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Known Structure

When the structure is known, the problem is to learn the parameters for the graph from the database. We illustrate this with an example taken from [17].

Consider the flipping of a thumbtack. It can either land on the ``head'' or ``tail''. We have a database D with outcomes, and want to learn the distribution of the variable which we can call $\theta$.

It can be shown that:

\begin{displaymath}
P(\theta \mid h \; heads,t \; tails, \epsilon)=c\theta^h(1-\theta)^t
P(\theta \mid \epsilon)
\end{displaymath}

Where $\epsilon$ is the background knowledge, h the number of heads, t the number of tails, and c is a normalization constant. This means that once we have assessed the prior distribution for $\theta$, we can determine the posterior distribution for any database. The order of which we observe outcomes is irrelevant, and h and t are a sufficient statistic for the database.

This formula can now be used for estimating the parameters in this situation with only one variable. In more complex situations with more variables the formulas will differ.



Torgeir Dingsoyr
2/26/1998