Bayesian decision making under soft probabilities

Abstract

Bayesian decision models use probability theory as a commonly technique to handling uncertainty and arise in a variety of important practical applications for estimation and prediction as well as offering decision support. But the deficiencies mainly manifest in the two aspects: First, it is often difficult to avoid subjective judgment in the process of quantization of priori probabilities. Second, applying point-valued probabilities in Bayesian decision making is insufficient to capture non-stochastically stable information. Soft set theory as an emerging mathematical tool for dealing with uncertainty has yielded fruitful results. One of the key concepts involved in the theory named soft probability which is as an immediate measurement over a statistical base can be capable of dealing with various types of stochastic phenomena including not stochastically stable phenomena, has been recently introduced to represent statistical characteristics of a given sample in a more natural and direct manner. Motivated by the work, this paper proposes a hybrid methodology that integrates soft probability and Bayesian decision theory to provide decision support when stochastically stable samples and exact values of probabilities are not available. According to the fact that soft probability is as a special case of interval probability which is mathematically proved in the paper, thus the proposed methodology is thereby consistent with Bayesian decision model with interval probability. In order to demonstrate the proof of concept, the proposed methodology has been applied to a numerical case study regarding medical diagnosis.