# Belief Revision

### Accuracy-Memory Tradeoffs and Phase Transitions in Belief Propagation

The analysis of Belief Propagation and other algorithms for the {\em reconstruction problem} plays a key role in the analysis of community detection in inference on graphs, phylogenetic reconstruction in bioinformatics, and the cavity method in statistical physics. We prove a conjecture of Evans, Kenyon, Peres, and Schulman (2000) which states that any bounded memory message passing algorithm is statistically much weaker than Belief Propagation for the reconstruction problem. More formally, any recursive algorithm with bounded memory for the reconstruction problem on the trees with the binary symmetric channel has a phase transition strictly below the Belief Propagation threshold, also known as the Kesten-Stigum bound. The proof combines in novel fashion tools from recursive reconstruction, information theory, and optimal transport, and also establishes an asymptotic normality result for BP and other message-passing algorithms near the critical threshold.

### Decrement Operators in Belief Change

While research on iterated revision is predominant in the field of iterated belief change, the class of iterated contraction operators received more attention in recent years. In this article, we examine a non-prioritized generalisation of iterated contraction. In particular, the class of weak decrement operators is introduced, which are operators that by multiple steps achieve the same as a contraction. Inspired by Darwiche and Pearl's work on iterated revision the subclass of decrement operators is defined. For both, decrement and weak decrement operators, postulates are presented and for each of them a representation theorem in the framework of total preorders is given. Furthermore, we present two types of decrement operators which have a unique representative.

### Markov versus quantum dynamic models of belief change during evidence monitoring

Two different dynamic models for belief change during evidence monitoring were evaluated: Markov and quantum. They were empirically tested with an experiment in which participants monitored evidence for an initial period of time, made a probability rating, then monitored more evidence, before making a second rating. The models were qualitatively tested by manipulating the time intervals in a manner that provided a test for interference effects of the first rating on the second. The Markov model predicted no interference whereas the quantum model predicted interference. A quantitative comparison of the two models was also carried out using a generalization criterion method: the parameters were fit to data from one set of time intervals, and then these same parameters were used to predict data from another set of time intervals. The results indicated that some features of both Markov and quantum models are needed to accurately account for the results.

### Probabilistic Belief Revision with Structural Constraints

Experts (human or computer) are often required to assess the probability of uncertain events. When a collection of experts independently assess events that are structurally interrelated, the resulting assessment may violate fundamental laws of probability. Such an assessment is termed incoherent. In this work we investigate how the problem of incoherence may be affected by allowing experts to specify likelihood models and then update their assessments based on the realization of a globally-observable random sequence. Papers published at the Neural Information Processing Systems Conference.

### On Convergence Rate of the Gaussian Belief Propagation Algorithm for Markov Networks

Gaussian Belief Propagation (BP) algorithm is one of the most important distributed algorithms in signal processing and statistical learning involving Markov networks. It is well known that the algorithm correctly computes marginal density functions from a high dimensional joint density function over a Markov network in a finite number of iterations when the underlying Gaussian graph is acyclic. It is also known more recently that the algorithm produces correct marginal means asymptotically for cyclic Gaussian graphs under the condition of walk summability. This paper extends this convergence result further by showing that the convergence is exponential under the walk summability condition, and provides a simple bound for the convergence rate.

### Iterated Belief Base Revision: A Dynamic Epistemic Logic Approach

AGM's belief revision is one of the main paradigms in the study of belief change operations. In this context, belief bases (prioritised bases) have been largely used to specify the agent's belief state - whether representing the agent's `explicit beliefs' or as a computational model for her belief state. While the connection of iterated AGM-like operations and their encoding in dynamic epistemic logics have been studied before, few works considered how well-known postulates from iterated belief revision theory can be characterised by means of belief bases and their counterpart in a dynamic epistemic logic. This work investigates how priority graphs, a syntactic representation of preference relations deeply connected to prioritised bases, can be used to characterise belief change operators, focusing on well-known postulates of Iterated Belief Change. We provide syntactic representations of belief change operators in a dynamic context, as well as new negative results regarding the possibility of representing an iterated belief revision operation using transformations on priority graphs.

### A Generalisation of AGM Contraction and Revision to Fragments of First-Order Logic

AGM contraction and revision assume an underlying logic that contains propositional logic. Consequently, this assumption excludes many useful logics such as the Horn fragment of propositional logic and most description logics. Our goal in this paper is to generalise AGM contraction and revision to (near-)arbitrary fragments of classical first-order logic. To this end, we first define a very general logic that captures these fragments. In so doing, we make the modest assumptions that a logic contains conjunction and that information is expressed by closed formulas or sentences. The resulting logic is called first-order conjunctive logic or FC logic for short. We then take as the point of departure the AGM approach of constructing contraction functions through epistemic entrenchment, that is the entrenchment-based contraction. We redefine entrenchment-based contraction in ways that apply to any FC logic, which we call FC contraction. We prove a representation theorem showing its compliance with all the AGM contraction postulates except for the controversial recovery postulate. We also give methods for constructing revision functions through epistemic entrenchment which we call FC revision; which also apply to any FC logic. We show that if the underlying FC logic contains tautologies then FC revision complies with all the AGM revision postulates. Finally, in the context of FC logic, we provide three methods for generating revision functions via a variant of the Levi Identity, which we call contraction, withdrawal and cut generated revision, and explore the notion of revision equivalence. We show that withdrawal and cut generated revision coincide with FC revision and so does contraction generated revision under a finiteness condition.

### Iterated Belief Revision Under Resource Constraints: Logic as Geometry

We propose a variant of iterated belief revision designed for settings with limited computational resources, such as mobile autonomous robots. The proposed memory architecture---called the {\em universal memory architecture} (UMA)---maintains an epistemic state in the form of a system of default rules similar to those studied by Pearl and by Goldszmidt and Pearl (systems $Z$ and $Z^+$). A duality between the category of UMA representations and the category of the corresponding model spaces, extending the Sageev-Roller duality between discrete poc sets and discrete median algebras provides a two-way dictionary from inference to geometry, leading to immense savings in computation, at a cost in the quality of representation that can be quantified in terms of topological invariants. Moreover, the same framework naturally enables comparisons between different model spaces, making it possible to analyze the deficiencies of one model space in comparison to others. This paper develops the formalism underlying UMA, analyzes the complexity of maintenance and inference operations in UMA, and presents some learning guarantees for different UMA-based learners. Finally, we present simulation results to illustrate the viability of the approach, and close with a discussion of the strengths, weaknesses, and potential development of UMA-based learners.

### Adams Conditioning and Likelihood Ratio Transfer Mediated Inference

Bayesian inference as applied in a legal setting is about belief transfer and involves a plurality of agents and communication protocols. A forensic expert (FE) may communicate to a trier of fact (TOF) first its value of a certain likelihood ratio with respect to FE's belief state as represented by a probability function on FE's proposition space. Subsequently FE communicates its recently acquired confirmation that a certain evidence proposition is true. Then TOF performs likelihood ratio transfer mediated reasoning thereby revising their own belief state. The logical principles involved in likelihood transfer mediated reasoning are discussed in a setting where probabilistic arithmetic is done within a meadow, and with Adams conditioning placed in a central role.

### Rethinking Epistemic Logic with Belief Bases

We introduce a new semantics for a logic of explicit and implicit beliefs based on the concept of multi-agent belief base. Differently from existing Kripke-style semantics for epistemic logic in which the notions of possible world and doxastic/epistemic alternative are primitive, in our semantics they are non-primitive but are defined from the concept of belief base. We provide a complete axiomatization and prove decidability for our logic via a finite model argument. We also provide a polynomial embedding of our logic into Fagin & Halpern's logic of general awareness and establish a complexity result for our logic via the embedding.