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

Weitkämper, Felix

doi:10.1017/s1471068421000314

Cited by 9 publications

(7 citation statements)

References 14 publications

(17 reference statements)

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“…We generalise the concept of projectivity to this setting and show that the main results of Schulte (2018, 2020) and Weitkämper (2021) carry over. In particular, we introduce AHK representations for structured input and prove an analogue of the representation theorem of Jaeger and Schulte (2020) We also show a oneto-one correspondence between families of distributions and distributions on a countably infinite domain.…”

Section: Introductionmentioning

confidence: 89%

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Projectivity revisited

Weitkämper¹

2022

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The behaviour of statistical relational representations across differently sized domains has become a focal area of research from both a modelling and a complexity viewpoint. In 2018, Jaeger and Schulte suggested projectivity of a family of distributions as a key property, ensuring that marginal inference is independent of the domain size. However, Jaeger and Schulte assume that the domain is characterised only by its size. This contribution extends the notion of projectivity from families of distributions indexed by domain size to functors taking extensional data from a database. This makes projectivity available for the large range of applications taking structured input. We transfer the known attractive properties of projective families of distributions to the new setting. Furthermore, we prove a correspondence between projectivity and distributions on countably infinite domains, which we use to unify and generalise earlier work on statistical relational representations in infinite domains. Finally, we use the extended notion of projectivity to define a further strengthening, which we call σ-projectivity, and which allows the use of the same representation in different modes while retaining projectivity.

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Section: Introductionmentioning

confidence: 89%

“…A PLP induces a projective family of distributions if it is determinate (Muggleton and Feng, 1990;Weitkämper, 2021), that is, if any variable occurring in the body of a clause also occurs in the head of the same clause.…”

Section: Projectivity On Unstructured Domainsmentioning

confidence: 99%

Projectivity revisited

Weitkämper¹

2022

Preprint

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“…Hence, a complete characterization of projectivity in most SRL languages is still an open challenge. In [19], Weitkamper has shown that the characterization of projectivity provided by Jaeger and Schulte [4], for probabilistic logic programs under distribution semantics, is indeed complete. In this work, we will extend this characterization to two variable fragment of Markov Logic Networks.…”

Section: Related Workmentioning

confidence: 99%

On Projectivity in Markov Logic Networks

Malhotra¹,

Serafini²

2022

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Markov Logic Networks (MLNs) define a probability distribution on relational structures over varying domain sizes. Like most relational models, MLNs do not admit consistent marginal inference over varying domain sizes. Furthermore, MLNs learned on a fixed domain do not generalize to domains of varied sizes. In recent works, connections have emerged between domain size dependence, lifted inference, and learning from a sub-sampled domain. The central idea of these works is the notion of projectivity. The probability distributions ascribed by projective models render the marginal probabilities of sub-structures independent of the domain cardinality. Hence, projective models admit efficient marginal inference. Furthermore, projective models potentially allow efficient and consistent parameter learning from sub-sampled domains. In this paper, we characterize the necessary and sufficient conditions for a two-variable MLN to be projective. We then isolate a special model in this class of MLNs, namely Relational Block Model (RBM). We show that, in terms of data likelihood maximization, RBM is the best possible projective MLN in the two-variable fragment. Finally, we show that RBMs also admit consistent parameter learning over sub-sampled domains.

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“…In the context of transferring models between domains of different size, the asymptotic behaviour of queries in Markov logic networks and relational logistic regression was analysed [30], leading to work on new domain-size aware formalisms perceived to have better scaling properties [28,33]. In [16] those families of probability distributions which are in a sense invariant to domain size were completely characterised, and in [34] the asymptotic behaviour of probabilistic logic programs, another main branch of statistical relational artificial intelligence was determined. Theorems 6.9 and 9.6 have as corollaries that if a coP LA + -network defines the probability distribution then the probability of any event defined by either a coP LA + -formula or a safe (with respect to the network) CP L-formula converges as the domain size tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

On the relative asymptotic expressivity of inference frameworks

Koponen¹,

Weitkämper²

2022

Preprint

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Let σ be a first-order signature and let Wn be the set of all σ-structures with domain [n] = {1, . . . , n}. We can think of each structure in Wn as representing a "possible state of the world (or the context of interest)", or simply a "possible world" as we will say, following common informal terminology in the field of Statistical Relational Artificial Intelligence. By an inference framework we mean a class F of pairs (P, L), where P = (Pn : n = 1, 2, 3, . . .) and each Pn is a probability distribution on Wn, and L is a logic with truth values in the unit interval [0, 1].The inference frameworks that we consider will contain pairs (P, L) where P is determined by a so-called probabilistic graphical model, a concept used in AI and machine learning, and L is a logic with expressive capabilities of interest when analysing data sets, so for example, we consider logics which can express statements about, for example, (conditional) probabilities or (arithmetic or geometric) averages.From the point of view of probabilistic and logical expressivity one may consider an inference framework as optimal if it allows any pair (P, L) where P = (Pn : n = 1, 2, 3, . . .) is a sequence of probability distributions on Wn and L is a logic. But from the point of view of using a pair (P, L) from such an inference framework for making inferences on Wn when n is large we face the problem of computational complexity. This motivates looking for an "optimal" trade-off between expressivity and computational efficiency. The issue of computational complexity also arises when learning a probabilistic graphical model as one may want to use a formal language (logic) for describing events that are relevant for learning the graphical model. Learning a more complex graphical model, which in turn determines a sequence (Pn : n = 1, 2, 3, . . .) of more complex probability distributions on Wn, generally requires more computational resources.We define a notion that an inference framework is "asymptotically at least as expressive" as another inference framework. This relation is a preorder and we describe a (strict) partial order on the equivalence classes of some inference frameworks that in our opinion are natural in the context of machine learning and artificial intelligence, illustrated by Figure 1. The results have bearing on issues concerning efficient learning and probabilistic inference, but are also new instances of results in finite model theory about "almost sure elimination" of extra syntactic features (e.g quantifiers) beyond the connectives. Often such a result has a logical convergence law as a corollary.

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

Cited by 9 publications

References 14 publications

Projectivity revisited

Projectivity revisited

On Projectivity in Markov Logic Networks

On the relative asymptotic expressivity of inference frameworks

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