I want to point out a characteristic of Bayesian practice that is implicit in several of the answers to L. Zadeh's question, and can reconcile several of the responses so far. In short, a Bayesian approach engages the decision-maker in an interactive "game" that eventually discovers the decision-choice. From the Bayesian point of view the question what to do about incomplete information (really an incomplete model) raises the question, why was the game interrupted just at this point? Implicit in the Bayesian formulation is that I, the analyst, pick the questions I want to ask. And the decision maker understands that its worth participating in the game because they have something riding on the outcome of their decision.
Rolf Haenni particularly lucid response includes a nice example where he wrote: > Let me illustrate this by a simple example. Suppose we know that: > 1) A implies X > 2) p(A) = 0.1 > What can be said about X? ... > > What would Bayesians do in such a case? Well not to be glib, the short answer is, if the decision maker volunteered this information so far, what impediment is there to her revealing the rest of her mental model? We see this investigative flavor of the analysis in some of the other discussions: In David Poole's response: > The captain is just supposed to answer "near" questions. We > don't ask her theoretical questions, just "are we now near > land"?... a further indication of this kind of interaction in a fragment from Kathryn Blackmond Laskey: > ...I would be well-advised to consult other > sources ... These fragments, admittedly taken out of context, describe an analyst in an active role, who is engaged in eliciting information and is not constrained with just using the information at hand. So, if the Bayesian presumes access to the persons involved in the decision, then without that access clearly "P is indeterminate", to use L. Zadeh's term. Well but, one might ask, what if the analyst is REQUIRED to make a decision on an incomplete basis? As a Bayesian then I might ask, what are the rules for this new game? For instance, in the new game does the unavailability of information imply something about its contents? Is there really a decision maker from whom I can elicit the problem or is she just a hypothetical construct? Perhaps the game aspect is somewhat tangential to just the question L. Zadeh proposed. It is instructive however to anyone who wants to apply a Bayesian method in a practical setting -- for them to know that a large part of their day-to-day effort as analyst will be to engage (interrogate might be a better term) the problem's stakeholders and not to be constrained to use just the information that comes across their desk.
