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 {\em para\-metrized 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 $\mathbf{W}_D$ of all first-order structures (for a suitable signature) with domain $D$. Using a formal logic we can describe events on $\mathbf{W}_D$. By combining a logic and a PPGM we can also define a probability distribution $\mathbb{P}_D$ on $\mathbf{W}_D$ and use it to compute the probability of an event. We consider a logic, denoted $PLA$, with truth values in the unit interval, which uses aggregation functions, such as arithmetic mean, geometric mean, maximum and minimum instead 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. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence $\varphi$, converge as the size of $D$ tends to infinity. The convergence result is obtained by showing that every formula $\varphi(x_1, \ldots, x_k)$ which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula $\psi(x_1, \ldots, x_k)$ without aggregation functions.
翻译:有向图模型中部分连续聚合函数的渐进消除
翻译后的摘要:
在统计关系人工智能中,一种将AI和机器学习的逻辑和统计学派别相结合的分支,使用参数化概率图模型(PPGM)来模拟(条件)随机变量之间的依存关系并对“可能的世界”上的事件进行概率推断。带有基础域D(一组对象)的可能世界集合可以用所有具有域D的第一阶结构(对于适当的签名)的集合Ω_D来表示。使用一个形式逻辑,我们可以描述对Ω_D上的事件。通过组合逻辑和PPGM,我们还可以定义Ω_D上的概率分布P_D,并用它来计算一个事件的概率。我们考虑一个逻辑,记为PLA,其真值在单位区间内,使用聚合函数(例如算术平均数、几何平均数、最大值和最小值)代替量词。然而,我们面临计算效率问题,这个问题是将统计关系AI中的方法更广泛地应用于实际应用的障碍。我们通过证明在对PPGM和句子Φ的某些假设下,所述概率将在D的大小趋于无穷大时收敛。通过展示每个仅包含“可接受”聚合函数(如算术平均数、几何平均数、最大值和最小值)的公式Φ(x1,……,xk)都是渐进等价于一个没有聚合函数的公式ψ(x1,……,xk)来获得收敛结果。