Asymptotic elimination of partially continuous aggregation functions in directed graphical models

Abstract

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.