An intelligent agent uses known facts, including statistical knowledge, to assign depees of belief to assertions it is uncertain about. We investigate three principled techniques for doing this. All three are applications of the principle of indifference, because they assign equal degree of belief to all basic "situations" consistent with the knowledge base. They differ because there are competing intuitions about what the basic situations are. Various natural patterns of reasoning, such as the preference for the most specific statistical data available, turn out to follow from some or all of the techniques. This is an improvement over earlier theories, such as work on direct inference and reference classes, which arbitrarily postulate these patterns without offering any deeper explanations or guarantees of consistency. The three methods we investigate have surprising characterisations: there are connections to the principle of maximum entropy, a principle of maximal independence, and a "center of mass" principle. There are also unexpected connections between the three, that help us understand why the specific language chosen (for the knowledge base) is much more critical in inductive reasoning of the sort we consider than it is in traditional deductive reasoning.