In this paper, we cast the problem of combinatorial auction design in a Bayesian framework in order to incorporate prior information into the auction process and minimize the number of rounds. We develop a generative model of bidder valuations and market prices such that clearing prices become maximum a posteriori estimates given observed bidder valuations. The model then forms the basis of an auction process which alternates between refining estimates of bidder valuations, and computing candidate clearing prices. We provide an implementation of the auction using assumed density filtering to estimate valuations and expectation maximization to compute prices. An empirical evaluation over a range of valuation domains demonstrates that our Bayesian auction mechanism is very competitive against a conventional combinatorial clock auction, even under the most favorable choices of price increment for this baseline.