Bayesian fundamentals

Bayesian fundamentals are reviewed here because several chapters in this volume apply these methods in complex ways to assessing uncertainty. The goal is to create enough understanding so that methods described in later chapters can be fully appreciated. 

The frequentist interval is often interpreted as if it were the Bayesian interval, but it is fundamentally defined by the probability of the data values given the parameter and not the probability of the parameter given the data. 

Chapter 3 provides an introduction to the identification of mathematical models for reactive systems and an extensive review of the methods for estimating the relevant adjustable parameters. The chapter is initiated with a comparison between Bayesian approach and Poppers falsificationism. The aim is to establish a few fundamental ideas on the reliability of scientific knowledge, which is based on the comparison between alternative models and the experimental results, and is limited by the nonexhaustive nature of the available theories and by the unavoidable experimental errors. 

Most of the methods for analyzing data from supersaturated designs have been adapted from methods tailored for saturated or unsaturated designs. This might be a mistake as supersaturated designs are fundamentally different. Consider from first principles how we should carry out frequentist inference and Bayesian analysis. 

Jeffreys (1961) advanced Bayesian theory by giving an unprejudiced prior density p(6, S) for suitably differentiable models. His result, given in Chapter 5 and used below, is fundamental in Bayesian estimation. 

Bayesian statistics has at its heart the following fundamental equality... 

In my view, the program would have been more responsive to the actual needs of the policy makers had it been structured and funded in a more Bayesian fashion. It could then have asked and addressed fundamental... 

Ben-Naim, A. On the so-called Gibbs paradox, and on the real paradox. Entropy 9, 132-136 (2007). [electronic] http //www.mdpi.org/entropy/papers/e9030132.pdf Jaynes, E.T The Gibbs paradox. In C.R. Smith, G.J. Erickson, P.O. Neudorfer (eds.) Maximum Entropy and Bayesian Methods, Fundamental Theories of Physics, vol. 50, pp. 1-22. Kluwer Academic Publishers, Dordrecht, Holland (1992)... 

A. J. M. Garrett, in Ma.ximum Entropy and Bayesian Methods. Cambridge. England. 1988 Fundamental Theories of Physics (ed. J. Skilling), pp. 107-116, Kluwer Academics Publishers, Dordrecht. 1989. 

Bayesian decision theory is a fundamental statistical approach to the problem of classification. This approach is based on quantifying the trade-offs between various classification decisions using probability and the costs that accompany such decisions. It makes the assumption that the decision problem is posed in probabilistic terms and that all of the relevant probability values are known. 

From the classical perspective, the decision maker is concerned with determining the likelihood that a hypothesis is true. Bayesian inference is the best-known technique, but signal detection theory, and fundamentally different approaches such as the Dempster-Schafer method, have seen application. Each of these approaches is discussed below. 

It remains a quadratic function but the coefficients are uncertain. Equation (2.164) is applicable to all modal frequencies for Timoshenko beam models. Therefore, it is proposed to bridge the squared fundamental frequency and the ambient temperature by a quadratic function, and the coefficients can be estimated by Bayesian analysis. 

In this section, Bayesian analysis is performed to identify the uncertain coefficients of the quadratic function. The effective temperature T is assumed to be different from the measured values since there is measurement noise in the data acquisition process and the temperature in different parts of the building could also be non-uniform. The difference is assumed Gaussian with zero mean and variance a. In this study, this standard deviation is taken to be aj = 0.5°C since the difference between the indoor and outdoor temperature measurement was around 1 °C and the average value was used as the measured temperature T for the wth day. On the other hand, the squared fundamental frequency is identified by the Bayesian spectral density approach to be presented in Chapter 3. Therefore, the uncertain parameters include the coefficients of thequadratic function and the effective temperatures 0 = [l>o, b, b2, T, Ti,7 ], where iV is the number of data points. The data include the measurements of the temperature and the identified squared fundamental frequencies X> = [li,. .., j, 2> >... 

The Bayesian spectral density approach for parametric identification and model updating regression analysis are applied. During the monitoring period, four typhoons flitted over Macao. The structural behavior under such violent wind excitation is treated as discordance and the measurements obtained under these events are not taken into account for the analysis. By excluding these fifteen days of measurements, there are 168 pairs of identified squared fundamental frequency and measured temperature in the data set. Figure 2.28(a) shows the variation of the identified squared fundamental frequencies with their associated uncertainties represented by a confidence interval that is bounded by the plus or minus three standard derivations from the estimated values. It is noticed that this confidence interval contains 99.7% of the probability. Since the confidence intervals are narrow compared with the variation... 

Another case is investigated with a very short duration of measurement, namely T = 60 s, so it contains roughly 38 fundamental periods of the oscillator. The Bayesian spectral density approach is used for its identification with the frequency index set /C = 1,2,..., 45. Figure 3.13 shows the conditional updated PDFs of and with all other parameters fixed at their optimal values. It is obvious that the conditional PDFs are non-Gaussian so the Gaussian... 

Another case is investigated with a very short period of measurement, namely T = 5 s, so it contains less than four fundamental periods of the oscillator. The Bayesian time-domain method is used for the identification and Figure 4.5 shows the conditional PDF of and f with all other parameters fixed at their optimal values. The solid lines show the conditional posterior PDF obtained by the Bayesian method and the dashed lines show the Gaussian approximation. It is clear that the posterior PDF is non-Gaussian. This confirms that the Bayesian time-domain approach is capable to offer the correct inference without assuming the type of the posterior PDF. In the case of a non-Gaussian posterior PDF, statistical moments, such as the variances of the estimates, can be computed by direct Monte Carlo simulation. The results are shown in Table 4.2 in the same fashion as Table 4.1. The computed uncertainty obtained here is reasonable by judging the normalized distance of the estimates. 

Even though the spectral density approach requires computation of the inverse and determinant of a number of matrices, the size of these matrices is only No y. No. They are significantly smaller than the NgNp x NgNp matrix Eyj j (= E22) required in the time-domain approach. Comparison of the computational efficiency between the two methods depends on the number of the elements in the frequency index set and the number of data points in a fundamental period. The ratio of the computations required by the Bayesian spectral density approach and the Bayesian time-domain approach can be approximated by ... 

Although the Bayesian method can be used to avoid a statistical trap in selecting a model class, the most important issue is still on the fundamental understanding of the underlying system or phenomenon so that good model class candidates can be constructed. Otherwise, satisfactory identification results can never be anticipated among the poor candidates. 

In the so-called probability of frequency approach (Kaplan Garrick 1981, Aven 2003), relative frequency-based probabilities are used to describe aleatory uncertainty and subjective probabilities to describe epistemic uncertainty. The probability of frequency approach differs fundamentally in philosophy but not much in practice fiwm a standard Bayesian approach (Aven 2003). In the Bayesian approach all uncertainty is epistemic, and probability is always considered an expression of belief it is not a property of the world in the way that a relative frequency-based probability is. The notion of aleatory uncertainty, sometimes just referred to as variation in the Bayesian approach (Aven 2003), is captured by the concept of chmce, defined as the limit of a relative frequency in an exchangeable, infinite Bernoulli series (Lind-ley 2006). A chance distribution is then the limit of... 

At the smaller scales—and this is the fundamental change—we are dealing with and manipulating large amounts of discrete stochastic, Bayesian, and Boolean information. The key word is information. In the past, we manipulated continuum descriptions, which are averages. Those were nice, but now we must manipulate discrete data—heterogeneous data. Who has been thinking about that for the last 20 or 30 years These problems form the core of computer science. 

The overall goal of Bayesian inference is knowing the posterior. The fundamental idea behind nearly all statistical methods is that as the sample size increases, the distribution of a random sample from a population approaches the distribution of the population. Thus, the distribution of the random sample from the posterior will approach the true posterior distribution. Other inferences such as point and interval estimates of the parameters can be constructed from the posterior sample. For example, if we had a random sample from the posterior, any parameter could be estimated by the corresponding statistic calculated from that random sample. We could achieve any required level of accuracy for our estimates by making sure our random sample from the posterior is large enough. Existing exploratory data analysis (EDA) techniques can be used on the sample from the posterior to explore the relationships between parameters in the posterior. 

A Bayesian system identificadmi approach provides a fundamental mathematical framework for quantifying the uncertainties and their effects on the identification results. This article describes the basics of Bayesian operational modal analysis, covering issues on formulation, efficient computations, interpretation of results, the quantification of identification uncertainties, and their management... 

In the context of Bayesian system identification, the spread of the posterior PDE is a direct fundamental quantification of the remaining uncertainty associated with the modal parameters for a given assumed identification model and in the presence of the measured data. Since the posterior PDF is typically unimodal and it can be approximated by a joint Gaussian PDF, the uncertainty of the modal parameters can be quantified by the covariance matrix, which is called the posterior covariance matrix. The posterior covariance matrix is the inverse of the Hessian matrix of the NLLF, and it can be calculated for a given set of data. Clearly it depends on the particular set of data. 

A Bayesian approach for modal identiflcaticm provides a fundamental means for processing the information contained in the data to make inference on the modal parameters consistent... 

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