Idempotent conjunctive and disjunctive combination of belief functions by distance minimization

The notions of less and more informative are often formalized through partial orderings extending usual set-inclusion, yet the only two informative partial orders that provide a straightforward idempotent rule leading to a unique result are those based on the conjunctive and disjunctive weight functions. In this article, we show that other partial orders can achieve a similar goal when the problem is slightly relaxed into a distance optimization one. Building upon previous work, this paper investigates the use of distances compatible with informative partial orders to determine a unique solution to the combination problem. The obtained operators are conjunctive/disjunctive, idempotent and commutative, but lack associativity. They are, however, quasi-associative allowing sequential combinations at no extra complexity. Some experiments demonstrate interesting discrepancies as compared to existing approaches, notably with the aforementioned rules relying on weight functions.