We consider an experiment with k two-level factors and insufficient resources to observe all 2k possible runs. The candidate designs are 2k-p fractional factorials. To approach the problem, we do not assume a parametric model and instead think of experimental observations as realizations of a stationary Gaussian process X operating on the design space. Pre-experimental knowledge is formally incorporated in the prior distribution of X , making the approach Bayesian. However, instead of demanding a precise prior for X, we seek designs that are optimal for general families of processes. We define several families of interest and prove each family is closed under various operations. In evaluating designs, we examine criteria such as D-, A-, G-, E-, and c-optimality, paying closest attention to D-optimality. Within a family of processes, we consider different ways to bring a distribution towards near-independence and near-dependence, then characterize the asymptotically optimal fractional factorial. Often the maximum resolution-minimum aberration design is found optimal in all cases. However, for some k and p, a second design turns out to be optimal for certain subfamilies of priors.