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Probabilistic Theorem Proving
"... Many representation schemes combining firstorder logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logic ..."
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Many representation schemes combining firstorder logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logical structure into account. We propose the first method that has the full power of both graphical model inference and firstorder theorem proving (in finite domains with Herbrand interpretations). We first define probabilistic theorem proving, their generalization, as the problem of computing the probability of a logical formula given the probabilities or weights of a set of formulas. We then show how this can be reduced to the problem of lifted weighted model counting, and develop an efficient algorithm for the latter. We prove the correctness of this algorithm, investigate its properties, and show how it generalizes previous approaches. Experiments show that it greatly outperforms lifted variable elimination when logical structure is present. Finally, we propose an algorithm for approximate probabilistic theorem proving, and show that it can greatly outperform lifted belief propagation. 1
Lifted Inference Seen from the Other Side: The Tractable Features
"... Lifted Inference algorithms for representations that combine firstorder logic and graphical models have been the focus of much recent research. All lifted algorithms developed to date are based on the same underlying idea: take a standard probabilistic inference algorithm (e.g., variable eliminatio ..."
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Cited by 9 (1 self)
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Lifted Inference algorithms for representations that combine firstorder logic and graphical models have been the focus of much recent research. All lifted algorithms developed to date are based on the same underlying idea: take a standard probabilistic inference algorithm (e.g., variable elimination, belief propagation etc.) and improve its efficiency by exploiting repeated structure in the firstorder model. In this paper, we propose an approach from the other side in that we use techniques from logic for probabilistic inference. In particular, we define a set of rules that look only at the logical representation to identify models for which exact efficient inference is possible. Our rules yield new tractable classes that could not be solved efficiently by any of the existing techniques. 1
Declarative Programming for Agent Applications
"... This paper introduces the computational model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higherorder functions, modal computation, probabilistic computation, and some theoremproving capa ..."
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Cited by 1 (1 self)
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This paper introduces the computational model of a declarative programming language intended for agent applications. Features supported by the language include functional and logic programming idioms, higherorder functions, modal computation, probabilistic computation, and some theoremproving capabilities. The need for these features is motivated and examples are given to illustrate the central ideas.
Probabilities on Sentences in an Expressive Logic
, 2012
"... 1 Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higherorder logic are ideally suited for repre ..."
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1 Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higherorder logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truthvalues. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wishlist, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wishlist into technical requirements for a prior probability
On Lifting the Gibbs Sampling Algorithm
"... Firstorder probabilistic models combine the power of firstorder logic, the de facto tool for handling relational structure, with probabilistic graphical models, the de facto tool for handling uncertainty. Lifted probabilistic inference algorithms for them have been the subject of much recent resea ..."
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Firstorder probabilistic models combine the power of firstorder logic, the de facto tool for handling relational structure, with probabilistic graphical models, the de facto tool for handling uncertainty. Lifted probabilistic inference algorithms for them have been the subject of much recent research. The main idea in these algorithms is to improve the accuracy and scalability of existing graphical models’ inference algorithms by exploiting symmetry in the firstorder representation. In this paper, we consider blocked Gibbs sampling, an advanced MCMC scheme, and lift it to the firstorder level. We propose to achieve this by partitioning the firstorder atoms in the model into a set of disjoint clusters such that exact lifted inference is polynomial in each cluster given an assignment to all other atoms not in the cluster. We propose an approach for constructing the clusters and show how it can be used to trade accuracy with computational complexity in a principled manner. Our experimental evaluation shows that lifted Gibbs sampling is superior to the propositional algorithm in terms of accuracy, scalability and convergence. 1
Advances in Lifted Importance Sampling
"... We consider lifted importance sampling (LIS), a previously proposed approximate inference algorithm for statistical relational learning (SRL) models. LIS achieves substantial variance reduction over conventional importance sampling by using various lifting rules that take advantage of the symmetry i ..."
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We consider lifted importance sampling (LIS), a previously proposed approximate inference algorithm for statistical relational learning (SRL) models. LIS achieves substantial variance reduction over conventional importance sampling by using various lifting rules that take advantage of the symmetry in the relational representation. However, it suffers from two drawbacks. First, it does not take advantage of some important symmetries in the relational representation and may exhibit needlessly high variance on models having these symmetries. Second, it uses an uninformative proposal distribution which adversely affects its accuracy. We propose two improvements to LIS that address these limitations. First, we identify a new symmetry in SRL models and define a lifting rule for taking advantage of this symmetry. The lifting rule reduces the variance of LIS. Second, we propose a new, structured approach for constructing and dynamically updating the proposal distribution via adaptive sampling. We demonstrate experimentally that our new, improved LIS algorithm is substantially more accurate than the LIS algorithm.
Exploiting Logical Structurein Lifted Probabilistic Inference
"... Representations that combine firstorder logic and probability have been the focus of much recent research. Lifted inference algorithms for them avoid grounding out the domain, bringing benefits analogous to those of resolution theorem proving in firstorder logic. However, all lifted probabilistic ..."
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Representations that combine firstorder logic and probability have been the focus of much recent research. Lifted inference algorithms for them avoid grounding out the domain, bringing benefits analogous to those of resolution theorem proving in firstorder logic. However, all lifted probabilistic inference algorithms to date treat potentials as black boxes, and do not take advantage of their logical structure. As a result, inference with them is needlessly inefficient compared to the logical case. We overcome this by proposing the first lifted probabilistic inference algorithm that exploits determinism and context specific independence. In particular, we show that AND/OR search can be lifted by introducing POWER nodes in addition to the standard AND and OR nodes. Experimental tests show the benefits of our approach.
HIGHERORDER LOGIC LEARNING AND λPROGOL
"... Abstract. We present our research produced about Higherorder Logic Learning (HOLL), which consists of adapting Firstorder Logic Learning (FOLL), like Inductive Logic Programming (ILP), within a Higherorder Logic (HOL) context. We describe a first working implementation of λProgol, a HOLL system a ..."
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Abstract. We present our research produced about Higherorder Logic Learning (HOLL), which consists of adapting Firstorder Logic Learning (FOLL), like Inductive Logic Programming (ILP), within a Higherorder Logic (HOL) context. We describe a first working implementation of λProgol, a HOLL system adapting the ILP system Progol and the HOL formalism λProlog. We compare λProgol and Progol on the learning of recursive theories showing that HOLL can, in these cases, outperform FOLL. Introduction, Problem Description and Background Much of logicbased Machine Learning research is based on Firstorder Logic (FOL) and Prolog, including Inductive Logic Programming (ILP). As such, learning higherorder theories is not possible for such a system, and even some firstorder tasks are not handled well, like “learning firstorder recursive theories ” which “is a difficult learning task ” in a normal
Unifying Probability and Logic for Learning
"... Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem head on. Uncertain knowledge can be modeled by using graded probabilities rather tha ..."
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Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem head on. Uncertain knowledge can be modeled by using graded probabilities rather than binary truthvalues, but so far a completely satisfactory integration of logic and probability has been lacking. In particular the inability of confirming universal hypotheses has plagued most if not all systems so far. We address this problem head on. The main technical problem to be discussed is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wishlist, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii)