  The property of perfectness plays an important role in the theory of Bayesian
networks. First, the existence of perfect distributions for arbitrary sets of
variables and directed acyclic graphs implies that various methods for reading
independence from the structure of the graph (e.g., Pearl, 1988; Lauritzen,
Dawid, Larsen & Leimer, 1990) are complete. Second, the asymptotic reliability
of various search methods is guaranteed under the assumption that the
generating distribution is perfect (e.g., Spirtes, Glymour & Scheines, 2000;
Chickering & Meek, 2002). We provide a lower-bound on the probability of
sampling a non-perfect distribution when using a fixed number of bits to
represent the parameters of the Bayesian network. This bound approaches zero
exponentially fast as one increases the number of bits used to represent the
parameters. This result implies that perfect distributions with fixed-length
representations exist. We also provide a lower-bound on the number of bits
needed to guarantee that a distribution sampled from a uniform Dirichlet
distribution is perfect with probability greater than 1/2. This result is
useful for constructing randomized reductions for hardness proofs.
