Updating Uncertain Evidence - Conditionalization and Complexity in Multi-Layer Belief Models
Giuseppe Manes

Abstract:
In (Pinto Prieto, 2024) it is shown how Dempster-Shafer theory can be combined with Topological models of evidence into a unified Multi-Layer Belief Model that efficiently handles the combination of uncertain, partial and mutually contradictory pieces of evidence. More precisely, the topological structure captures specific kinds of doxastic attitudes towards evidence, while a belief function reconstructed on top of this structure represents the uncertainty of the evidence. This model has been shown to achieve results analogous to both Dempster-Shafer theory and Topological models of evidence, while being strictly more powerful, as it subsumes the essential properties of both frameworks. However, as originally defined, the model does not include any updating procedure for conditionalization, i.e. a way for revising the model after learning new certain evidence. This is the problem we address in this work.
Both in Dempster-Shafer theory (Shafer, 1976) and in Topological models of evidence (Özgün, 2017) there exist different ways to update beliefs. Drawing on the intuitions behind those approaches, we define an updating procedure for the Multi-Layer Belief Model that captures conditionalization of evidence while preserving the model’s expressivity and its ability to recover the foundational frameworks, now also at the dynamic level.
We approach the problem from two main dimensions. First epistemically, we investigate how the update respects the agent’s original evidential standards and how it relates to established notions of conditional belief. Then computationally, we analyze the complexity of the update algorithms and provide a concrete implementation. Thus, the Multi-Layer Belief Model, extended with conditionalization, offers a unified framework for reasoning with uncertain, partial, and possibly inconsistent evidence under dynamic information.
More concretely, in Chapter 1 we first establish the basic concepts underlying the frameworks we consider and analyze how they model an agent’s doxastic demands. Then, in Chapter 2, we define how to update each layer of the model upon learning a proposition Φ ⊆ X . For the qualitative layer, we restricted its elements via a public announcement-like operation, and the frame of justification is recomputed on the latter (Section 2.2). At the level tof the quantitative layer the update is performed using a novel merging function δ' that combines the weights of evidence pieces sharing the same intersection with Φ (Section 2.3). Finally, the bridging layer recomputes the evidence allocation function and the final belief on the updated structure (Section 2.5). Furthermore, we show that the procedure satisfies four natural desiderata: restriction of the state space, renormalization of the mass function, preservation of the agent’s doxastic standards, and iterability (Section 2.6). All the updating definitions have been encoded in explicit polynomial-time algorithms, for which we proved correctness, and we provide an implementation available in a public repository (Manes, 2026).
Finally, in Chapter 3 we relate our conditionalization to the conditioning methods of Dempster-Shafer theory and Topological models of evidence. Under suitable choices of the frame of justification and the evidence allocation function, our procedure recovers Dempster-Shafer conditioning via Dempster’s rule of combination, both at the level of the evidence set and at the level of the final basic probability assignment (Section 3.1), and replicates the belief dynamics of public announcement and conditional belief in topological models (Section 3.2).