Deterministic versus Probabilistic Methods

In most deterministic identification methods, important observable quantities are measured (e.g., modal frequencies of a building for stiffness identification) and the uncertain model parameters are obtained by minimizing a goodness-of-fit/error function of these measurements 

Probabilistic approaches, particularly the Bayesian approach, utilize the complete information of the data for the statistical inference if the appropriate likelihood function is constructed whereas deterministic methods usually bypass this issue. As a result, probabilistic methods allow for the quantification of the uncertainty of the parametric estimation. Furthermore, using the appropriate probability distribution ensures the correctness of the optimal parameters. Although Bayesian inference is attractive for allowing direct quantification of the uncertainty of the parameter estimation, there are main challenges in developing Bayesian methods  

In some applications, it is difficult to obtain the likelihood function with an appropriate choice of the type of probability distribution. This is not a trivial task since the probability distribution of the random variables in establishing the likelihood function may be complicated. For example, consider a random process x and its measurements at different time steps with equal spacing xi,X2. x. The auto-correlation function can be estimated by  

In the probabilistic approach, the solution is not simply the optimal parameters but also the probability density function that describes the complete picture of the uncertainty. It is a challenging task to demonstrate the representation of the updated PDF since it has 

Bayesian Methods for Structural Dynamics and Civil Engineering 

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Any analysis of risk should recognize these distinctions in all of their essential features. A typical approach to acute risk separates the stochastic nature of discrete causal events from the deterministic consequences which are treated using engineering methods such as mathematical models. Another tool if risk analysis is a risk profile that graphs the probability of occurrence versus the severity of the consequences (e.g., probability, of a fish dying or probability of a person contracting liver cancer either as a result of exposure to a specified environmental contaminant). In a way, this profile shows the functional relationship between the probabilistic and the deterministic parts of the problem by showing probability versus consequences. 

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