On the Consistent Recovery of Joint Distributions from Conditionals

Self-supervised learning methods that mask parts of the input data and train models to predict the missing components have led to significant advances in machine learning. These approaches learn conditional distributions $p(x_T \mid x_S)$ simultaneously, where $x_S$ and $x_T$ are subsets of the observed variables. In this paper, we examine the core problem of when all these conditional distributions are consistent with some joint distribution, and whether common models used in practice can learn consistent conditionals. We explore this problem in two settings. First, for the complementary conditioning sets where $S \cup T$ is the complete set of variables, we introduce the concept of path consistency, a necessary condition for a consistent joint. Second, we consider the case where we have access to $p(x_T \mid x_S)$ for all subsets $S$ and $T$. In this case, we propose the concepts of autoregressive and swap consistency, which we show are necessary and sufficient conditions for a consistent joint. For both settings, we analyze when these consistency conditions hold and show that standard discriminative models \emph{may fail to satisfy them}. Finally, we corroborate via experiments that proposed consistency measures can be used as proxies for evaluating the consistency of conditionals $p(x_T \mid x_S)$, and common parameterizations may find it hard to learn true conditionals.