Studies on simulation input uncertainty are often built on the availability of input data. In this paper, we investigate an inverse problem where, given only the availability of output data, we nonparametrically calibrate the input models and other related performance measures of interest. We propose an optimization-based framework to compute statistically valid bounds on input quantities. The framework utilizes constraints that connect the statistical information of the real-world outputs with the input-output relation via a simulable map. We analyze the statistical guarantees of this approach from the view of data-driven distributionally robust optimization, and show how they relate to the function complexity of the constraints arising in our framework. We investigate an iterative procedure based on a stochastic quadratic penalty method to approximately solve the resulting optimization. We conduct numerical experiments to demonstrate our performances in bounding the input models and related quantities.