Reasoning about an entity's preferences (be it a user of an application, an individual targeted for marketing, or a group of people whose choices are of interest) has a long history in different areas of study. In this talk, we adopt the point of view that grows out of the intersection of databases and knowledge representation, where preferences are usually represented as strict partial orders over the set of tuples in a database or the consequences of a knowledge base. We introduce order-based Markov models (OMMs), which flexibly combine such preferences with probabilistic uncertainty. Their applications are clear in domains such as the Social Semantic Web, where users often express preferences in an incomplete manner and through different means, often in contradiction with each other. We show that the basic problems associated with reasoning with OMMs (computing the probability of a possible world or a given query) are #P-complete, and then explore ways to make these computations tractable by: (i) leveraging results from Order Theory to obtain a fully polynomial-time randomized approximation scheme (FPRAS); (ii) studying a fragment of the language of OMMs for which exact computations can be performed in fixed-parameter polynomial time; and (iii) variable elimination methods for other fragments, along with an empirical evaluation of their tractability.
The work presented in this talk has been carried out in conjunction with Maria Vanina Martinez (UNS Bahia Blanca and CONICET, Argentina), Thomas Lukasiewicz (University of Oxford, UK), and David Poole (University of British Columbia, Canada).