A Bayesian analysis for factorial experiments is presented, using finite mixture distributions to model the main effects and interactions. This allows both estimation and an analogue of hypothesis testing in a posterior analysis using a single prior specification. A detailed formulation based on this approach is provided for the case of the two-way model with replication, allowing interactions. Issues in formulating a suitable prior are discussed in detail, and, in the context of two illustrative applications, we discuss implementation, presentation of posterior distributions, sensitivity and performance of the Markov chain Monte Carlo methods that are used.