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#

Summary and Future Work

We have described a graphical representation for probabilistic
dependencies similar to the Bayesian network called a dependency
network. Like a Bayesian network, a dependency network has a graph
and a probability component. In its consistent form, the graph
component is a cyclic directed graph such that a node's parents render
that node independent of all other nodes in the network. As in a
Bayesian network, the probability component consists of the
probability of a node given its parents for each node--the local
distributions.
In practice, for computational reasons, we learn the structure and
parameters of a dependency network for a given domain by independently
performing a classification/regression for each variable in the domain
with inputs consisting of all variables except the target variable.
The parameterized model for each variable is the local distribution
for that variable; and the structure of the network reflects any
independencies discovered in the classification/regression process
(via feature selection). As a result of this learning procedure, the
dependency network is usually inconsistent--that is, it is not the
case that the local distributions can be obtained via inference from a
single joint distribution for the domain. Nonetheless, because each
local distribution is learned from the same data, the local
distributions are ``almost'' consistent when there is adequate data.
Consequently, as a useful heuristic, we can apply the machinery of
Gibbs sampling to this network to extract a joint distribution for the
domain and to answer probabilistic queries. Experiments on real data
show this approach to yield accurate predictions.
In addition to their application to probabilistic inference, we have
shown that dependency networks are useful for collaborative filtering
(the task of predicting preferences) and for the visualization of
acausal predictive relationships. In fact, Microsoft has included
dependency networks in two of its products--SQL Server 2000 and
Commerce Server 2000--for both the collaborative filtering and data
visualization tasks.
The intent of our paper has been to introduce the basic concepts and
applications of dependency networks. Consequently, there is
significant additional work to be done. For example, many of the
results described in this paper can be extended to domains that
include continuous variables. In addition, more work is needed to
characterize those situations in which the joint distribution defined
by an (inconsistent) dependency network is insensitive to errors in
the learned local distributions. As another example, experimental
work is needed to examine the predictive accuracy of dependency
networks across a variety of domains using alternative methods for
classification and regression. It may also be useful to consider
pseudo-Gibbs sampling methods that resample variables in random rather
than fixed order.
Finally, we note that the representation itself can be generalized.
Recall that a dependency network is useful for collaborative filtering
primarily because the network stores in its local distributions
precisely the probabilistic quantities needed by the ranking
algorithm. In general, we can construct a ``query network'' that
directly learns probabilities corresponding to a set of given queries.
As an illustration, suppose we have a domain consisting of variables
and , and we know we will be answering the query
. We can learn this distribution directly from data by
performing a series of (independent) classifications/regressions. We
can construct the classifications/regressions and
and use multiplication to answer the query. Alternatively,
we may construct the classifications/regressions and
and use a pseudo-Gibbs sampler to answer the query. In
either case, with sufficient data, the conditional probabilities
learned will be ``almost'' consistent with the true distribution, and
are likely to produce accurate answers to the query.

** Next:** Acknowledgments
** Up:** Dependency Networks for Inference,
** Previous:** Data Visualization
Journal of Machine Learning Research,
2000-10-19