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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.