Conditional probability logic, lifted Bayesian networks, and almost sure quantifier elimination

Koponen, Vera

doi:10.1016/j.tcs.2020.08.006

Cited by 9 publications

(40 citation statements)

References 12 publications

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“…Candidates for this are for instance Keisler's (1985) logic with probability quantifiers or Koponen's (2020) conditional probability logic. Appropriate asymptotic quantifier elimination results have been shown in both settings (Koponen 2020;Keisler and Lotfallah 2009), allowing an immediate application of our results there.…”

Section: Further Workmentioning

confidence: 59%

An Asymptotic Analysis of Probabilistic Logic Programming, with Implications for Expressing Projective Families of Distributions

Weitkämper¹

2021

Theory and Practice of Logic Programming

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Probabilistic logic programming is a major part of statistical relational artificial intelligence, where approaches from logic and probability are brought together to reason about and learn from relational domains in a setting of uncertainty. However, the behaviour of statistical relational representations across variable domain sizes is complex, and scaling inference and learning to large domains remains a significant challenge. In recent years, connections have emerged between domain size dependence, lifted inference and learning from sampled subpopulations. The asymptotic behaviour of statistical relational representations has come under scrutiny, and projectivity was investigated as the strongest form of domain size dependence, in which query marginals are completely independent of the domain size. In this contribution we show that every probabilistic logic program under the distribution semantics is asymptotically equivalent to an acyclic probabilistic logic program consisting only of determinate clauses over probabilistic facts. We conclude that every probabilistic logic program inducing a projective family of distributions is in fact everywhere equivalent to a program from this fragment, and we investigate the consequences for the projective families of distributions expressible by probabilistic logic programs.

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Section: Further Workmentioning

confidence: 59%

An Asymptotic Analysis of Probabilistic Logic Programming, with Implications for Expressing Projective Families of Distributions

Weitkämper¹

2021

Theory and Practice of Logic Programming

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“…Very limited convergence results with respect to logical expressibility, covering only boolean combinations of atomic formulas, for some other types of PPGMs, including (domain-aware) Markov logic networks and (domain-aware) relational logistic regression networks have been obtained by Poole et al [15], Mittal et al [11] and Weitkämper [17], and other related results have been found by Jain et al [6], Jaeger and Schulte [5], and Weitkämper [18]. For lifted Bayesian networks in the sense of [9] (discussed below in Section 4.2) there is also a probabilistic convergence result, proved by almost sure elimination of a certain type of quantifiers, applicable to every CPL-formula ϕ which does not refer to a finite set of real numbers which only depends on the lifted Bayesian network that determines the asymptotic probability distribution and the complexity of ϕ. This result generalizes a convergence result of Keisler and Lotfallah [7] which they proved by almost sure elimination of relative frequence quantifiers.…”

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confidence: 84%

“…The first main result, Theorem 5.5, says that if a P LA(σ)-network has the property that no aggregation formula (of the network) contains an aggregation function, then every formula of P LA(σ) with only admissible aggregation functions is asymptotically equivalent to a formula without aggregation functions. The second main result, Theorem 5.6, is a generalization of Theorem 5.5 to lifted Bayesian networks satisfying the conditions of the main results in [9]. As explained by Remark 7.13, the asymptotically equivalent formula without aggregation functions can be computed from the original formula by only using the P LA(σ)-network (or lifted Bayesian network) that induces the probability distribution, so the computation is independent of the size of the domain.…”

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confidence: 93%

“…The idea of generalizing Mostowski style quantifiers to the context of many valued logics, at least for logics with finitely many truth values, goes back (at least) to Rescher [16] and it is a small step to translate his notion of generalized quantifier in many valued logic to the notion of an aggregation function (representing the corresponding generalized quantifier). A particular instance of the relationship between generalized quantifiers and aggregation function is given by considering the conditional probability logic (CP L) which is studied in [9]. CP L is a 2-valued logic which extends first-order logic and with which one can express, with a generalized quantifier, that a (conditional) probability is larger than a given threshold, or larger than another (conditional) probability.…”

mentioning

confidence: 99%

“…The first notion of Bayesian network that we consider is the the notion of a P LA(σ)-network (Definition 4.3) which is roughly the same as Jaeger's Relational Bayesian Network [4]. The other is the notion of a lifted Bayesian network in the sense of [9,Definition 3.8].…”

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confidence: 99%

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Asymptotic elimination of partially continuous aggregation functions in directed graphical models

Koponen¹,

Weitkämper²

2021

Preprint

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In Statistical Relational Artificial Intelligence, a branch of AI and machine learning which combines the logical and statistical schools of AI, one uses the concept parametrized probabilistic graphical model (PPGM) to model (conditional) dependencies between random variables and to make probabilistic inferences about events on a space of "possible worlds". The set of possible worlds with underlying domain D (a set of objects) can be represented by the set WD of all first-order structures (for a suitable signature) with domain D. Using a formal logic we can describe events on WD. By combining a logic and a PPGM we can also define a probability distribution PD on WD and use it to compute the probability of an event. We consider a logic, denoted P LA, with truth values in the unit interval, which uses aggregation functions instead of quantifiers. This is motivated by the fact that aggregation functions such as arithmetic mean, geometric mean, maximum and minimum are important tools in analysis of data and also by the fact that aggregation functions can play the role of quantifiers.However we face the problem of computational efficiency and this problem is an obstacle to the wider use of methods from Statistical Relational AI in practical applications. The brute force way of computing, for S ⊆ [0, 1] and a P LA-sentence ϕ, the probability that the value of ϕ belongs to S needs an amount of time which grows exponentially in the size of D. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence ϕ, converge as the size of D tends to infinity. The convergence result is obtained by showing that every formula ϕ(x1, . . . , x k ) which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula ψ(x1, . . . , x k ) without aggregation functions. This means that for every ε > 0 the probability that, for some parameters a1, . . . , a k ∈ D, the values of ϕ(a1, . . . , a k ) and ψ(a1, . . . , a k ) differ by more than ε approaches 0 as the domain size tends to infinity. The proof provides a method of computing the probability that the value of ϕ(a1, . . . , a k ) belongs to S with an amount of time which is independent of D and only depends on ϕ and the PPGM.

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Functional Lifted Bayesian Networks: Statistical Relational Learning and Reasoning with Relative Frequencies

Weitkämper

2024

Lecture Notes in Computer Science

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Conditional probability logic, lifted Bayesian networks, and almost sure quantifier elimination

Cited by 9 publications

References 12 publications

An Asymptotic Analysis of Probabilistic Logic Programming, with Implications for Expressing Projective Families of Distributions

An Asymptotic Analysis of Probabilistic Logic Programming, with Implications for Expressing Projective Families of Distributions

Asymptotic elimination of partially continuous aggregation functions in directed graphical models

Functional Lifted Bayesian Networks: Statistical Relational Learning and Reasoning with Relative Frequencies

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