Let be a general parameter taking values in a parameter space . Let '' '' = ' be a partition of into

Let θ be a general parameter taking values in a parameter space „¦. Let „¦'' ˆª „¦'' = „¦' be a partition of „¦ into two disjoint sets „¦' and „¦''. We want to choose between two decisions: d' says that θ ˆˆ „¦', and d'' says that θ ˆˆ „¦''. We have the following loss function:

We have two choices for expressing this decision problem as a hypothesis-testing problem. One choice would be to define H0 : θ ˆˆ „¦' and H1 : θ ˆˆ „¦''. The other choice would be to define H0 : θ ˆˆ „¦'' and H1 : θ ˆˆ „¦'. In this problem, we show that the Bayes test makes the same decision regardless of which hypothesis we call the null and which we call the alternative.
a. For each choice, say how we would define each of the following in order to make this problem fit the hypothesis-testing framework described in this section: w0, w1, d0, d1, „¦0, and „¦1.
b. Now suppose that we can observe data X = x and compute the posterior distribution of θ, ξ(θ|x). Show that, for each of the two setups constructed in the previous part, the Bayes test chooses the same decision d' or d''. That is, observing x leads to choosing d' in the first setup if and only if observing x leads to choosing d' in the second setup. Similarly, observing x leads to choosing d'' in the first setup if and only if observing x leads to choosing d'' in the second setup.

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