Download 2 Graphical Models in a Nutshell

2.4
Learning
51
A simple approach is simply to take some fixed Dirichlet distribution, e.g.,
Dirichlet (α, α, α, . . . , α), for every parameter, where α is a predetermined constant.
A typical choice is α = 1. This prior is often referred to as the K2 prior, referring
to the name of the system where it was first used.
A more sophisticated approach is called the BDe prior. We elicit a prior distribution P over the entire probability space and an equivalent sample size M for
the set of imaginary samples. We then set the parameters as follows:
αxi |pai = M · P (xi , pai ).
This choice avoids certain inconsistencies exhibited by the K2 prior. We can
represent P as a Bayesian network, whose structure can represent our prior about
the domain structure. Most simply, when we have no prior knowledge, we set P to be the uniform distribution, i.e., the empty Bayesian network with a uniform
marginal distribution for each variable.
The BDe score turns out to satisfy an important property. Two networks are
said to be I-equivalent if they encode the same set of independence statements.
Hence based on observed independencies we cannot distinguish between I-equivalent
networks. This suggests that based on observing data cases, we do not expect to
distinguish between equivalent networks. The BDe score has the desirable property
that I-equivalent networks have the same score, or are score-equivalent.
2.4.3.2
Search
We now have a well-defined optimization problem. Our input is
training set D;
scoring function (including priors, if needed);
a set GG of possible network structures (incorporating any prior knowledge).
Our desired output is a network structure (from the set of possible structures) that
maximizes the score.
It turns out that, for this discussion, we can ignore the specific choice of score.
Our search algorithms will apply unchanged to all three of these scores.
An important property of the scores that affects the efficiency of search is their
decomposability. A score is decomposable if we can write the score of a network
structure G:
FamScore(Xi | PaGi : D)
score(G : D) =
i
All of the scores we have considered are decomposable. Another property that is
shared by all these scores is score equivalence; if G is independence-equivalent to
G , then score(G : D) = score(G : D).