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Probability Review
Published in Charles Yoe, Principles of Risk Analysis, 2019
Probability is the chance that something will or will not happen. There are two schools of thought on the nature of probability. They go by many names, but we'll call them the frequentist and subjectivist schools. The frequentist approach to probability is based loosely on the notion that true probability values are “out there” and that we can discover them through data. Specifically, we can calculate the long-run expected frequency of occurrence. The probability of an event A, P(A), is equal to n/N, where n is the number of times event A occurs in N opportunities. It is the frequency with which A occurs out of the number of times it could occur. So, the annual probability of a hurricane hitting your community is estimated by the number of years a hurricane strikes out of the number of years observed. The probability of an accident per vehicle mile is the number of accidents divided by the number of vehicle miles. The frequentist view of probability works quite well with repeatable events.
Evaluation and Incorporation of Uncertainties in Geotechnical Engineering
Published in Chong Tang, Kok-Kwang Phoon, Model Uncertainties in Foundation Design, 2021
There are two distinct “philosophies” of inference: frequentist and Bayesian. The fundamental difference between frequentist and Bayesian approaches is how the concept of probability is interpreted. The frequentist approach defines an event’s probability as the limit of its relative frequency in a large number of trials. From a Bayesian viewpoint, probability is related to the degree of belief about the value of an unknown parameter that is a measure of the plausibility of an event given incomplete knowledge. Frequentist inference is based on sampling theory in which random samples are taken from a population to ascertain the underlying parameters of interest (e.g. mean, COV or correlation). From a frequentist viewpoint, unknown parameters are often assumed to have fixed but unknown values that are not capable of being treated as random variates. Hence, probabilities cannot be associated with these unknown parameters. On the contrary, Bayesian inference assigns probabilities to represent the belief that given values of the parameter are true. While “probabilities” are involved in both approaches for inference, the probabilities are associated with different entities. The result of a Bayesian approach can be a probability distribution for what is known about the parameters, while the result of a frequentist approach is either a “true or false” conclusion from a significance test or a conclusion from a confidence interval. Because it is easy to implement in the computational sense, currently, the frequentist approach dominates the characterization of geotechnical data, as outlined in DNVGL-RP-C207 (DNV 2017).
Bayesian Learning Approach
Published in Mark Chang, Artificial Intelligence for Drug Development, Precision Medicine, and Healthcare, 2020
There are differences regarding the concept of probability, Bayesian probability versus frequentist probability. As we discussed earlier, frequentists would consider data X is random and the parameter θ (such as the effect of a drug on a given patient population) is fixed, while Bayesians consider both X and parameter Θ to be random. Here, the Bayesian model parameter Θ is the knowledge of frequentist parameter θ. In other words, the effect θ is fixed but unknown to us. But our knowledge of the effect of the drug is changing as the data accumulates.
Uncertainty quantification for characterization of rock elastic modulus based on P-velocity
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Jian Liu, Quan Jiang, Dingping Xu, Hong Zheng, Fengqiang Gong, Jie Xin
According to the conception of the frequentist framework, the data are random samples from repeated trials, and the parameters of the model are fixed but unknown values to be estimated. The most commonly used method for parameter estimation of regression models is the least-squares (LS) method. The basic idea of the LS method is to ensure that the predicted values are close enough to the real values, the real parameters should minimise the sum of squares of the residuals (the difference between the observed value and the fitted value). The sum of squares of the residuals can be expressed as where is the observed value and is the predicted value. The parameters that can minimise the sum of squares of the residuals, , are the parameters of the least-square estimations that give the best fit.
Characterisation of geotechnical model uncertainty
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2019
For a design model in geotechnical engineering, a set of model factors (population) will be obtained from a database. Statistical inference concerns estimating unknown parameters that describe the characteristics of the population (e.g. mean, COV, probability distribution function, and even higher order information involving dependencies between variables, etc.). Inferences are made using data and a statistical model that links the data to the parameters. There are two distinct “philosophies” to inference – frequentist and Bayesian. The fundamental difference between frequentist and Bayesian approaches is how the concept of probability is interpreted. The frequentist approach interprets probability as the relative frequency in a large number of trials, while the Bayesian approach interprets probability as a measure of the degree of belief about the value of an unknown parameter. Currently, the frequentist approach dominates the characterisation of model factors, as it is easy to implement in the computational sense.
A computerized hybrid Bayesian-based approach for modelling the deterioration of concrete bridge decks
Published in Structure and Infrastructure Engineering, 2019
Eslam Mohammed Abdelkader, Tarek Zayed, Mohamed Marzouk
The main significant difference between the Bayesian inference and frequentist inference is the capability of the Bayesian inference to include additional information in the form of prior distribution (Rudas, 2008). Kelly et al. (2010) illustrated that the main distinctive feature of the Bayesian inference is its capability to consider information from different sources into the inference model. Thus, the Bayesian inference integrates the old knowledge and the new knowledge into an evidence-based state of knowledge distribution. Bayesian inference is based on interpreting the probability as ‘a rational, and conditional measure of uncertainty’, which nearly matches the sense of the word probability in the ordinary language (Bernardo, 2003).