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Let
be the
evidence. Then the probability of the hidden variable given evidence
is obtained by applying Bayes' rule as follows:
In particular, when we observe all but the choice
variable, i.e.,
and
, we obtain the
posterior probability distribution of
:
![\begin{displaymath}
Pr[z=k\vert V=x]\;\equiv\;Pr[z=k\vert x]\;=\;
\frac{\lambda_k T^k(x)}{\sum_{k'}\lambda_{k'}T^{k'}(x)}.
\end{displaymath}](img123.png) |
(4) |
The probability distribution of a given subset of
given the evidence is
Thus the result is again a mixture of the results of inference
procedures run on the component trees.
Journal of Machine Learning Research
2000-10-19