We consider the inverse sensitivity analysis of a map from a set of parameters and data to a quantity of interest. We are particularly interested in implicitly-defined maps, e.g. involving the solution of a differential equation. The inverse problem is to describe the random variation in the input that leads to an imposed or observed random variation in the output quantity. We formulate this as an ill-posed inverse problem for an integral equation using the Law of Total Probability. We then describe a computational method for computing solutions that has two stages. In the first part, we approximate the unique set-valued solution to the inverse of the integral equation using derivative information. In the second part, we apply basic ideas from measure theory to compute the approximate probability measure on the parameter and data space that solves the integral equation. We discuss convergence of the method, and explain how to use the method to compute the probability of events in the input (parameter) space. The talk is illustrated with a number of examples. We also discuss briefly the numerical analysis (accuracy) of the method and the consideration of multiple quantities of interest and data assimilation.