The numerical approximation of expectations of quantities depending on random parameters is one of the key tasks in uncertainty quantification. While this task is nowadays well understood for functions with random input parameters, the goal of this project is to consider the expectation of pseudodifferential operators with order below a critical threshold. To do so, we need to overcome the curse of dimensionality for numerical quadrature, the nonlocality of the pseudodifferential operators, and the fact that those operators are often only accessible from input-output pairs.