Scenario-based risk evaluation

Risk measures such as expected shortfall (ES) and value-at-risk (VaR) have been prominent in banking regulation and financial risk management. Motivated by practical considerations in the assessment and management of risks, including tractability, scenario relevance and robustness, we consider theoretical properties of scenario-based risk evaluation. We establish axiomatic characterisations of scenario-based risk measures that are comonotonic-additive or coherent, and we obtain a novel ES-based representation result. We propose several novel scenario-based risk measures, including various versions of Max-ES and Max-VaR, and study their properties. The theory is illustrated with financial data examples.

The Universal Approximation Property

The universal approximation property of various machine learning models is currently only understood on a case-by-case basis, limiting the rapid development of new theoretically justified neural network architectures and blurring our understanding of our current models’ potential. This paper works towards overcoming these challenges by presenting a characterization, a representation, a construction method, and an existence result, each of which applies to any universal approximator on most function spaces of practical interest. Our characterization result is used to describe which activation functions allow the feed-forward architecture to maintain its universal approximation capabilities when multiple constraints are imposed on its final layers and its remaining layers are only sparsely connected. These include a rescaled and shifted Leaky ReLU activation function but not the ReLU activation function. Our construction and representation result is used to exhibit a simple modification of the feed-forward architecture, which can approximate any continuous function with non-pathological growth, uniformly on the entire Euclidean input space. This improves the known capabilities of the feed-forward architecture.

Locally finite complexes, modules and generalized information systems

Simplicial complexes (here briefly complexes) are set systems on an arbitrary set which are object of study in many areas of both mathematics and theoretical computer science. Usually, they are investigated over finite sets. However, in general, when we consider an arbitrary set [Formula: see text] (not necessarily finite) and a complex [Formula: see text] on [Formula: see text], the most natural property related to finiteness is the following: for any subset [Formula: see text] of [Formula: see text], if [Formula: see text] for all finite subsets [Formula: see text] of [Formula: see text], then [Formula: see text]. We call locally finite any complex [Formula: see text] having such a property. Bearing in mind some motivations and constructions derived from the analysis of information systems in rough set theory, in this paper we associate with any locally finite complex [Formula: see text] a corresponding pre-closure operator [Formula: see text] and, through it, we establish several properties of [Formula: see text]. Next, we investigate the main features of the specific sub-class of locally finite complexes [Formula: see text] for which [Formula: see text] is a closure operator. We call these complexes closable and exhibit a particular family of closable locally finite complexes using left-modules on rings with identity. Finally, we establish a representation result according to which we can associate a pairing structure with any closable locally finite complex.

Ambiguity and Awareness: A Coherent Multiple Priors Model

Ambiguity in the ordinary language sense means that available information is open to multiple interpretations. We model this by assuming that individuals are unaware of some possibilities relevant to the outcome of their decisions and that multiple probabilities may arise over an individual’s subjective state space depending on which of these possibilities are realized. We formalize a notion of coherent multiple priors and derive a representation result that with full awareness corresponds to the usual unique (Bayesian) prior but with less than full awareness generates multiple priors. When information is received with no change in awareness, each element of the set of priors is updated in the standard Bayesian fashion (that is, full Bayesian updating). An increase in awareness, however, leads to an expansion of the individual’s subjective state and (in general) a contraction in the set of priors under consideration.

Fitzpatrick transform of monotone relations in Hadamard spaces

In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.

Two Sides of the Same Coin: Belief Revision and Enforcing Arguments

We study a type of change on knowledge bases inspired by the dynamics of formal argumentation systems, where the goal is to enforce acceptance of certain arguments. We put forward that enforcing acceptance of arguments can be viewed as a member of the wider family of belief change operations, and that an axiomatic treatment of it is therefore desirable. In our case, laying down axioms enables a precise account of the close connection between enforcing arguments and belief revision. Our analysis of enforcing arguments proceeds by (i) axiomatizing it as an operation in propositional logic and providing a representation result in terms of rankings on sets of interpretations, (ii) showing that it stands in close relationship to belief revision, and (iii) using it as a gateway towards a principled treatment of enforcement in abstract argumentation.

Belief Update in the Horn Fragment

In line with recent work on belief change in fragments of propositional logic, we study belief update in the Horn fragment. We start from the standard KM postulates used to axiomatize belief update operators; these postulates lend themselves to semantic characterizations in terms of partial (resp. total) preorders on possible worlds. Since the Horn fragment is not closed under disjunction, the standard postulates have to be adapted for the Horn fragment. Moreover, a restriction on the preorders (i.e., Horn compliance) and additional postulates are needed to obtain sensible characterizations for the Horn fragment, and this leads to our main contribution: a representation result which shows that the class of update operators captured by Horn compliant partial (resp. total) preorders over possible worlds is precisely that given by the adapted and augmented Horn update postulates. With these results at hand, we provide concrete Horn update operators and are able to shed light on Horn revision operators based on partial preorders.

Trust as a Precursor to Belief Revision

Belief revision is concerned with incorporating new information into a pre-existing set of beliefs. When the new information comes from another agent, we must first determine if that agent should be trusted. In this paper, we define trust as a pre-processing step before revision. We emphasize that trust in an agent is often restricted to a particular domain of expertise. We demonstrate that this form of trust can be captured by associating a state partition with each agent, then relativizing all reports to this partition before revising. We position the resulting family of trust-sensitive revision operators within the class of selective revision operators of Ferme and Hansson, and we prove a representation result that characterizes the class of trust-sensitive revision operators in terms of a set of postulates. We also show that trust-sensitive revision is manipulable, in the sense that agents can sometimes have incentive to pass on misleading information.