  Three classes of algorithms to learn the structure of Bayesian networks from
data are common in the literature: constraint-based algorithms, which use
conditional independence tests to learn the dependence structure of the data;
score-based algorithms, which use goodness-of-fit scores as objective functions
to maximise; and hybrid algorithms that combine both approaches.
Constraint-based and score-based algorithms have been shown to learn the same
structures when conditional independence and goodness of fit are both assessed
using entropy and the topological ordering of the network is known (Cowell,
2001).
  In this paper, we investigate how these three classes of algorithms perform
outside the assumptions above in terms of speed and accuracy of network
reconstruction for both discrete and Gaussian Bayesian networks. We approach
this question by recognising that structure learning is defined by the
combination of a statistical criterion and an algorithm that determines how the
criterion is applied to the data. Removing the confounding effect of different
choices for the statistical criterion, we find using both simulated and
real-world complex data that constraint-based algorithms are often less
accurate than score-based algorithms, but are seldom faster (even at large
sample sizes); and that hybrid algorithms are neither faster nor more accurate
than constraint-based algorithms. This suggests that commonly held beliefs on
structure learning in the literature are strongly influenced by the choice of
particular statistical criteria rather than just by the properties of the
algorithms themselves.
