Oblivious Bounds on the Probability of Boolean Functions

This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an exact characterization of optimal oblivious bounds, i.e. when the new probabilities are chosen independent of the probabilities of all other variables. Our motivation comes from the weighted model counting problem (or, equivalently, the problem of computing the probability of a Boolean function), which is #P-hard in general. By performing several dissociations, one can transform a Boolean formula whose probability is difficult to compute, into one whose probability is easy to compute, and which is guaranteed to provide an upper or lower bound on the probability of the original formula by choosing appropriate probabilities for the dissociated variables. Our new bounds shed light on the connection between previous relaxation-based and model-based approximations and unify them as concrete choices in a larger design space. We also show how our theory allows a standard relational database management system (DBMS) to both upper and lower bound hard probabilistic queries in guaranteed polynomial time.Comment: 34 pages, 14 figures, supersedes: http://arxiv.org/abs/1105.281

10 Years of Probabilistic Querying – What Next?

Over the past decade, the two research areas of probabilistic databases and probabilistic programming have intensively studied the problem of making structured probabilistic inference scalable, but — so far — both areas developed almost independently of one another. While probabilistic databases have focused on describing tractable query classes based on the structure of query plans and data lineage, probabilistic programming has contributed sophisticated inference techniques based on knowledge compilation and lifted (first-order) inference. Both fields have developed their own variants of — both exact and approximate — top-k algorithms for query evaluation, and both investigate query optimization techniques known from SQL, Datalog, and Prolog, which all calls for a more intensive study of the commonalities and integration of the two fields. Moreover, we believe that natural-language processing and information extraction will remain a driving factor and in fact a longstanding challenge for developing expressive representation models which can be combined with structured probabilistic inference — also for the next decades to come

Approximate Lifted Inference with Probabilistic Databases

This paper proposes a new approach for approximate evaluation of #P-hard queries with probabilistic databases. In our approach, every query is evaluated entirely in the database engine by evaluating a fixed number of query plans, each providing an upper bound on the true probability, then taking their minimum. We provide an algorithm that takes into account important schema information to enumerate only the minimal necessary plans among all possible plans. Importantly, this algorithm is a strict generalization of all known results of PTIME self-join-free conjunctive queries: A query is safe if and only if our algorithm returns one single plan. We also apply three relational query optimization techniques to evaluate all minimal safe plans very fast. We give a detailed experimental evaluation of our approach and, in the process, provide a new way of thinking about the value of probabilistic methods over non-probabilistic methods for ranking query answers.Comment: 12 pages, 5 figures, pre-print for a paper appearing in VLDB 2015. arXiv admin note: text overlap with arXiv:1310.625

Learning Tuple Probabilities in Probabilistic Databases

Learning the parameters of complex probabilistic-relational models from labeled training data is a standard technique in machine learning, which has been intensively studied in the subfield of Statistical Relational Learning (SRL), but---so far---this is still an under-investigated topic in the context of Probabilistic Databases (PDBs). In this paper, we focus on learning the probability values of base tuples in a PDB from query answers, the latter of which are represented as labeled lineage formulas. Specifically, we consider labels in the form of pairs, each consisting of a Boolean lineage formula and a marginal probability that comes attached to the corresponding query answer. The resulting learning problem can be viewed as the inverse problem to confidence computations in PDBs: given a set of labeled query answers, learn the probability values of the base tuples, such that the marginal probabilities of the query answers again yield in the assigned probability labels. We analyze the learning problem from a theoretical perspective, devise two optimization-based objectives, and provide an efficient algorithm (based on Stochastic Gradient Descent) for solving these objectives. Finally, we conclude this work by an experimental evaluation on three real-world and one synthetic dataset, while competing with various techniques from SRL, reasoning in information extraction, and optimization

On the Semantics of Queries over Graphs with Uncertainty

We study the semantics of queries over uncertain graphs, which are directed graphs in which each edge is associated with a value in [0,1] representing its cer- tainty. In this work, we consider the certainty values as probabilities and show the challenges involved in evaluating the reachability and transitive closure queries over uncertain/probabilistic graphs. As the evaluation method, we adopted graph re- duction from automata theory used for finding regular expressions for input finite state machines. However, we show that different order of eliminating nodes may yield different certainty associated with the results. We then formulate the notion of "correct" results for queries over uncertain graphs, justified based on the notion of common sub-expressions, and identify common paths and avoid their redundant multiple contributions during the reduction. We identify a set of possible patterns to facilitate the reduction process. We have implemented the proposed ideas for answering reachability and transitive closure queries. We evaluated the effectiveness of the proposed solutions using a library of many uncertain graphs with different sizes and structures. We believe the proposed ideas and solution techniques can yield query processing tools for uncertain data management systems

Efficient querying and learning in probabilistic and temporal databases

Probabilistic databases store, query, and manage large amounts of uncertain information. This thesis advances the state-of-the-art in probabilistic databases in three different ways: 1. We present a closed and complete data model for temporal probabilistic databases and analyze its complexity. Queries are posed via temporal deduction rules which induce lineage formulas capturing both time and uncertainty. 2. We devise a methodology for computing the top-k most probable query answers. It is based on first-order lineage formulas representing sets of answer candidates. Theoretically derived probability bounds on these formulas enable pruning low-probability answers. 3. We introduce the problem of learning tuple probabilities which allows updating and cleaning of probabilistic databases. We study its complexity, characterize its solutions, cast it into an optimization problem, and devise an approximation algorithm based on stochastic gradient descent. All of the above contributions support consistency constraints and are evaluated experimentally.Probabilistische Datenbanken können große Mengen an ungewissen Informationen speichern, anfragen und verwalten. Diese Doktorarbeit treibt den Stand der Technik in diesem Gebiet auf drei Arten vorran: 1. Ein abgeschlossenes und vollständiges Datenmodell für temporale, probabilistische Datenbanken wird präsentiert. Anfragen werden mittels Deduktionsregeln gestellt, welche logische Formeln induzieren, die sowohl Zeit als auch Ungewissheit erfassen. 2. Ein Methode zur Berechnung der k Anworten höchster Wahrscheinlichkeit wird entwickelt. Sie basiert auf logischen Formeln erster Stufe, die Mengen an Antwortkandidaten repräsentieren. Beschränkungen der Wahrscheinlichkeit dieser Formeln ermöglichen das Kürzen von Antworten mit niedriger Wahrscheinlichkeit. 3. Das Problem des Lernens von Tupelwahrscheinlichkeiten für das Aktualisieren und Bereiningen von probabilistischen Datenbanken wird eingeführt, auf Komplexität und Lösungen untersucht, als Optimierungsproblem dargestellt und von einem stochastischem Gradientenverfahren approximiert. All diese Beiträge unterstützen Konsistenzbedingungen und wurden experimentell analysiert