The basic process for reduced order modeling is to generate or measure training data, use the training data to construct a reduced order model (ROM) forthe data space, generate or measure test data, then project the test data onto to the ROM. In order to quantify uncertainties with respect to the use of the ROM, it is necessary to understand the errors and uncertainties associated with each step in the process, but simply propagating errors from one step to the next in quadrature overestimates the total error in many applications. In this work we present an extension of Probabilistic Principal Component Analysis (PPCA) for computing a reduced order model coupled to a Bayesian model for projecting onto the ROM, which allows for an estimate of the uncertainty associated with use of the ROM, accounting for the correlations in the errors at each stage of the process. The efficacy of the approach is demonstrated with applications to real-time radiation transport estimation from simulated distributed radiation sensor systems. Co-authors: Indika Udagedara, Clarkson University; Brian Helenbrook, Clarkson University; Jared Catenacci, Nevada National Security Site; Stephen Mitchell, Nevada National Security Site