Bayesian decision making with soft probabilities under statistical regularity

Yuan Zou; Yuan Zou

doi:10.3934/math.2026447

AIMS Mathematics

2026, Volume 11, Issue 4: 10883-10907. doi: 10.3934/math.2026447

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Bayesian decision making with soft probabilities under statistical regularity

Yuan Zou ^{1,2
,
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1.
School of Economics, Chongqing Technology and Business University, Chongqing 400067, China
2.
Chongqing Fintech Institute, Chongqing 401300, China

Received: 05 February 2026 Revised: 27 March 2026 Accepted: 07 April 2026 Published: 20 April 2026
MSC : Primary 62C10; Secondary 60A10

Soft set theory, originated by Molodtsov in 1999, is a general mathematical technique for modeling uncertainties via parameterization tools. A fundamental branch of this theory, soft probability, serves as an immediate measurement on a statistical base that takes a parameterized family of subintervals of the unit interval as its value. It can handle any stochastic events, including non-stable ones. To ensure soft probabilities accurately capture the statistical characteristics of stochastic information, the minimum sample size for their construction is determined using two statistical regularity hypotheses. Moreover, soft dependency guides attribute selection to enhance decision robustness against parameter perturbations. Hence, a novel Bayesian decision-making model integrating soft probabilities under statistical regularity hypotheses is proposed. Key steps of the method include: First, verifying these hypotheses on the initial database to determine the minimum sample size for soft probability construction; second, designing a soft dependency-based attribute selection procedure; third, calculating each decision alternative's soft posterior risk and identifying the optimal one by comparing interval-valued possibility degrees; and finally a medical diagnosis case study, combined with critical parameter sensitivity analysis and comparisons with Naive Bayes, logistic regressions (without regularization, L1 regularization and L2 regularization), decision tree, and support vector machine, demonstrates the method's feasibility and effectiveness.
- soft probabilities,
- statistical regularity,
- soft dependency,
- Bayes' rule,
- decision making
Citation: Yuan Zou. Bayesian decision making with soft probabilities under statistical regularity[J]. AIMS Mathematics, 2026, 11(4): 10883-10907. doi: 10.3934/math.2026447

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Abstract

Soft set theory, originated by Molodtsov in 1999, is a general mathematical technique for modeling uncertainties via parameterization tools. A fundamental branch of this theory, soft probability, serves as an immediate measurement on a statistical base that takes a parameterized family of subintervals of the unit interval as its value. It can handle any stochastic events, including non-stable ones. To ensure soft probabilities accurately capture the statistical characteristics of stochastic information, the minimum sample size for their construction is determined using two statistical regularity hypotheses. Moreover, soft dependency guides attribute selection to enhance decision robustness against parameter perturbations. Hence, a novel Bayesian decision-making model integrating soft probabilities under statistical regularity hypotheses is proposed. Key steps of the method include: First, verifying these hypotheses on the initial database to determine the minimum sample size for soft probability construction; second, designing a soft dependency-based attribute selection procedure; third, calculating each decision alternative's soft posterior risk and identifying the optimal one by comparing interval-valued possibility degrees; and finally a medical diagnosis case study, combined with critical parameter sensitivity analysis and comparisons with Naive Bayes, logistic regressions (without regularization, L1 regularization and L2 regularization), decision tree, and support vector machine, demonstrates the method's feasibility and effectiveness.

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