scholarly journals A Bayesian Approach to Morphological Models Characterization

2021 ◽  
Vol 40 (3) ◽  
pp. 171-180
Author(s):  
Bruno Figliuzzi ◽  
Antoine Montaux-Lambert ◽  
François Willot ◽  
Grégoire Naudin ◽  
Pierre Dupuis ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chains ◽  
Objective Function ◽  
Bayesian Approach ◽  
Posterior Distribution ◽  
Heterogeneous Materials ◽  
Optimal Parameters ◽  
Mcmc Algorithm ◽  
Definition Of ◽  
The Bayesian Approach

Morphological models are commonly used to describe microstructures observed in heterogeneous materials. Usually, these models depend upon a set of parameters that must be chosen carefully to match experimental observations conducted on the microstructure. A common approach to perform the parameters determination is to try to minimize an objective function, usually taken to be the discrepancy between measurements computed on the simulations and on the experimental observations, respectively. In this article, we present a Bayesian approach for determining the parameters of morphological models, based upon the definition of a posterior distribution for the parameters. A Monte Carlo Markov Chains (MCMC) algorithm is then used to generate samples from the posterior distribution and to identify a set of optimal parameters. We show on several examples that the Bayesian approach allows us to properly identify the optimal parameters of distinct morphological models and to identify potential correlations between the parameters of the models.

Markov Chain Monte Carlo

2015 ◽  
Author(s):  
N. Thompson Hobbs ◽  
Mevin B. Hooten
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Analysis ◽  
Posterior Distribution ◽  
Mcmc Algorithm ◽  
Bayesian Analyses ◽  
Seminal Paper ◽  
Formal Treatment ◽  
High Level

This chapter explains how to implement Bayesian analyses using the Markov chain Monte Carlo (MCMC) algorithm, a set of methods for Bayesian analysis made popular by the seminal paper of Gelfand and Smith (1990). It begins with an explanation of MCMC with a heuristic, high-level treatment of the algorithm, describing its operation in simple terms with a minimum of formalism. In this first part, the chapter explains the algorithm so that all readers can gain an intuitive understanding of how to find the posterior distribution by sampling from it. Next, the chapter offers a somewhat more formal treatment of how MCMC is implemented mathematically. Finally, this chapter discusses implementation of Bayesian models via two routes—by using software and by writing one's own algorithm.

scholarly journals Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach

2019 ◽  
Vol 42 (2) ◽  
pp. 143-166 ◽  
Author(s):  
Renato Santos Silva ◽  
Fernando Ferraz Nascimento
Keyword(s):  
Monte Carlo ◽  
Order Statistics ◽  
Bayesian Approach ◽  
Extreme Value Theory ◽  
Likelihood Estimation ◽  
Value Theory ◽  
Extreme Value ◽  
Environmental Data ◽  
Heavy Tailed ◽  
The Bayesian Approach

Extreme Value Theory (EVT) is an important tool to predict efficient gains and losses. Its main areas of analyses are economic and environmental. Initially, for that form of event, it was developed the use of patterns of parametric distribution such as Normal and Gamma. However, economic and environmental data presents, in most cases, a heavy-tailed distribution, in contrast to those distributions. Thus, it was faced a great difficult to frame extreme events. Furthermore, it was almost impossible to use conventional models, making predictions about non-observed events, which exceed the maximum of observations. In some situations EVT is used to analyse only the maximum of some dataset, which provide few observations, and in those cases it is more effective to use the r largest-order statistics. This paper aims to propose Bayesian estimators' for parameters of the r largest-order statistics. During the research, it was used Monte Carlo simulation to analyze the data, and it was observed some properties of those estimators, such as mean, variance, bias and Root Mean Square Error (RMSE). The estimation of the parameters provided inference for its parameters and return levels. This paper also shows a procedure to the choice of the r-optimal to the r largest-order statistics, based on the Bayesian approach applying Markov chains Monte Carlo (MCMC). Simulation results reveal that the Bayesian approach has a similar performance to the Maximum Likelihood Estimation, and the applications were developed using the Bayesian approach and showed a gain in accurary compared with otherestimators.

scholarly journals Bayesian Confidence Intervals for Coefficients of Variation of PM10 Dispersion

2021 ◽  
Vol 5 (2) ◽  
pp. 139-154
Author(s):  
Warisa Thangjai ◽  
Sa-Aat Niwitpong ◽  
Suparat Niwitpong
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Confidence Intervals ◽  
Bayesian Approach ◽  
Normal Distributions ◽  
Coefficients Of Variation ◽  
The Difference ◽  
Log Normal ◽  
Log Normal Distribution ◽  
The Bayesian Approach

Herein, we propose the Bayesian approach for constructing the confidence intervals for both the coefficient of variation of a log-normal distribution and the difference between the coefficients of variation of two log-normal distributions. For the first case, the Bayesian approach was compared with large-sample, Chi-squared, and approximate fiducial approaches via Monte Carlo simulation. For the second case, the Bayesian approach was compared with the method of variance estimates recovery (MOVER), modified MOVER, and approximate fiducial approaches using Monte Carlo simulation. The results show that the Bayesian approach provided the best approach for constructing the confidence intervals for both the coefficient of variation of a log-normal distribution and the difference between the coefficients of variation of two log-normal distributions. To illustrate the performances of the confidence limit construction approaches with real data, they were applied to analyze real PM10 datasets from the Nan and Chiang Mai provinces in Thailand, the results of which are in agreement with the simulation results. Doi: 10.28991/esj-2021-01264 Full Text: PDF

USING FULL PROBABILITY MODELS TO COMPUTE PROBABILITIES OF ACTUAL INTEREST TO DECISION MAKERS

2001 ◽  
Vol 17 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Frank E. Harrell ◽  
Ya-Chen Tina Shih
Keyword(s):  
Bayesian Approach ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Likelihood Function ◽  
Scientific Evidence ◽  
A Priori ◽  
Relevant Information ◽  
Decision Makers ◽  
Probability Models ◽  
The Bayesian Approach

The objective of this paper is to illustrate the advantages of the Bayesian approach in quantifying, presenting, and reporting scientific evidence and in assisting decision making. Three basic components in the Bayesian framework are the prior distribution, likelihood function, and posterior distribution. The prior distribution describes analysts' belief a priori; the likelihood function captures how data modify the prior knowledge; and the posterior distribution synthesizes both prior and likelihood information. The Bayesian approach treats the parameters of interest as random variables, uses the entire posterior distribution to quantify the evidence, and reports evidence in a “probabilistic” manner. Two clinical examples are used to demonstrate the value of the Bayesian approach to decision makers. Using either an uninformative or a skeptical prior distribution, these examples show that the Bayesian methods allow calculations of probabilities that are usually of more interest to decision makers, e.g., the probability that treatment A is similar to treatment B, the probability that treatment A is at least 5% better than treatment B, and the probability that treatment A is not within the “similarity region” of treatment B, etc. In addition, the Bayesian approach can deal with multiple endpoints more easily than the classic approach. For example, if decision makers wish to examine mortality and cost jointly, the Bayesian method can report the probability that a treatment achieves at least 2% mortality reduction and less than $20,000 increase in costs. In conclusion, probabilities computed from the Bayesian approach provide more relevant information to decision makers and are easier to interpret.

scholarly journals Hierarchical Bayesian Estimation for Stationary Autoregressive Models Using Reversible Jump MCMC Algorithm

2018 ◽  
Vol 7 (4.30) ◽  
pp. 64
Author(s):  
Supar Man ◽  
Mohd Saifullah Rusiman
Keyword(s):  
Bayesian Approach ◽  
Posterior Distribution ◽  
Autoregressive Model ◽  
Prior Distribution ◽  
Likelihood Function ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Reversible Jump ◽  
Hierarchical Bayesian ◽  
Mcmc Algorithm

The autoregressive model is a mathematical model that is often used to model data in different areas of life. If the autoregressive model is matched against the data then the order and coefficients of the autoregressive model are unknown. This paper aims to estimate the order and coefficients of an autoregressive model based on data. The hierarchical Bayesian approach is used to estimate the order and coefficients of the autoregressive model. In the hierarchical Bayesian approach, the order and coefficients of the autoregressive model are assumed to have a prior distribution. The prior distribution is combined with the likelihood function to obtain a posterior distribution. The posterior distribution has a complex shape so that the Bayesian estimator is not analytically determined. The reversible jump Markov Chain Monte Carlo (MCMC) algorithm is proposed to obtain the Bayesian estimator. The performance of the algorithm is tested by using simulated data. The test results show that the algorithm can estimate the order and coefficients of the autoregressive model very well. Research can be further developed by comparing with other existing methods.

Probabilistic multiscale modeling of fracture in heterogeneous materials and structures

2020 ◽  
Vol 86 (7) ◽  
pp. 45-54
Author(s):  
A. M. Lepikhin ◽  
N. A. Makhutov ◽  
Yu. I. Shokin
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Fracture Mechanics ◽  
Multiscale Modeling ◽  
Virtual Work ◽  
Numerical Models ◽  
Heterogeneous Materials ◽  
Heterogeneous Structure ◽  
Generalized Coordinates ◽  
Heterogeneous Structures

The probabilistic aspects of multiscale modeling of the fracture of heterogeneous structures are considered. An approach combining homogenization methods with phenomenological and numerical models of fracture mechanics is proposed to solve the problems of assessing the probabilities of destruction of structurally heterogeneous materials. A model of a generalized heterogeneous structure consisting of heterogeneous materials and regions of different scales containing cracks and crack-like defects is formulated. Linking of scales is carried out using kinematic conditions and multiscale principle of virtual forces. The probability of destruction is formulated as the conditional probability of successive nested fracture events of different scales. Cracks and crack-like defects are considered the main sources of fracture. The distribution of defects is represented in the form of Poisson ensembles. Critical stresses at the tops of cracks are described by the Weibull model. Analytical expressions for the fracture probabilities of multiscale heterogeneous structures with multilevel limit states are obtained. An approach based on a modified Monte Carlo method of statistical modeling is proposed to assess the fracture probabilities taking into account the real morphology of heterogeneous structures. A feature of the proposed method is the use of a three-level fracture scheme with numerical solution of the problems at the micro, meso and macro scales. The main variables are generalized forces of the crack propagation and crack growth resistance. Crack sizes are considered generalized coordinates. To reduce the dimensionality, the problem of fracture mechanics is reformulated into the problem of stability of a heterogeneous structure under load with variations of generalized coordinates and analysis of the virtual work of generalized forces. Expressions for estimating the fracture probabilities using a modified Monte Carlo method for multiscale heterogeneous structures are obtained. The prospects of using the developed approaches to assess the fracture probabilities and address the problems of risk analysis of heterogeneous structures are shown.

scholarly journals The Bayesian Approach and the Results of the ISAR-REACT 5 Trial

2021 ◽  
Vol 14 (2) ◽  
pp. 231-232
Author(s):  
Adnan Kastrati ◽  
Alexander Hapfelmeier
Keyword(s):  
Bayesian Approach ◽  
The Bayesian Approach

scholarly journals AB0140 A HIGH-CONFIDENCE DEFINITION OF THERAPEUTIC RESPONSE IN RHEUMATOID ARTHRITIS USING A MONTE CARLO SIMULATION APPROACH

2021 ◽  
Vol 80 (Suppl 1) ◽  
pp. 1097.2-1098
Author(s):  
V. Strand ◽  
S. Cohen ◽  
L. Zhang ◽  
T. Mellors ◽  
A. Jones ◽  
...  
Keyword(s):  
Rheumatoid Arthritis ◽  
Monte Carlo ◽  
Disease Activity ◽  
Health Assessment Questionnaire ◽  
Health Assessment ◽  
High Confidence ◽  
Response Status ◽  
Definition Of ◽  
Fidelity Score ◽  
Assessment Questionnaire

Background:Therapy choice and therapy change depend on the ability to accurately assess patients’ disease activity. The clinical assessments used to evaluate treatment response in rheumatoid arthritis have inherent variability, normally considered as measurement error, intra-observer variability or within subject variability. Each contribute to variability in deriving response status as defined by composite measures such as the ACR or EULAR criteria, particularly when a one-time observed measurement lies near the boundary defining response or non-response. To select an optimal therapeutic strategy in the burgeoning age of precision medicine in rheumatology, achieve the lowest disease activity and maximize long-term health outcomes for each patient, improved treatment response definitions are needed.Objectives:Develop a high-confidence definition of treatment response and non-response in rheumatoid arthritis that exceeds the expected variability of subcomponents in the composite response criteria.Methods:A Monte Carlo simulation approach was used to assess ACR50 and EULAR response outcomes in 100 rheumatoid arthritis patients who had been treated for 6 months with a TNF inhibitor therapy. Monte Carlo simulations were run with 2000 iterations implemented with measurement variability derived for each clinical assessment: tender joint count, swollen joint count, Health Assessment Questionnaire disability index (HAQ-DI), patient pain assessment, patient global assessment, physician global assessment, serum C-reactive protein level (CRP) and disease activity score 28-joint count with CRP.1-3 Each iteration of the Monte Carlo simulation generated one outcome with a value of 0 or 1 indicating non-responder or responder, respectively.Results:A fidelity score, calculated separately for ACR50 and EULAR response, was defined as an aggregated score from 2000 iterations reported as a fraction that ranges from 0 to 1. The fidelity score depicted a spectrum of response covering strong non-responders, inconclusive statuses and strong responders. A fidelity score around 0.5 typified a response status with extreme variability and inconclusive clinical response to treatment. High-fidelity scores were defined as >0.7 or <0.3 for responders and non-responders, respectively, meaning that the simulated clinical response status label among all simulations agreed at least 70% of the time. High-confidence true responders were considered as those patients with high-fidelity outcomes in both ACR50 and EULAR outcomes.Conclusion:A definition of response to treatment should exceed the expected variability of the clinical assessments used in the composite measure of therapeutic response. By defining high-confidence responders and non-responders, the true impact of therapeutic efficacy can be determined, thus forging a path to development of better treatment options and advanced precision medicine tools in rheumatoid arthritis.References:[1]Cheung, P. P., Gossec, L., Mak, A. & March, L. Reliability of joint count assessment in rheumatoid arthritis: a systematic literature review. Semin Arthritis Rheum43, 721-729, doi:10.1016/j.semarthrit.2013.11.003 (2014).[2]Uhlig, T., Kvien, T. K. & Pincus, T. Test-retest reliability of disease activity core set measures and indices in rheumatoid arthritis. Ann Rheum Dis68, 972-975, doi:10.1136/ard.2008.097345 (2009).[3]Maska, L., Anderson, J. & Michaud, K. Measures of functional status and quality of life in rheumatoid arthritis: Health Assessment Questionnaire Disability Index (HAQ), Modified Health Assessment Questionnaire (MHAQ), Multidimensional Health Assessment Questionnaire (MDHAQ), Health Assessment Questionnaire II (HAQ-II), Improved Health Assessment Questionnaire (Improved HAQ), and Rheumatoid Arthritis Quality of Life (RAQoL). Arthritis Care Res (Hoboken) 63 Suppl 11, S4-13, doi:10.1002/acr.20620 (2011).Disclosure of Interests:Vibeke Strand Consultant of: Abbvie, Amgen, Arena, BMS, Boehringer Ingelheim, Celltrion, Galapagos, Genentech/Roche, Gilead, GSK, Ichnos, Inmedix, Janssen, Kiniksa, Lilly, Merck, Novartis, Pfizer, Regeneron, Samsung, Sandoz, Sanofi, Setpoint, UCB, Stanley Cohen: None declared, Lixia Zhang Shareholder of: Scipher Medicine Corporation, Employee of: Scipher Medicine Corporation, Ted Mellors Shareholder of: Scipher Medicine Corporation, Employee of: Scipher Medicine Corporation, Alex Jones Shareholder of: Scipher Medicine Corporation, Employee of: Scipher Medicine Corporation, Johanna Withers Shareholder of: Scipher Medicine Corporation, Employee of: Scipher Medicine Corporation, Viatcheslav Akmaev Shareholder of: Scipher Medicine Corporation, Employee of: Scipher Medicine Corporation

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