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On the relative asymptotic expressivity of inference frameworks

Vera Koponen, Felix Weitkämper

TL;DR

It is proved that two new results about the asymptotic equivalence of formulas where each result has a convergence law as a corollary can be formulated as results about the relative asymptotic expressivity of inference frameworks.

Abstract

We consider logics with truth values in the unit interval $[0,1]$. Such logics are used to define queries and to define probability distributions. In this context the notion of almost sure equivalence of formulas is generalized to the notion of asymptotic equivalence. We prove two new results about the asymptotic equivalence of formulas where each result has a convergence law as a corollary. These results as well as several older results can be formulated as results about the relative asymptotic expressivity of inference frameworks. An inference framework $\mathbf{F}$ is a class of pairs $(\mathbb{P}, L)$, where $\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$, $\mathbb{P}_n$ are probability distributions on the set $\mathbf{W}_n$ of all $σ$-structures with domain $\{1, \ldots, n\}$ (where $σ$ is a first-order signature) and $L$ is a logic with truth values in the unit interval $[0, 1]$. An inference framework $\mathbf{F}'$ is asymptotically at least as expressive as an inference framework $\mathbf{F}$ if for every $(\mathbb{P}, L) \in \mathbf{F}$ there is $(\mathbb{P}', L') \in \mathbf{F}'$ such that $\mathbb{P}$ is asymptotically total variation equivalent to $\mathbb{P}'$ and for every $\varphi(\bar{x}) \in L$ there is $\varphi'(\bar{x}) \in L'$ such that $\varphi'(\bar{x})$ is asymptotically equivalent to $\varphi(\bar{x})$ with respect to $\mathbb{P}$. This relation is a preorder. If, in addition, $\mathbf{F}$ is at least as expressive as $\mathbf{F}'$ then we say that $\mathbf{F}$ and $\mathbf{F}'$ are asymptotically equally expressive. Our third contribution is to systematize the new results of this paper and several previous results in order to get a preorder on a number of inference systems that are of relevance in the context of machine learning and artificial intelligence.

On the relative asymptotic expressivity of inference frameworks

TL;DR

It is proved that two new results about the asymptotic equivalence of formulas where each result has a convergence law as a corollary can be formulated as results about the relative asymptotic expressivity of inference frameworks.

Abstract

We consider logics with truth values in the unit interval . Such logics are used to define queries and to define probability distributions. In this context the notion of almost sure equivalence of formulas is generalized to the notion of asymptotic equivalence. We prove two new results about the asymptotic equivalence of formulas where each result has a convergence law as a corollary. These results as well as several older results can be formulated as results about the relative asymptotic expressivity of inference frameworks. An inference framework is a class of pairs , where , are probability distributions on the set of all -structures with domain (where is a first-order signature) and is a logic with truth values in the unit interval . An inference framework is asymptotically at least as expressive as an inference framework if for every there is such that is asymptotically total variation equivalent to and for every there is such that is asymptotically equivalent to with respect to . This relation is a preorder. If, in addition, is at least as expressive as then we say that and are asymptotically equally expressive. Our third contribution is to systematize the new results of this paper and several previous results in order to get a preorder on a number of inference systems that are of relevance in the context of machine learning and artificial intelligence.
Paper Structure (16 sections, 41 theorems, 115 equations, 1 figure)

This paper contains 16 sections, 41 theorems, 115 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Connectives and aggregation functions
  4. Logics
  5. Some notation and concepts concerning first-order logic
  6. Probabilistic logics with aggregation functions
  7. Conditional probability logic
  8. Probabilistic graphical models, sequences of probability distributions, and asymptotic elimination of aggregation functions
  9. Sequences of probability distributions and asymptotic equivalence
  10. L(sigma)-Networks
  11. Lifted Bayesian networks
  12. Asymptotic elimination of strongly admissible aggregation functions
  13. Saturation and convergence of atomic types: finishing the proof of Theorem \ref{['elimination of strongly admissible aggregation functions']}
  14. Almost sure elimination of safe $CPL$-formulas when the distribution is induced by a $coPLA^+$-network
  15. Relative asymptotic expressivity of inference frameworks
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $Z$ be the sum of $n$ independent binary random variables, each one with probability $p$ of having the value 1. For every $\varepsilon > 0$ there is $c_\varepsilon > 0$, depending only on $\varepsilon$, such that the probability that $|Z - pn| > \varepsilon p n$ is less than $2 e^{-c_\varepsilon

Figures (1)

  • Figure 1: The inference frameworks are defined in Definition \ref{['concrete inference frameworks']}. The symbol $\simeq$ denotes asymptotic equal expressivity. A path upwards means that the the upper inference framework is asymptotically more expressive (i.e. $\prec$ holds). The absence of a path "upwards" between two inference framewords means that the inference frameworks are incomparable with respect to asymptotic expressivity.

Theorems & Definitions (92)

  • Lemma 2.1
  • Corollary 2.2
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4: Convergence testing sequence
  • Definition 3.5: Admissibility and continuity
  • Proposition 3.6
  • Example 3.7: More (strongly) admissible aggregation functions
  • Definition 3.8: Functional representations of sequences
  • ...and 82 more