AUTOMATED DERIVATION OF CONDITIONAL PROBABILITIES IN DECISION TREES.
FA Sonnenberg, MP Wellman, MH Eckman, C Fleming, SL Marshall, and SG
Pauker.  New England Medical Center, Boston, MA and Massachusetts
Institute of Technology, Cambridge, MA.
   In many decision trees, probabilities must be conditioned upon events
which are modeled by branches of nodes of the tree.  An example is the
probability of a diagnosis (the conditioned event) at a point in the
tree distal to a branch which models a given test result (the
conditioning event).  In general, a conditional probability is equal to
the probability of the conditioned event given the conditioning data.
In practice, probability revisions can become extremely complicated and
are often a source of errors in decision trees.  We have implemented a
method for automatically revising probabilities within a decision tree
program operating on an AI workstation.
  Our method makes use of a probability tree (a representation of the
event space for the decision problem) which consists of chance nodes
whose branches represent all events on which probabilities may be
conditioned (for example, a test result or the presence or absence of
disease).  Each path through the probability tree represents a unique
combination of events.  The path probability, the product of all
individual probabilities along a path, is equal to the probability of
the occurrence of the combination of events corresponding to that path.
The overall probability of any event is equal to the sum of the path
probabilities for all paths which include that event.  The probability
revision facility uses the event space model to derive arbitrary
conditional probabilities specified in terms of the primitive events.
Revised conditional probabilities can be represented as symbolic
expressions thus ensuring appropriate linkage of quantities during
sensitivity analysis.
  Probability trees can be constructed in whatever ordering seems most
natural to the analyst or fits best with the available data and
probability assessments.  A decision tree, in contrast, imposes an
ordering of events based on informational and temporal dependencies that
may be at odds with conveniently available data.  Probability trees and
decision trees used in combination, allow the analyst to isolate
complicated components of the probabilistic model and use them in their
most convenient form.  This will promote easier and less errorful
construction of decision trees.

