A decision-based approach to establish non-informative prior sample space for decision-making in geotechnical applications

Bayes’ theorem is widely adopted for risk-informed decision-making in natural hazards (which often have limited data), but the prior sample space based on the existing methods may lead to inconsistent, irrational, and not defensible results. Therefore, Decision Entropy Theory (DET) is under development to improve the assessment of small probabilities when limited information is available for non-informative prior sample space in assisting Bayesian decision-making. The key idea to establish a non-informative prior sample space with DET is that the value of new information is as uncertain as possible, or the entropy of the new information is maximized. The mathematical formulation includes prior decision analysis by maximizing the relative entropy of the value of perfect information and pre-posterior decision analysis by maximizing the relative entropy of the value of imperfect information given each value of perfect information. The goal of this research includes (1) apply the theory to simple problems to demonstrate and study its rigorous implementation, evaluate possible approximations to reduce the computational effort required to implement it rigorously, and develop insight into the results; (2) propose and characterize the likelihood functions to represent subjective judgment for small-probability events in the decision analysis; and (3) demonstrate the application of the theory to real-world cases histories. From this research, the following conclusions are drawn: (1) results of illustrative decision analysis examples show that the non-informative prior probabilities obtained from DET are sensical and address concerns that have been raised about other approaches to establish non-informative prior probabilities that do not consider their impact on decision making; moreover, the DET-based non-informative prior is invariant to transformations of uncertain variables as it depends on the decisions rather than how the states of nature are defined; (2) an approximation to the rigorous DET reduces the computation effort considerably (many orders of magnitude), provides reasonable results for the prior decision and value of perfect information, but is less able to approximate the value of imperfect information; (3) likelihood functions proposed for fractional occurrence models with the Binomial distribution, Poisson distribution, and Multinomial distribution have a maximum at the estimated fraction of occurrences and a Fisher information quantity that is inversely proportional to the estimated fraction and proportional to the length of the record used to estimate the fraction; and (4) the non-informative prior probabilities obtained with DET for the dam case history provide useful insight into the potential impacts of not making assumptions beyond what is actually known. When uncertainty in frequencies of overtopping and the chance of dam failure given overtopping (fragility) are included, the decision to rehabilitate the dam is justified with a cost of dam breach that is more than 100 times smaller than when this uncertainty is neglected and more than 10 times smaller than when uncertainty in the hazard but not the fragility is neglected. In addition, the maximum value of obtaining additional information about frequencies of hazard and fragility is 35% of the cost of rehabilitation. The theory will be advanced in the future by developing more efficient algorithms that optimize the time complexity and space complexity for the numerical implementation of DET and applying it to more complicated and realistic problems.