On the relative asymptotic expressivity of inference frameworks

Let $\sigma$ be a first-order signature and let $\mathbf{W}_n$ be the set of all $\sigma$-structures with domain $\{1, \ldots, n\}$. By an inference framework we mean a class $\mathbf{F}$ of pairs $(\mathbb{P}, L)$, where $\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$ and $\mathbb{P}_n$ is a probability distribution on $\mathbf{W}_n$, 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 another 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 and we describe a partial order on the equivalence classes of some inference frameworks that seem natural in the context of machine learning and artificial intelligence. Several previous results about asymptotic (or almost sure) equivalence of formulas or convergence in probability can be formulated in terms of relative asymptotic strength of inference frameworks. We incorporate these results in our classification of inference frameworks and prove two new results. Both concern sequences of probability distributions defined by directed graphical models that use ``continuous'' aggregation functions. The first considers queries expressed by a logic with truth values in $[0, 1]$ which employs continuous aggregation functions. The second considers queries expressed by a two-valued conditional logic that can express statements about relative frequencies.