Julie asks,
> Is there any algorithms of Bayesian Network to work directly on the
> mixture of continuous and categorical variables?
[...]
I know of a few possible approaches. One is to discretize all the
continuous variables. Another is to approximate the distributions
of the continuous variables by conditional Gaussians; exact
algorithms are known for loopy Bayesian networks with discrete and
conditional Gaussian continuous variables, although IIRC no
discrete variable can be a child of a continuous variable.
If the distributions involved admit a partial result which is
exact, you should exploit that. For example, the sum of several
variables with arbitrary continuous distributions is the
convolution of the densities, for which a fast, accurate approx-
imation can be computed via the fast Fourier transform. But many
or most probabilistic relations won't have such easy answers.
So what I did for the problems I worked on in my dissertation was
this: compute exact results to the extent that such are known,
then apply approximations (via numerical integration) for cases
that don't have exact results available. The basic algorithm was
Pearl's polytree algorithm, since the required operations could
be represented in terms of simple integrals (even if the results
are difficult to compute). Doubtless someone could take the same
exact/approximate approach for other algorithms. I handled loops
by conditioning.
A key feature of this scheme is that the Bayesian network can be
represented using the original distributions from the problem
domain, and at least some of the results will also have some
ordinary parametric form, rather than being uninterpretable
approximations. These considerations were important in the
problem domain in which I was working.
I used an adaptive quadrature algorithm in one dimension and a
quasi-Monte Carlo algorithm in two or more dimensions. The result-
ing densities are represented by mixtures of Gaussians or monotone
cubic splines, depending on the day of the week. :)
I built up a catalog of partial results, some of which are exact
and some of which are good approximations (better in a specific
case than the general calculation by numerical integration). The
catalog is in an appendix of my dissertation.
I found that the mixed exact/approximate strategy worked well for
the problems I tried, which were all in an engineering domain.
You can find more details in my dissertation: see
http://civil.colorado.edu/~dodier/publications.html
and for the source code I developed and further notes,
http://civil.colorado.edu/~dodier/sonero-mirror/index.html
Hope this helps,
Robert Dodier
This archive was generated by hypermail 2b29 : Fri Apr 07 2000 - 07:21:24 PDT