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.

翻译：设 $\sigma$ 为一阶签名，$\mathbf{W}_n$ 为所有定义域为 $\{1, \ldots, n\}$ 的 $\sigma$-结构的集合。推理框架是指一类 $\mathbf{F}$，其中每个元素为 $(\mathbb{P}, L)$ 对，$\mathbb{P} = (\mathbb{P}_n : n = 1, 2, 3, \ldots)$ 且 $\mathbb{P}_n$ 是 $\mathbf{W}_n$ 上的概率分布，$L$ 是真值位于单位区间 $[0, 1]$ 的逻辑系统。若对于每个 $(\mathbb{P}, L) \in \mathbf{F}$，存在 $(\mathbb{P}', L') \in \mathbf{F}'$，使得 $\mathbb{P}$ 与 $\mathbb{P}'$ 渐近全变差等价，且对于每个 $\varphi(\bar{x}) \in L$，存在 $\varphi'(\bar{x}) \in L'$，使得 $\varphi'(\bar{x})$ 关于 $\mathbb{P}$ 与 $\varphi(\bar{x})$ 渐近等价，则称推理框架 $\mathbf{F}'$ 渐近至少与 $\mathbf{F}$ 具有相同表达性。该关系构成一个预序，我们描述了在机器学习和人工智能背景下看似自然的一些推理框架等价类上的偏序。关于公式渐近（或几乎必然）等价性或依概率收敛的若干先前结果，均可表述为推理框架的相对渐近强度。我们将这些结果纳入推理框架的分类中，并证明两个新结论。两者均涉及由使用"连续"聚合函数的有向图模型定义的概率分布序列。第一个结果考虑使用连续聚合函数的真值在 $[0, 1]$ 中的逻辑所表达的查询。第二个结果考虑可表达关于相对频率陈述的二值条件逻辑所表达的查询。