7.2 The evolution of the set of priors
The fact that all elements of M (P*) may be updated consistently does not
fully resolve the problems associated with updating multiple priors. Halpern
(2003) makes the point that deriving the Bayesian posterior for each prior
separately is problematic, given that, in general, observed signals will have
different likelihood for each of the different priors. It is natural to consider
whether there are conditions under which this problem will not arise, and, if
so, what are the implications for the structure of knowledge.
Examining the consistency condition above, we observe that there is no
problem with propositions ,p'' E P-г,. By additivity, the condition includes
the special cases
μ (p ^ p'') = μ (p) ^ μ (p'')
and
μ (p' ^ p'') = μ (p') ^ μ (p")
For the first choose p E Pг, p' E P* ,p'' E P',, apply the condition first with
p,p',p'', then with p, ^p',p'' and add. Similarly for the second.
On the other hand, by hypothesis, the elements of P* yield distinct con-
ditional probabilities for members of P'.Hence, we cannot expect a criterion
that will apply for all possible histories and time periods.
Suppose instead, we require that for all elements p E P^l ∩ N(t), p' E P*
μ (p ^ p') = μ (p) ^ μ (p')
Then considered propositions for which the truth value is observed at time t
are independent of all elements of P*, and therefore do not affect the weight-
ing that might be given to different priors.
Under the conditions discussed above, neither considered nor unconsid-
ered propositions provide any information about P* at time t. Hence, if the
set of multiple priors is to evolve, it can only do so if the truth value of some
p' E P* is revealed at time t, that is, if P* ∩ N(t) is nonempty.
8 Example
Suppose
у = βx + z + w
and that it is implicitly known that E [w] = 0 and that x, w and z are
independent.
Now suppose there exist one or more alternative hypotheses about the
value of z, each of which induces a prior distribution on у for given x. Then
13