The name is absent

above, an induced measure on P^t, which will be denoted μ_k and we define

It is easy to check that this definition agrees with that given above for the
case P * = {p', p'}

7.1 Consistent updating with multiple priors

The definition of consistent updating with a unique measure μ can be ex-
tended in a straightforward fashion to the case of a given λ∈M (P*) . For
each k,i,t we may define μ^l_{t k} as above and set

X = ∑ Wh
к

and similarly for μt+ _{1 k} and λ^+ _{1 k} For all i,t,p,k,

μ^ltlι,k ^{(E (}P⁾⁾ = μ^lt+ι,k^{(E (}P⁾⁾

then for all i,t,p

⅛+ 1 (E (p)) = ‰ (E (p))

so that consistent Bayesian updating for each μ_k is sufficient to ensure con-
sistent Bayesian updating for all λ ∈ M (P*) . Necessity is trivial.

Now, by the definition of μ_k, we obtain: ⁵

Proposition 2 Consistent Bayesian updating for all X ∈ M (P*) holds if
and only if, for any p,p',p'' such that p ∈ P^t, p' ∈ P * ,p" ∈ P ',

μ (p Λ p' Λ p") = μ (p Λ p') Λ μ (p'')

⁵By additivity, the condition includes the special cases

P ⁽p ^λ p") = P (p) ^λ P (p")

and

p (p' Λ p") = p (p') Λ p (p").

For the first choose p ∈ P∖ p' ∈ P*,p'' ∈ P ^l,, apply the condition first with p,p',p'',
then with p, -∣p', p'' and add. Similarly for the second.

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