Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 4

A beta distribution is a convenient prior for a probability $p\in (0,1)$ and is conjugate to the binomial distribution of the death count. Match its mean $m=0.05$ and variance $v=0.02^2=0.0004$. For $p\sim \mathrm{Beta}(\alpha ,\beta )$,The moment matching for a beta prior formula givesConditional on $p$, use $D\sim \mathrm{Binomial}(90,p)$ and the observed count $D=9$. Multiplying its likelihood $p^9(1-p{)}^{81}$ by the prior density gives the Bayesian posterior through Beta-binomial conjugacy:Under squared-error loss, take its posterior mean as the point estimate. A 95% equal-tail credible interval is given by its 0.025 and 0.975 quantiles. Numerically,Matching two moments does not uniquely determine a prior distribution; the beta family is a justified convenient choice, not a conclusion forced by those moments. Its transfer to this hospital presumes that the historical between-hospital distribution is relevant to the present risk and case mix.

The posterior mean is a weighted average of the sample proportion and prior mean:Thus it is pulled from 10% toward the prior's 5%, giving about 7.17%. This is an affine shrinkage estimator for a binomial proportion. The prior carries substantial concentration relative to the 90 observations, so the posterior is also more precise under this model: its standard deviation is about $0.01785$, compared with the binomial plug-in standard error $\sqrt{0.1(0.9)/90}=0.03162$. The posterior 95% credible interval $[0.04076,0.11035]$ is narrower and shifted down compared with the simple Wald interval $[0.03802,0.16198]$ based on $9/90$ alone.

The point estimate shrinks toward the historical mean, and the interval is narrower under the informative prior. A credible interval and a frequentist confidence interval have different probability interpretations; the comparison does not make the prior's relevance automatic.

For any prior density $\pi (p)$, the Bayes estimate under squared-error loss is the posterior mean:For the matched beta distribution, the integrals are beta integrals and their ratio isThe posterior mean minimizes posterior expected squared error because $\mathbb{E}[(p-a{)}^2\mid D]=\mathrm{Var}(p\mid D)+(a-\mathbb{E}[p\mid D]{)}^2$. The phrase Bayes point estimate depends on the loss: under absolute-error loss it would be a posterior median, while the posterior mode is a different summary and here equals $(14.8875-1)/(207.75-2)\approx 0.06750$.

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.

Use a hierarchical Bayesian model with binomial observation errors and a common distribution for the underlying risks:Give $(\alpha ,\beta )$ a shared hyperprior, or parameterize them by mean $m=\alpha /(\alpha +\beta )$ and concentration $\kappa =\alpha +\beta$ and put priors on $m\in (0,1)$ and $\kappa >0$. Conditional on these hyperparameters, the risks are independent and identically distributed; after integrating them out they are exchangeable random variables with shared uncertainty. The data estimate the overall risk level and between-hospital variation. Conditional Beta-binomial conjugacy givesand full Bayes point estimates under squared-error loss areAn Empirical Bayes method instead estimates the hyperparameters from the marginal beta-binomial distribution likelihoodand inserts their fitted values in the conditional posterior means. Both approaches provide partial pooling, stronger for smaller hospitals. Full Bayes also propagates hyperparameter uncertainty into the credible intervals. If systematic covariates are needed, one can place exchangeable residual effects in a logistic regression model rather than assuming a common unconditional risk distribution.