= Solution
<Exchangeability> means that the joint prior law of the hospital risks is unchanged by relabelling hospitals; it permits different realized $p_j$. We should not assume unconditional <exchangeability> when known hospital characteristics predict systematic risk differences. Examples include severity and age of patients, elective versus emergency operations, procedure mix, specialist referral status, or a relationship between surgical volume and risk. Mixing such groups in one unconditional prior confounds these differences with unexplained hospital variation.
Use <covariates> or clinically comparable strata first; only remaining hospital effects might reasonably be <exchangeable random variables>. This is <conditional exchangeability of hospital risks>. Different sample sizes alone do not refute <exchangeability> of the underlying risks, although a known association between size and risk would matter. Nor does variation in the observed death proportions refute <exchangeability>: sampling variability and genuine random heterogeneity are both allowed.
\b[Do not pool hospitals as exchangeable risks when their known <case mix>, procedures or other relevant characteristics distinguish their expected risk distributions.]
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