An intelligent agent uses known facts, including
statistical knowledge, to assign <I>degrees of belief</I> to
assertions it is uncertain about. We investigate three principled
techniques for doing this. All three are applications of the
<I>principle of indifference</I>, 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.<br>
 
The three methods we investigate have surprising characterizations:
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.<p>