  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.
