The problem considered in this paper is detection and estimation of multiple radiation sources using a time series of radiation counts from a collection of sensors. A Bayesian framework is adopted. Source detection is approached as a model selection problem in which competing models are compared using partial Bayes factors. Given the number of sources, the posterior mean is the minimum mean square error estimator of the source parameters. Exact calculation of the partial Bayes factors and the posterior mean is not possible due to the presence of intractable integrals. Importance sampling using progressive correction is proposed as a computationally efficient method for approximating these integrals. Previously proposed algorithms have been restricted to one or two sources. A simulation analysis shows that the proposed methods can detect and accurately estimate the parameters of four sources with reasonable computational expense.