We develop a framework for studying optimization problems over distributions. In the worst case these problems are typically NP-hard, even to approximate, and are usually addressed in practice by a range of heuristics. To understand the effectiveness of such approaches, we define a class of algorithms, called statistical algorithms, that captures many natural algorithms, eg local search, MCMC, and simulated annealing. We show that for specific optimization problems over distributions, namely, for variants of MAX-XOR-SAT, k-clique, and moment maximization, the complexity of any statistical algorithm grows exponentially in their input parameters. Our techniques are inspired by (and generalize) the statistical query model in learning theory, and in parallel to that concept, we can prove lower bounds on statistical algorithms for optimization problems over distributions using a single parameter that we call statistical dimension. Although this does not fully rule out efficient algorithms for such problems, it indicates the limitations of known heuristics.