The analysis of network connections, diffusion processes and cascades requires evaluating properties of the diffusion network. Properties of interest often involve variables that are not explicitly observed in real world diffusions. Connection strengths in the network and diffusion paths of infections over the network are examples of such hidden variables. These hidden variables therefore need to be estimated for these properties to be evaluated. In this paper, we propose and study this novel problem in a Bayesian framework by capturing the posterior distribution of these hidden variables given the observed cascades, and computing the expectation of these properties under this posterior distribution. We identify and characterize interesting network diffusion properties whose expectations can be computed exactly and efficiently, either wholly or in part. For properties that are not 'nice' in this sense, we propose a Gibbs Sampling framework for Monte Carlo integration. In detailed experiments using various network diffusion properties over multiple synthetic and real datasets, we demonstrate that the proposed approach is significantly more accurate than a frequentist plug-in baseline. We also propose a map-reduce implementation of our framework and demonstrate that this can analyze cascades with millions of infections in minutes. © 2014 ACM.