4. Exchangeable prior distributions: suppose it is known a priori that the 2.J p

Question

4. Exchangeable prior distributions: suppose it is known a priori that the 2.J parameters 0,, , 2/ are clustered into two groups, with exactly half being drawn from a N(1,1) distribution, and the other half being drawn from a N() distribution, but we have not observed which parameters come from which distribution (a) Are 1, .. . , 2/ exchangeable under this prior distribution? (b) Show that this distribution cannot be written as a mixture of independent and iden tically distributed components (c) Why can we not simply take the limit as J oo and get a counterexample to de Finetti's theorem? See Exercise 8.10 for a related problem.

Explanation / Answer

Ans a: It doesnt provide with the distribution so you be sure whether or not distribution is exchangeable.

Definition of exchangeability:

Lets assume we have a sequence of x1, x2,.....xn.

Then we say that these quantities are exchangeable if P(x1, x2,....xn) = P(xp1, xp2,.......,xpn)

where p represent any permutation of x1, x2, ....,xn.

It will be exchangeable if and only if we can write down the same joint probability of any permuted sequence.

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