# A method to establish non-informative prior probabilities for risk-based decision analysis

2008-08

## Abstract

In Bayesian decision analysis, uncertainty and risk are accounted for with probabilities for the possible states, or states of nature, that affect the outcome of a decision. Application of Bayes’ theorem requires non-informative prior probabilities, which represent the probabilities of states of nature for a decision maker under complete ignorance. These prior probabilities are then subsequently updated with any and all available information in assessing probabilities for making decisions. The conventional approach for the non-informative probability distribution is based on Bernoulli’s principle of insufficient reason. This principle assigns a uniform distribution to uncertain states when a decision maker has no information about the states of nature. The principle of insufficient reason has three difficulties: it may inadvertently provide a biased starting point for decision making, it does not provide a consistent set of probabilities, and it violates reasonable axioms of decision theory. The first objective of this study is to propose and describe a new method to establish non-informative prior probabilities for decision making under uncertainty. The proposed decision-based method is focuses on decision outcomes that include preference in decision alternatives and decision consequences. The second objective is to evaluate the logic and rationality basis of the proposed decision-based method. The decision-based method overcomes the three weaknesses associated with the principle of insufficient reason, and provides an unbiased starting point for decision making. It also produces consistent non-informative probabilities. Finally, the decision-based method satisfies axioms of decision theory that characterize the case of no information (or complete ignorance). The third and final objective is to demonstrate the application of the decision-based method to practical decision making problems in engineering. Four major practical implications are illustrated and discussed with these examples. First, the method is practical because it is feasible in decisions with a large number of decision alternatives and states of nature and it is applicable to both continuous and discrete random variables of finite and infinite ranges. Second, the method provides an objective way to establish non-informative prior probabilities that capture a highly nonlinear relationship between states of nature. Third, we can include any available information through Bayes’ theorem by updating the non-informative probabilities without the need to assume more than is actually contained in the information. Lastly, two different decision making problems with the same states of nature may have different non-informative probabilities.

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