Another extension of Formulation 9.5 is to allow multiple observations, , , , , before making a decision. Each is assumed to belong to an observation space, . A strategy, , now depends on all observations:
Under the nondeterministic model, is specified for each and . The set is replaced by
Under the probabilistic model, is specified instead. It is often assumed that the observations are conditionally independent given . This means for any , , and such that , . The condition in (9.26) is replaced by . Applying Bayes' rule, and using the conditional independence of the 's given , yields
Conditional independence allows a dramatic simplification that avoids the full specification of . Sometimes the conditional independence assumption is used when it is incorrect, just to exploit this simplification. Therefore, a method that uses conditional independence of observations is often called naive Bayes.
Steven M LaValle 2012-04-20