A Bayesian analysis for 2 to the k power factorial arrangements of treatments is presented in this paper. To perform the analysis, an experimenter must specify prior distributions on an orthogonal set of linear functions representing the main effects and interactions and on a function representing the grand mean. The solution is relatively straightforward and not unduly complex numerically. The independence of the functions and the fact that they represent the primary concerns in a factorial arrangement aids in the conceptualization of prior distributions. The independence also facilitates the computation of likelihoods since univariate normal distributions must be evaluated rather than a multivariate normal distribution. An experimenter may desire posterior distributions on other comparisons of the treatment means that are not included in the orthogonal set. For these situations a method is presented whereby posterior distributions for any Scheffe type contrast of the means can be generated without great difficulty. Although the solution given in this paper is specifically for 2 to the k power factorial arrangements, the method is immediately generalizable to any factorial. (Author/CTM)