The previous section formally defined probability in terms of cumulative distribu- tion function and probability density function. What is the meaning of this probabil- ity? Though the concepts of probability are well-established in the literature, there is considerable disagreement among researchers on the interpretation of probability.
There are two major interpretations based on physical and subjective probabilities respectively. It is essential to understand the difference between these two interpre- tations before delving deeper into this dissertation; this is mainly because the latter philosophy is widely used in this research work. In fact, this dissertation advocates the
latter philosophy because of its ability to integrate the various sources of uncertainty across multiple levels of models that represent the overall engineering system.
2.3.1 Physical Probability
Physical probabilities [4], also referred to objective or frequentist probabilities, are related to random physical systems such as rolling dice, tossing coins, roulette wheels, etc. Each trial of the experiment leads to an event (which is a subset of the sample space), and in the long run of repeated trials, each event tends to occur at a persistent rate, and this rate is referred to as the “relative frequency”. These relative frequencies are expressed and explained in terms of physical probabilities. Thus, physical probabilities are defined only in the context of random experiments. The theory of classical statistics is based on physical probabilities. Within the realm of physical probabilities, there are two types of interpretations: von Mises’ frequentist [5]
and Popper’s propensity [6]; the former is more easily understood and widely used.
In the context of physical probabilities, the mean of a random variable, sometimes referred to as the population mean, is deterministic. It is meaningless to talk about the PDF of this mean. In fact, for any type of parameter estimation, the underlying parameter is assumed to be deterministic and only an estimate of this parameter is obtained. The uncertainty in the parameter estimate is addressed through confi- dence intervals. The interpretation of confidence intervals is, at times, confusing and misleading, and the uncertainty in the parameter estimate cannot be used for further uncertainty quantification. For example, if the uncertainty in the elastic modulus was estimated using a simple axial test, this uncertainty cannot be used for quantifying the response in a plate made of the same material. This is a serious limitation, since it is not possible to propagate uncertainty after parameter estimation, which is often necessary in the case of model-based quantification of uncertainty in the system-level
response. Another disadvantage of this approach is that, when a quantity is not ran- dom, but unknown, then the tools of probability cannot be used to represent this type of uncertainty (epistemic). The second interpretation of probability, i.e. the subjective interpretation, overcomes these limitations.
2.3.2 Subjective Probability
Subjective probabilities [7] can be assigned to any “statement”. It is not necessary that the concerned statement is in regard to an event which is a possible outcome of a random experiment. In fact, subjective probabilities can be assigned even in the absence of random experiments. The Bayesian methodology is based on subjective probabilities, which are simply considered to be degrees of belief and quantify the extent to which the “statement” is supported by existing knowledge and available evidence. Calvetti and Somersalo [8] explain that “randomness” in the context of physical probabilities is equivalent to “lack of information” in the context of subjective probabilities.
In this approach, even deterministic quantities can be represented using probabil- ity distributions which reflect the subjective degree of the analyst’s belief regarding such quantities. As a result, probability distributions can be assigned to parameters that need to be estimated, and therefore, this interpretation facilitates uncertainty propagation after parameter estimation; this is helpful for uncertainty integration across multiple models.
For example, consider the case where a variable is assumed to be normally dis- tributed and it is desired to estimate the mean and the standard deviation based on available point data. If sufficient data were available, then it is possible to uniquely estimate these distribution parameters. However, in some cases, data may be sparse and therefore, it may be necessary to quantify the uncertainty in these distribution
parameters. Note that this uncertainty is an example of epistemic uncertainty; the quantities may be estimated deterministically with enough data. The former philos- ophy based on physical probabilities inherently assumes that these distribution pa- rameters are deterministic and expresses the uncertainty through confidence intervals on mean and standard deviation. It is not possible to propagate this description of uncertainty through a mathematical model. On the other hand, the Bayesian method- ology can calculate probability distributions for the distribution parameters, which can be easily used in uncertainty propagation. Therefore, the Bayesian methodology provides a framework in which epistemic uncertainty can be also addressed using probability theory, in contrast with the frequentist approach.
The fundamentals of Bayesian philosophy are well-established in several text- books [9–12], and the Bayesian approach is being increasingly applied to engineering problems in recent times, especially to solve statistical inverse problems. In this dissertation, the Bayesian methodology is extensively used to integrate not only the different types and sources of uncertainty, but also to integrate multiple models which represent the overall system under study.