  In Bayesian network structure learning (BNSL), we need the prior probability
over structures and parameters. If the former is the uniform distribution, the
latter determines the correctness of BNSL. In this paper, we compare BDeu
(Bayesian Dirichlet equivalent uniform) and Jeffreys' prior w.r.t. their
consistency. When we seek a parent set $U$ of a variable $X$, we require
regularity that if $H(X|U)\leq H(X|U')$ and $U\subsetneq U'$, then $U$ should
be chosen rather than $U'$. We prove that the BDeu scores violate the property
and cause fatal situations in BNSL. This is because for the BDeu scores, for
any sample size $n$,there exists a probability in the form
$P(X,Y,Z)={P(XZ)P(YZ)}/{P(Z)}$ such that the probability of deciding that $X$
and $Y$ are not conditionally independent given $Z$ is more than a half. For
Jeffreys' prior, the false-positive probability uniformly converges to zero
without depending on any parameter values, and no such an inconvenience occurs.
