Real-world problems often require reasoning about hybrid beliefs, over both discrete and continuous random variables. Yet, such a setting has hardly been investigated in the context of planning. Moreover, existing online Partially Observable Markov Decision Processes (POMDPs) solvers do not support hybrid beliefs directly. In particular, these solvers do not address the added computational burden due to an increasing number of hypotheses with the planning horizon, which can grow exponentially. As part of this work, we present a novel algorithm, Hybrid Belief Monte Carlo Planning (HB-MCP) that utilizes the Monte Carlo Tree Search (MCTS) algorithm to solve a POMDP while maintaining a hybrid belief. We illustrate how the upper confidence bound (UCB) exploration bonus can be leveraged to guide the growth of hypotheses trees alongside the belief trees. We then evaluate our approach in highly aliased simulated environments where unresolved data association leads to multi-modal belief hypotheses.

Probability kinematics is a leading paradigm in probabilistic belief change. It is based on the idea that conditional beliefs should be independent from changes of their antecedents' probabilities. In this paper, we propose a re-interpretation of this paradigm for Spohn's ranking functions which we call Generalized Ranking Kinematics as a new principle for iterated belief revision of ranking functions by sets of conditional beliefs. This general setting also covers iterated revision by propositional beliefs. We then present c-revisions as belief change methodology that satisfies Generalized Ranking Kinematics.

In this article, we consider iteration principles for contraction, with the goal of identifying properties for contractions that respect conditional beliefs. Therefore, we investigate and evaluate four groups of iteration principles for contraction which consider the dynamics of conditional beliefs. For all these principles, we provide semantic characterization theorems and provide formulations by postulates which highlight how the change of beliefs and of conditional beliefs is constrained, whenever that is possible. The first group is similar to the syntactic Darwiche-Pearl postulates. As a second group, we consider semantic postulates for iteration of contraction by Chopra, Ghose, Meyer and Wong, and by Konieczny and Pino P\'erez, respectively, and we provide novel syntactic counterparts. Third, we propose a contraction analogue of the independence condition by Jin and Thielscher. For the fourth group, we consider natural and moderate contraction by Nayak. Methodically, we make use of conditionals for contractions, so-called contractionals and furthermore, we propose and employ the novel notion of $ \alpha $-equivalence for formulating some of the new postulates.

We present a novel offline-online method to mitigate the computational burden of the characterization of conditional beliefs in statistical learning. In the offline phase, the proposed method learns the joint law of the belief random variables and the observational random variables in the tensor-train (TT) format. In the online phase, it utilizes the resulting order-preserving conditional transport map to issue real-time characterization of the conditional beliefs given new observed information. Compared with the state-of-the-art normalizing flows techniques, the proposed method relies on function approximation and is equipped with thorough performance analysis. This also allows us to further extend the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional belief variables. On the one hand, we present novel heuristics to reorder and/or reparametrize the variables to enhance the approximation power of TT. On the other, we integrate the TT-based transport maps and the parameter reordering/reparametrization into layered compositions to further improve the performance of the resulting transport maps. We demonstrate the efficiency of the proposed method on various statistical learning tasks in ordinary differential equations (ODEs) and partial differential equations (PDEs).

The belief revision literature has largely focussed on the issue of how to revise one's beliefs in the light of information regarding matters of fact. Here we turn to an important but comparatively neglected issue: How might one extend a revision operator to handle conditionals as input? Our approach to this question of 'conditional revision' is distinctive insofar as it abstracts from the controversial details of how to revise by factual sentences. We introduce a 'plug and play' method for uniquely extending any iterated belief revision operator to the conditional case. The flexibility of our approach is achieved by having the result of a conditional revision by a Ramsey Test conditional ('arrow') determined by that of a plain revision by its corresponding material conditional ('hook'). It is shown to satisfy a number of new constraints that are of independent interest.

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].

Halpern 1990; Smets 1991; Yu and Arasta 1994), Dempster's conditional and Fagin-Halpern (FH) conditional can be considered the most widely used two DST conditional The flexibility and expressiveness of Dempster-Shafer (DS) A widely used approach for carrying out precise computation theoretic models make DS evidence theory (Dempster 1967; of the Dempster's conditional is a matrix calculus 1968; Shafer 1976) an ideal framework for reasoning and based algorithm which generates the Dempster's conditional decision making under uncertainty in Artificial Intelligence masses (Klawonn and Smets 1992; Smets 2002). Therefore, this specialization matrix-based method imposes Computing the DST belief functions and the DST conditionals, a prohibitive burden when dealing with larger FoDs.

Darwiche and Pearl's seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most of the suggestions made have resulted in a form of `reductionism' that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka `Independence', characteristic of `admissible' revision operators, remain commendably more modest. In this paper, we supplement both the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states that are associated with a `proper ordinal interval' (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of a large number of AGM era postulates, including Superexpansion, that are not sound for admissible operators in general.

Shenoy and Shafer {Shenoy:90} demonstrated that both for Dempster-Shafer Theory and probability theory there exists a possibility to calculate efficiently marginals of joint belief distributions (by so-called local computations) provided that the joint distribution can be decomposed (factorized) into a belief network. A number of algorithms exists for decomposition of probabilistic joint belief distribution into a bayesian (belief) network from data. For example Spirtes, Glymour and Schein{Spirtes:90b} formulated a Conjecture that a direct dependence test and a head-to-head meeting test would suffice to construe bayesian network from data in such a way that Pearl's concept of d-separation {Geiger:90} applies. This paper is intended to transfer Spirtes, Glymour and Scheines {Spirtes:90b} approach onto the ground of the Dempster-Shafer Theory (DST). For this purpose, a frequentionistic interpretation of the DST developed in {Klopotek:93b} is exploited. A special notion of conditionality for DST is introduced and demonstrated to behave with respect to Pearl's d-separation {Geiger:90} much the same way as conditional probability (though some differences like non-uniqueness are evident). Based on this, an algorithm analogous to that from {Spirtes:90b} is developed. The notion of a partially oriented graph (pog) is introduced and within this graph the notion of p-d-separation is defined. If direct dependence test and head-to-head meeting test are used to orient the pog then its p-d-separation is shown to be equivalent to the Pearl's d-separation for any compatible dag.

Plausibility models are Kripke models that agents use to reason about knowledge and belief, both of themselves and of each other. Such models are used to interpret the notions of conditional belief, degrees of belief, and safe belief. The logic of conditional belief contains that modality and also the knowledge modality, and similarly for the logic of degrees of belief and the logic of safe belief. With respect to these logics, plausibility models may contain too much information. A proper notion of bisimulation is required that characterises them. We define that notion of bisimulation and prove the required characterisations: on the class of image-finite and preimage-finite models (with respect to the plausibility relation), two pointed Kripke models are modally equivalent in either of the three logics, if and only if they are bisimilar. As a result, the information content of such a model can be similarly expressed in the logic of conditional belief, or the logic of degrees of belief, or that of safe belief. This, we found a surprising result. Still, that does not mean that the logics are equally expressive: the logics of conditional and degrees of belief are incomparable, the logics of degrees of belief and safe belief are incomparable, while the logic of safe belief is more expressive than the logic of conditional belief. In view of the result on bisimulation characterisation, this is an equally surprising result. We hope our insights may contribute to the growing community of formal epistemology and on the relation between qualitative and quantitative modelling.