}\mathrm{He}+{}^{4}\mathrm{He}\to {}^{7}\mathrm{Be}+\gamma $ to perform the extrapolation. At this order the EFT describes the capture process using an s-wave scattering length and effective range, the asymptotic properties of ${}^{7}\mathrm{Be}$ and its excited state, and short-distance contributions to the E1 capture amplitude. We extract the multi-dimensional posterior of all these parameters via a Bayesian analysis that uses capture data below 2 MeV. We find that properties of the ${}^{7}\mathrm{Be}$ ground and excited states are well constrained. The total S factor $S(0)\,={0.577}_{-0.016}^{+0.015}$ keV b, while the branching ratio for excited- to ground-state capture at zero energy, ${Br}(0)={0.406}_{-0.011}^{+0.013}$, both at 68% degree of belief. This S(0) is broadly consistent with other recent evaluations, and agrees with the previously recommended value $S(0)=0.56\pm 0.03\,\,\mathrm{keV}$ b, but has a smaller error bar. We also find significant constraints on ${}^{3}\mathrm{He}$–${}^{4}\mathrm{He}$ scattering parameters, and we obtain constraints on the angular distribution of capture gamma rays, which is important for interpreting experiments. The path forward for this reaction seems to lie with better measurements of the scattering phase shifts and S(E)'s angular dependence away from zero energy, together with better understanding of the asymptotic normalization coefficients of the ${}^{7}$Be bound states' wave functions. Lastly, data on these could further reduce the uncertainty on S(0).

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Recent advances in the quantification of uncertainties in reaction theory

Uncertainty quantification has become increasingly more prominent in nuclear physics over the past several years. In few-body reaction theory, there are four main sources that contribute to the uncertainties in the calculated observables: the effective potentials, approximations made to the few-body problem, structure functions, and degrees of freedom left out of the model space. In this work, we illustrate some of the features that can be obtained when modern statistical tools are applied in the context of nuclear reactions. This work consists of a summary of the progress that has been made in quantifying theoretical uncertainties in this domain, focusing primarily on those uncertainties coming from the effective optical potential as well as their propagation within various reaction theories. We use, as the central example, reactions on the doubly-magic stable nucleus 40 Ca, namely neutron and proton elastic scattering and single-nucleon transfer 40 Ca(d,p) 41 Ca. First, we show different optimization schemes used to constrain the optical potential from differential cross sections and other experimental constraints; we then discuss how these uncertainties propagate to the transfer cross section, comparing two reaction theories. Finally, we finish by laying out our future plans.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Quantifying bias due to non-Gaussian foregrounds in an optimal reconstruction of CMB lensing and temperature power spectra

We estimate the magnitude of the bias due to non-Gaussian extragalactic foregrounds on the optimal reconstruction of the cosmic microwave background (CMB) lensing potential and temperature power spectra. The reconstruction is performed using a Bayesian inference method known as the marginal unbiased score expansion (MUSE). We apply MUSE to a minimum variance combination of multifrequency maps drawn from the Agora publicly available simulations of the lensed CMB and correlated extragalactic foreground emission. Taking noise levels appropriate to the SPT-3G D1 release, we find non-Gaussian foregrounds may bias the MUSE reconstruction of the lensing potential amplitude at the level of (0.7 ± 0.3)σ when using modes up to ℓ max = 3500. We do not detect a statistically significant bias, finding a value of (-0.4 ± 0.3)σ, when restricted to lower angular multipoles, ℓ max = 3000. This work is a first step toward understanding the impact of extragalactic foregrounds on optimal reconstructions of CMB temperature and lensing potential power spectra.

Statistical sampling techniques↗
DOE OSTI2026

Frequentist cosmological constraints from full-shape clustering measurements in DESI DR1

We present a frequentist analysis of clustering measurements from Data Release 1 of the Dark Energy Spectroscopic Instrument (DESI) using the standard profile likelihood method. While Bayesian inferences for effective field theory models of galaxy clustering can be highly sensitive to prior choices for extended cosmological models, frequentist inferences are not susceptible to such effects. We compare frequentist and Bayesian constraints for the parameter set {σ 8 , H 0 , Ω m , w 0 , w a } using the full-shape power spectrum multipoles, post-reconstruction baryon acoustic oscillation (BAO) measurements, and external datasets from the CMB and type Ia supernovae measurements. The frequentist confidence intervals are significantly shifted relative to the Bayesian credible intervals for the w 0 w a CDM model, unless supernovae data are included. When DESI full-shape and BAO data are fit jointly, we obtain the following 1σ frequentist confidence intervals for ΛCDM (w 0 w a CDM): σ 8 = 0.863 +0.048 -0.040 , H 0 = 68.96 +0.81 -0.80 km s -1 Mpc -1 , Ω m = 0.3034 ± 0.0110 (σ 8 = 0.782 +0.060 -0.036 , H 0 = 63.7 +4.2 -2.0 km s -1 Mpc -1 , Ω m = 0.378 +0.024 -0.047 , w 0 = -0.16 +0.10 -0.50 , w a = -3.0 +1.7 ), corresponding to 0.8σ, 0.3σ, 0.7σ (2.1σ, 4.1σ, 6.5σ, 6.3σ, 6.6σ) shifts between the maximum likelihood estimate and the Bayesian posterior mean for ΛCDM (w 0 w a CDM) respectively.

Bayesian reasoning↗