The stochastic block model (SBM) has long been studied in machine learning and network science as a canonical model for clustering and community detection. In the recent years, new developments have demonstrated the presence of threshold phenomena for this model, which have set new challenges for algorithms. This was proved for two communities, but remained open from three communities. We prove this conjecture here, obtaining a more general result that applies to arbitrary SBMs with linear size communities. The developed algorithm is a linearized acyclic belief propagation (ABP) algorithm, which mitigates the effects of cycles while provably achieving the KS threshold in $O(n \ln n)$ time.

We start with the distinction of outcome- and belief-based Bayesian models of the sequential update of agents' beliefs and subjective reliability of sources (trust). We then focus on discussing the influential Bayesian model of belief-based trust update by Eric Olsson, which models dichotomic events and explicitly represents anti-reliability. After sketching some disastrous recent results for this perhaps most promising model of belief update, we show new simulation results for the temporal dynamics of learning belief with and without trust update and with and without communication. The results seem to shed at least a somewhat more positive light on the communicating-and-trust-updating agents. This may be a light at the end of the tunnel of belief-based models of trust updating, but the interpretation of the clear findings is much less clear.

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 primarily used to specify the agent's 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 dynamic epistemic logic. Particularly, it has been shown that some postulates can be characterised through transformations in priority graphs, while others may not be represented that way. This work investigates changes in the semantics of Dynamic Preference Logic that give rise to an appropriate syntactic representation for its models that allow us to represent and reason about iterated belief base change in this logic.

Base revision in classical logic is done by the removal of formulas. Exploiting the non-monotonicity of ASP allows one to propose other revision strategies, namely addition strategy or removal and/or addition strategy. These strategies allow one to define families of rule-based revision operators. The paper presents a semantic characterization of these families of revision operators in terms of answer sets. This semantic characterization allows for equivalently considering the evolution of syntactic logic programs and the evolution of their semantic content. It then studies the logical properties of the proposed operators and gives complexity results.

In this article, the epistemic-entrenchment and partial-meet characterizations of Parikh's relevance-sensitive axiom for belief revision, known as axiom (P), are provided. In short, axiom (P) states that, if a belief set $K$ can be divided into two disjoint compartments, and the new information $\varphi$ relates only to the first compartment, then the revision of $K$ by $\varphi$ should not affect the second compartment. Accordingly, we identify the subclass of epistemic-entrenchment and that of selection-function preorders, inducing AGM revision functions that satisfy axiom (P). Hence, together with the faithful-preorders characterization of (P) that has already been provided, Parikh's axiom is fully characterized in terms of all popular constructive models of Belief Revision. Since the notions of relevance and local change are inherent in almost all intellectual activity, the completion of the constructive view of (P) has a significant impact on many theoretical, as well as applied, domains of Artificial Intelligence.

Information theory provides a mathematical foundation to measure uncertainty in belief. Belief is represented by a probability distribution that captures our understanding of an outcome's plausibility. Information measures based on Shannon's concept of entropy include realization information, Kullback-Leibler divergence, Lindley's information in experiment, cross entropy, and mutual information. We derive a general theory of information from first principles that accounts for evolving belief and recovers all of these measures. Rather than simply gauging uncertainty, information is understood in this theory to measure change in belief. We may then regard entropy as the information we expect to gain upon realization of a discrete latent random variable. This theory of information is compatible with the Bayesian paradigm in which rational belief is updated as evidence becomes available. Furthermore, this theory admits novel measures of information with well-defined properties, which we explore in both analysis and experiment. This view of information illuminates the study of machine learning by allowing us to quantify information captured by a predictive model and distinguish it from residual information contained in training data. We gain related insights regarding feature selection, anomaly detection, and novel Bayesian approaches.

According to Boutillier, Darwiche and Pearl and others, principles for iterated revision can be characterised in terms of changing beliefs about conditionals. For iterated contraction a similar formulation is not known. This is especially because for iterated belief change the connection between revision and contraction via the Levi and Harper identity is not straightforward, and therefore, characterisation results do not transfer easily between iterated revision and contraction. In this article, we develop an axiomatisation of iterated contraction in terms of changing conditional beliefs. We prove that the new set of postulates conforms semantically to the class of operators like the ones given by Konieczny and Pino Pérez for iterated contraction. 1 Introduction For the three main classes of theory change, revision, expansion and contraction, different characterisations are known [12], which are heavily supported by the correspondence between revision and contraction via the Levi and Harper identities [13, 17].

W e argue the case for Gaussian Belief Propagation (GBP) as a strong algorithmic framework for the distributed, generic and incremental probabilistic estimation we need in Spatial AI as we aim at high performance smart robots and devices which operate within the constraints of real products. Processor hardware is changing rapidly, and GBP has the right character to take advantage of highly distributed processing and storage while estimating global quantities, as well as great flexibility. W e present a detailed tutorial on GBP, relating to the standard factor graph formulation used in robotics and computer vision, and give several simulation examples with code which demonstrate its properties.

In most classical models of belief change, epistemic states are represented by theories (AGM) or formulas (Katsuno-Mendelzon) and the new pieces of information by formulas. The Representation Theorem for revision operators says that operators are represented by total preorders. This important representation is exploited by Darwiche and Pearl to shift the notion of epistemic state to a more abstract one, where the paradigm of epistemic state is indeed that of a total preorder over interpretations. In this work, we introduce a 3-valued logic where the formulas can be identified with a generalisation of total preorders of three levels: a ranking function mapping interpretations into the truth values. Then we analyse some sort of changes in this kind of structures and give syntactical characterizations of them.

The problem of belief tracking in the presence of stochastic actions and observations is pervasive and yet computationally intractable. In this work we show however that probabilistic beliefs can be maintained in factored form exactly and efficiently across a number of causally closed beams, when the state variables that appear in more than one beam obey a form of backward determinism . Since computing marginals from the factors is still computationally intractable in general, and variables appearing in several beams are not always backward-deterministic, the basic formulation is extended with two approximations: forms of belief propagation for computing marginals from factors, and sampling of non-backward-deterministic variables for making such variables backward-deterministic given their sampled history. Unlike, Rao-Blackwellized particle-filtering, the sampling is not used for making inference tractable but for making the factorization sound . The resulting algorithm involves sampling and belief propagation or just one of them as determined by the structure of the model.