A generative probabilistic model for relational data consists of a family of
probability distributions for relational structures over domains of different
sizes. In most existing statistical relational learning (SRL) frameworks, these
models are not projective in the sense that the marginal of the distribution
for size-$n$ structures on induced sub-structures of size $k<n$ is equal to the
given distribution for size-$k$ structures. Projectivity is very beneficial in
that it directly enables lifted inference and statistically consistent learning
from sub-sampled relational structures. In earlier work some simple fragments
of SRL languages have been identified that represent projective models.
However, no complete characterization of, and representation framework for
projective models has been given. In this paper we fill this gap: exploiting
representation theorems for infinite exchangeable arrays we introduce a class
of directed graphical latent variable models that precisely correspond to the
class of projective relational models. As a by-product we also obtain a
characterization for when a given distribution over size-$k$ structures is the
statistical frequency distribution of size-$k$ sub-structures in much larger
size-$n$ structures. These results shed new light onto the old open problem of
how to apply Halpern et al.'s "random worlds approach" for probabilistic
inference to general relational signatures.

通过利用无限可交换阵列的表示定理，介绍了一类有向图形潜变量模型，精确对应于投影关系模型，此外，还获得了关于什么时候给定大小 - k 结构上的分布是在更大的 size-n 结构中的大小 - k 子结构的统计频率分布的表征。