a new paper: Uncertainty, Imprecise Probabilities and Interval Capacity Measures on a Product Space

[MARCELLO BASILI]

Dear Friends.

Luca Pratelli and I have a new paper about Interval capacity measures on a product space.

We write a first paper on a new approach to imprecise probabilities and set out a representation through interval measures based on the new notion of weak complementation. Unlike other approaches, we establish uncertainty in both a distributive algebra that is only weakly complemented and, equivalently, in a non-distributive complemented algebra (do you remember Quantum Mechanics?). Importantly, these two approaches lead to the same notion of interval probability measure for H, a given event, since it is defined in relation to its set of indecisive eventualities. H, the set of indecisive and not incompatible eventualities and the weak complement or incompatible eventualities of H  form Ω the generic non-empty set we consider.

In that paper we represent the problem of the barometer and umbrella (Keynes 1921, 28) as a situation where there are four relevant causes for raining:
     i) barometer low and no black clouds in the sky, denoted as eventuality ω₀₁;
    ii) barometer high, many black clouds in the sky, denoted as eventuality ω₁₀;
    iii) barometer low, many black clouds in the sky, denoted as eventuality ω₀₀;
    iv) barometer high, no black clouds in the sky, denoted as eventuality ω₁₁.

Such a dilemma is one of the most famous puzzle in economics.

We establish the partition Z=Z₀,Z₁ with Z₀={ω{0,1},ω{1,0}}, Z₁={ω{0,0},ω{1,1}} to describe the weak complementation and, consequently, uncertain events. In particular, the Law of the Exclude Middle does not held because the (weak) complementary event of {ω₁₀} is {ω₀₁} and Z₁ represents the set of indecisive eventualities of {ω₁₀}.

In such a case interval problaity measures depend on the width of the indecisive set, that can be assumed to rely on relaibility of barometer and glance.

In the new paper we elaborate on fundamental concepts such as imprecise probabilities, interval capacity measures and conditional interval probability measures. We illustrate the notion of stochastic dominance between two random variables based on interval probability measures and exhibit an example of the product of interval probability measures. As an example you can consider two agents in the barometer's dilemma.

Your comments and suggestions are very welcome.

Regards

Marcello