A sequential multi-prior integration and updating method for complex multi-level system based on Bayesian melding method

In engineering, there exist multiple priors about system and subsystems uncertainties, which should be integrated properly to analyze the system reliability. In the past research, an iterative updating procedure based on Bayesian Melding Method (I-BMM) was developed to merge and update multiple priors for the double-level system. However, the in-depth study in this paper shows that the original iterative procedure has no effect on the prior updating. Thus it is proposed that only a single BMM iteration process is needed following the original prior integration and updating formulation. BMM involves the sampling procedure for the probability density function (PDF) updating, wherein it is generally difficult to define the sampling number properly for obtaining accurate priors. To address this problem, a sequential prior integration and updating framework based on the original single BMM iteration process (S-BMM) is developed in this paper. In each cycle of prior updating, the sample number is sequentially added, and the difference between prior distributions obtained in the two consecutive cycles is measured with the symmetric Kullback-Leibler Divergence (SKLD). The sequential procedure is continued until the convergence to the accurate updated prior. The S-BMM framework for double-level systems is further extended for multi-level systems. Situations with some missing subsystem or component priors are also discussed. Finally, two numerical examples and one satellite engineering case are used to demonstrate and verify the proposed algorithms.

Discrete Arrow–Pratt indexes for risk and uncertainty

The Arrow–Pratt index, a gold standard in studies of risk attitudes, is not directly observable from choice data. Existing methods to measure it rely on parametric assumptions. We introduce a discrete Arrow–Pratt index, and its relative counterpart, that can be directly obtained from choices. Our approach is general: it is (i) non-parametric, (ii) applicable to both risk and uncertainty, (iii) and robust to probability transformation, non-additive beliefs and multiple priors. Our index can also be used to characterize various decision models through various simple consistency requirements. We analyze its properties and demonstrate how it can be measured.

Ambiguity and Awareness: A Coherent Multiple Priors Model

Ambiguity in the ordinary language sense means that available information is open to multiple interpretations. We model this by assuming that individuals are unaware of some possibilities relevant to the outcome of their decisions and that multiple probabilities may arise over an individual’s subjective state space depending on which of these possibilities are realized. We formalize a notion of coherent multiple priors and derive a representation result that with full awareness corresponds to the usual unique (Bayesian) prior but with less than full awareness generates multiple priors. When information is received with no change in awareness, each element of the set of priors is updated in the standard Bayesian fashion (that is, full Bayesian updating). An increase in awareness, however, leads to an expansion of the individual’s subjective state and (in general) a contraction in the set of priors under consideration.