Fuzzy-Probabilistic Inference System; Weighted Quantile Regression; Inference Mechanism; Linear Combination; Wasserstein Barycenter

This work studies the inference mechanism of fuzzy--probabilistic inference systems (FPIS), a class of rule-based models where antecedents encode fuzzy information and consequents represent conditional probability distributions of the output variable. A system of m rules is considered: if the input belongs to a fuzzy set $A_k$, then the output follows a probability distribution described by an empirical quantile function. The antecedents form a covering fuzzy partition of the universe, ensuring that every input has positive membership in at least one fuzzy set. In practice, uniform or generalized partitions are typically employed. Local quantile functions are estimated from data as weighted quantiles, with weights given by membership degrees. The inference mechanism produces an empirical quantile function for any input as a linear combination of these local quantile functions, using normalized membership weights. Fuzzy rule-based systems capture input-output relationships in a rough manner, while the inference mechanism refines this into a complete mapping usable in practice. Previous studies compared the standard weighted average of quantile functions with several alternatives on synthetic and real datasets. However, a theoretical analysis of these mechanisms, including the original weighted average and related $L_1$-based minimization approaches, remains open. This gap motivates a deeper investigation of the foundations of the inference mechanism for FPIS.

This work studies the inference mechanism of fuzzy--probabilistic inference systems (FPIS), a class of rule-based models where antecedents encode fuzzy information and consequents represent conditional probability distributions of the output variable. A system of m rules is considered: if the input belongs to a fuzzy set $A_k$, then the output follows a probability distribution described by an empirical quantile function. The antecedents form a covering fuzzy partition of the universe, ensuring that every input has positive membership in at least one fuzzy set. In practice, uniform or generalized partitions are typically employed.Local quantile functions are estimated from data as weighted quantiles, with weights given by membership degrees. The inference mechanism produces an empirical quantile function for any input as a linear combination of these local quantile functions, using normalized membership weights.Fuzzy rule-based systems capture input-output relationships in a rough manner, while the inference mechanism refines this into a complete mapping usable in practice. Previous studies compared the standard weighted average of quantile functions with several alternatives on synthetic and real datasets. However, a theoretical analysis of these mechanisms, including the original weighted average and related $L_1$-based minimization approaches, remains open. This gap motivates a deeper investigation of the foundations of the inference mechanism for FPIS.