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Computing query probability with incidence algebras
 In PODS
"... We describe an algorithm that evaluates queries over probabilistic databases using Mobius ’ inversion formula in incidence algebras. The queries we consider are unions of conjunctive queries (equivalently: existential, positive First Order sentences), and the probabilistic databases are tupleindepe ..."
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We describe an algorithm that evaluates queries over probabilistic databases using Mobius ’ inversion formula in incidence algebras. The queries we consider are unions of conjunctive queries (equivalently: existential, positive First Order sentences), and the probabilistic databases are tupleindependent structures. Our algorithm runs in PTIME on a subset of queries called ”safe ” queries, and is complete, in the sense that every unsafe query is hard for the class F P #P. The algorithm is very simple and easy to implement in practice, yet it is nonobvious. Mobius ’ inversion formula, which is in essence inclusionexclusion, plays a key role for completeness, by allowing the algorithm to compute the probability of some safe queries even when they have some subqueries that are unsafe. We also apply the same latticetheoretic techniques to analyze an algorithm based on lifted conditioning, and prove that it is incomplete. 1
The Dichotomy of Probabilistic Inference for Unions of Conjunctive Queries
"... We study the complexity of computing the probability of a query on a probabilistic database. The queries that we consider are unions of conjunctive queries, UCQ: equivalently, these are positive, existential First Order Logic sentences, or nonrecursive datalog programs. The databases that we consid ..."
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We study the complexity of computing the probability of a query on a probabilistic database. The queries that we consider are unions of conjunctive queries, UCQ: equivalently, these are positive, existential First Order Logic sentences, or nonrecursive datalog programs. The databases that we consider are tupleindependent. We prove the following dichotomy theorem. For every UCQ query, either its probability can be computed in polynomial time in the size of the database, or is hard for FP #P. Our result also has applications to the problem of computing the probability of positive, Boolean expressions, and establishes a dichotomy for such classes based on their structure. For the tractable case, we give a very simple algorithm that alternates between two steps: applying the inclusion/exclusion formula, and removing one existential variable. A key, and novel feature of this algorithm is that it avoids computing terms that cancel out in the inclusion/exclusion formula, in other words it only computes those terms whose Mobius function in an appropriate lattice is nonzero. We show that This simple feature is a key ingredient needed to ensure completeness. For the hardness proof, we give a reduction from the counting problem for positive, partitioned 2CNF, which is known to be #Pcomplete. The hardness proof is nontrivial, and uses techniques from logic and from classical algebra.
Informed source separation: A Bayesian tutorial
 Proceedings of the 13th European Signal Processing Conference (EUSIPCO 2005
, 2005
"... Source separation problems are ubiquitous in the physical sciences; any situation where signals are superimposed calls for source separation to estimate the original signals. In this tutorial I will discuss the Bayesian approach to the source separation problem. This approach has a specific advantag ..."
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Source separation problems are ubiquitous in the physical sciences; any situation where signals are superimposed calls for source separation to estimate the original signals. In this tutorial I will discuss the Bayesian approach to the source separation problem. This approach has a specific advantage in that it requires the designer to explicitly describe the signal model in addition to any other information or assumptions that go into the problem description. This leads naturally to the idea of informed source separation, where the algorithm design incorporates relevant information about the specific problem. This approach promises to enable researchers to design their own highquality algorithms that are specifically tailored to the problem at hand. 1. UNDERSTANDING THE PROBLEM
All questions answered
 Notices of the AMS, 49 (3), 318–324 http: //www.ams.org/notices/200203/feaknuth.pdf
, 2002
"... Abstract. The Boolean lattice of logical statements induces the free distributive lattice of questions. Inclusion on this lattice is based on whether one question answers another. Generalizing the zeta function of the question lattice leads to a valuation called relevance or bearing, which is a meas ..."
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Abstract. The Boolean lattice of logical statements induces the free distributive lattice of questions. Inclusion on this lattice is based on whether one question answers another. Generalizing the zeta function of the question lattice leads to a valuation called relevance or bearing, which is a measure of the degree to which one question answers another. Richard Cox conjectured that this degree can be expressed as a generalized entropy. With the assistance of yet another important result from Janos Aczél, I show that this is indeed the case, and that the resulting inquiry calculus is a natural generalization of information theory. This approach provides a new perspective on the Principle of Maximum Entropy. “A wise man’s question contains half the answer. ” Solomon Ibn Gabirol (10211058)
Revealing relationships among relevant climate variables with information theory
 EarthSun System Technology Conference 2005
, 2005
"... Abstract—A primary objective of the NASA EarthSun Exploration ..."
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Abstract—A primary objective of the NASA EarthSun Exploration
Measuring Questions: Relevance and its Relation to Entropy
, 2004
"... The Boolean lattice of logical statements induces the free distributive lattice of questions. Inclusion on this lattice is based... ..."
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Cited by 4 (4 self)
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The Boolean lattice of logical statements induces the free distributive lattice of questions. Inclusion on this lattice is based...
From Inference to Physics ∗
, 808
"... Entropic dynamics, a program that aims at deriving the laws of physics from standard probabilistic and entropic rules for processing information, is developed further. We calculate the probability for an arbitrary path followed by a system as it moves from given initial to final states. For an appro ..."
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Entropic dynamics, a program that aims at deriving the laws of physics from standard probabilistic and entropic rules for processing information, is developed further. We calculate the probability for an arbitrary path followed by a system as it moves from given initial to final states. For an appropriately chosen configuration space the path of maximum probability reproduces Newtonian dynamics. 1
On the Risk of Using Rényi’s Entropy for Blind Source Separation
"... Abstract—Recently, some researchers have suggested Rényi’s entropy in its general form as a blind source separation (BSS) objective function. This was motivated by two arguments: 1) Shannon’s entropy, which is known to be a suitable criterion for BSS, is a particular case of Rényi’s entropy, and 2) ..."
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Abstract—Recently, some researchers have suggested Rényi’s entropy in its general form as a blind source separation (BSS) objective function. This was motivated by two arguments: 1) Shannon’s entropy, which is known to be a suitable criterion for BSS, is a particular case of Rényi’s entropy, and 2) some practical advantages can be obtained by choosing another specific value for the Rényi exponent, yielding to, e.g., quadratic entropy. Unfortunately, by doing so, there is no longer guarantee that optimizing this generalized criterion would lead to recovering the original sources. In this paper, we show that Rényi’s entropy in its exact form (i.e., out of any consideration about its practical estimation or computation) might lead to not recovering the sources, depending on the source densities and on Rényi’s exponent value. This is illustrated on specific examples. We also compare our conclusions with previous works involving Rényi’s entropies for blind deconvolution. Index Terms—Blind source separation (BSS), contrast function, independent component analysis, Rényi’s entropy, Taylor expansion. I.
Autonomous Science Platforms and QuestionAsking Machines
"... Abstract—As we become increasingly reliant on remote science platforms, the ability to autonomously and intelligently perform data collection becomes critical. In this paper we view these platforms as questionasking machines and introduce a paradigm based on the scientific method, which couples the ..."
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Abstract—As we become increasingly reliant on remote science platforms, the ability to autonomously and intelligently perform data collection becomes critical. In this paper we view these platforms as questionasking machines and introduce a paradigm based on the scientific method, which couples the processes of inference and inquiry to form a modelbased learning cycle. Unlike modern autonomous instrumentation, the system is not programmed to collect data directly, but instead, is programmed to learn based on a set of models. Computationally, this learning cycle is implemented in software consisting of a Bayesian probabilitybased inference engine coupled to an entropybased inquiry engine. Operationally, a given experiment is viewed as a question, whose relevance is computed using the inquiry calculus, which is a natural ordertheoretic generalization of information theory. In simple cases, the relevance is proportional to the entropy. This data is then analyzed by the inference engine, which updates the state of knowledge of the instrument. This new state of knowledge is then used as a basis for future inquiry as the system continues to learn. This paper will introduce the learning methodology, describe its implementation in software, and demonstrate the process with a robotic explorer that autonomously and intelligently performs data collection to solve a searchandcharacterize problem. I.