Definition: In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.
Input: I propose a framework for an agent to change its probabilistic beliefs when a new piece of propositional information α is observed. Traditionally, belief change occurs by either a revision process or by an update process, depending on whether the agent is informed with α in a static world or, respectively, whether α is a ‘signal’ from the environment due to an event occurring. Boutilier suggested a unified model of qualitative belief change, which “combines aspects of revision and update, providing a more realistic characterization of belief change.” In this paper, I propose a unified model of quantitative belief change, where an agent’s beliefs are represented as a probability distribution over possible worlds. As does Boutilier, I take a dynamical systems perspective. The proposed approach is evaluated against several rationality postulated, and some properties of the approach are worked out. Information acquired can be due to evolution of the world or revelation about the world. That is, one may notice via some ‘signal’ generated by the changing environment that the environment has changed, or, one may be informed by an independent agent in a static environment that some ‘fact’ holds. In the present work, I deal with belief change of agents who handle uncertainty by maintaining a probability distribution over possible situations. The agents in this framework also have models for nondeterministic events, and noisy observations. Noisy observation models can model imperfect sensory equipment for receiving environmental signals, but they can also model untrustworthy informants in a static world. In this paper, I provide the work of Boutilier (1998) as background, because it has several connections with and was the seed for the present work. However, I do not intend simply to give a probabilistic version of his Generalized Update Semantics. Whereas Boutilier (1998) presents a model for unifying qualitative belief revision and update, I build on his work to present a unified model of belief revision and update in a stochastic (probabilistic) setting. I also take a dynamical systems perspective, like him. Due to my quantitative approach, an agent can maintain a probability distribution Copyright c © 2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. over the worlds it believes possible, using an expectation semantics of change. This is in contrast to Boutilier’s “generalized update” approach, which takes a most-plausible event and most-plausible world approach. Finally, my proposal requires a trade-off factor to mix the changes in probability distribution over possible worlds brought about due to the probabilistic belief revision process and, respectively, the probabilistic belief update process. Boutilier’s model has revision and update more tightly coupled. For this reason, his approach is better called “unified” while mine is called “hybrid”. The belief change community does not study probabilistic belief update; it is studied almost exclusively in frameworks employing Bayesian conditioning – for modeling events and actions in dynamical domains (e.g., DBNs, MDPs, POMDPs) (Koller and Friedman, 2009; Poole and Mackworth, 2010, e.g.). The part of my approach responsible for updating stays within the Bayesian framework, but combines the essential elements of belief update with unobservable events and belief update as partially observable Markov decision process (POMDP) state estimation. On the other hand, there is plenty of literature on probabilistic belief revision (Voorbraak, 1999; Grove and Halpern, 1998; Kern-Isberner, 2008; Yue and Liu, 2008, e.g.). The subject is both deep and broad. There is no one accepted approach and to argue which is the best is not the focus of this paper. I shall choose one reasonable method for probabilistic belief revision suitable to the task at hand. In the first section, Boutilier’s ‘generalized update’ is reviewed. Then, in the next section, I introduce stochastic update and stochastic revision, culminating in the ‘hybrid stochastic belief change’ (HSBC) approach. The final section presents an example inspired by Boutilier’s article (1998) and analyses the results. Some proofs of propositions are omitted to save space; they are available on request. Boutilier’s Generalized Update I use Boutilier’s notation and descriptions, except that I am more comfortable with α and β to represent sentences, instead of A and B. It is assumed that an agent has a deductively closed belief set K, a set of sentences drawn from some logical language reflecting the agent’s beliefs about the current state of the world. For ease of presentation, I assume ar X iv :1 60 4. 02 12 6v 1 [ cs .A I] 7 A pr 2 01 6 a logically finite, classical propositional language, denoted L (LCPL in Boutilier (1998)), and consequence operation Cn . The belief set K will often be generated by some finite knowledge base KB (i.e., K = Cn(KB)). The identically true and false propositions are denoted > and ⊥, respectively. Given a set of possible worlds W (or valuations over L) and α ∈ L, the set of α-worlds, that is, the elements of W satisfying α, is denoted by ||α||. The worlds satisfying all sentences in a set K is denoted ||K||.

On Stochastic Belief Revision and Update and their Combination