The Random Conditional Distribution for Higher-Order Probabilistic Inference

The need to condition distributional properties such as expectation, variance, and entropy arises in algorithmic fairness, model simplification, robustness and many other areas. At face value however, distributional properties are not random variables, and hence conditioning them is a semantic error and type error in probabilistic programming languages. On the other hand, distributional properties are contingent on other variables in the model, change in value when we observe more information, and hence in a precise sense are random variables too. In order to capture the uncertain over distributional properties, we introduce a probability construct -- the random conditional distribution -- and incorporate it into a probabilistic programming language Omega. A random conditional distribution is a higher-order random variable whose realizations are themselves conditional random variables. In Omega we extend distributional properties of random variables to random conditional distributions, such that for example while the expectation a real valued random variable is a real value, the expectation of a random conditional distribution is a distribution over expectations. As a consequence, it requires minimal syntax to encode inference problems over distributional properties, which so far have evaded treatment within probabilistic programming systems and probabilistic modeling in general. We demonstrate our approach case studies in algorithmic fairness and robustness.