Credal networks under epistemic irrelevance: The sets of desirable gambles approach

• We study credal networks under epistemic irrelevance for sets of desirable gambles.
• We characterise the so-called irrelevant natural extension of these networks.
• We obtain marginalisation properties for this irrelevant natural extension.
• We introduce an asymmetrical separation criterion, called AD-separation.
• We show that AD-separation implies epistemic irrelevance in the irr. nat. ext.

We present a new approach to credal networks, which are graphical models that generalise Bayesian networks to deal with imprecise probabilities. Instead of applying the commonly used notion of strong independence, we replace it by the weaker, asymmetrical notion of epistemic irrelevance. We show how assessments of epistemic irrelevance allow us to construct a global model out of given local uncertainty models, leading to an intuitive expression for the so-called irrelevant natural extension of a credal network. In contrast with Cozman [4], who introduced this notion in terms of credal sets, our main results are presented using the language of sets of desirable gambles. This has allowed us to derive some remarkable properties of the irrelevant natural extension, including marginalisation properties and a tight connection with the notion of independent natural extension. Our perhaps most important result is that the irrelevant natural extension satisfies a collection of epistemic irrelevancies that is induced by AD-separation, an asymmetrical adaptation of d-separation. Both AD-separation and the induced collection of irrelevancies are shown to satisfy all graphoid properties except symmetry.