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Draw $P\sim \mathrm{Beta}(\alpha ,\beta )$, $X\mid P\sim \mathrm{Binomial}(n,P)$ and $Q\mid X\sim \mathrm{Beta}(\alpha +X,\beta +n-X)$, with $\alpha ,\beta >0$ and integer $n\ge 0$. The joint density of $(P,X,Q)$ is symmetric in $P,Q$, making them exchangeable random variables with identical beta distribution marginal distributions. The posterior beta normalization constant depends on $X$ and must be retained. The correlation coefficient is $n/(n+\alpha +\beta )$: this constructs tunable dependence without altering either marginal.

Unknown hospital event probabilities can be modelled as exchangeable random variables conditional on shared hyperparameters, for example $P_j\mid m,\kappa \sim \mathrm{Beta}(m\kappa ,(1-m)\kappa )$ independently. Different sample sizes affect the binomial observations but do not by themselves invalidate exchangeability of the risks. Known systematic covariate differences, such as case mix or procedure type, should instead enter the risk model; only residual effects may then be exchangeable. Exchangeability permits heterogeneous realized risks and is not an assertion that all risks are equal.

// Target: biology.bigb

For an infinite sequence of real exchangeable random variables, conditional on its invariant sigma-algebra of an exchangeable sequence, the coordinates are independent and identically distributed. In particular, for Borel sets $A_0,\dots ,A_p$,Here is the core proof. Let ${\mathcal{S}}_m$ denote invariance under permutations of coordinates $0,\dots ,m$. Symmetrization as conditional expectation identifies the empirical average of each indicator with its conditional expectation on ${\mathcal{S}}_m$. The reverse martingale convergence theorem gives its limit conditional on $\mathcal{S}$. Multiplying the averages sums over all ordered coordinate tuples, including repetitions. Symmetrization of the product itself sums over the distinct tuples. The fraction of repeated tuples is at most $\left(\genfrac{}{}{0ex}{}{p+1}{2}\right)/(m+1)$, so these bounded averages have the same limit. This proves the displayed factorization. Permutation invariance also makes the individual conditional laws identical. A regular conditional distribution on the real line packages that common law as a random probability measure, giving the equivalent mixture-of-iid formulation.

For a real sequence of exchangeable random variables, every event $A$ in its invariant sigma-algebra of an exchangeable sequence agrees almost surely with an event of its tail sigma-algebra. Put $F=1_A$. Approximate $F$ in $L^1$ by its conditional expectation on the first $n+1$ coordinates. Swap that block with the next $n+1$ coordinates. Invariance of $A$ and the law shows that the conditional expectation on the second block is an equally good approximation. It is measurable with respect to the tail after index $n$. The L1 contraction of conditional expectation then bounds $\Vert F-\mathbb{E}[F\mid {\mathcal{G}}_n]{\Vert }_1$ by twice the approximation error. The reverse martingale convergence theorem gives $F=\mathbb{E}[F\mid {\bigcap }_n{\mathcal{G}}_n]$ almost surely. Thresholding the latter at $1/2$ supplies the required tail event. Together with the reverse inclusion, the two sigma-algebras have identical probability completions.

Conditional on the pooled number $s$ of successes in $m+n$ exchangeable random variables that are binary, all placements of those successes are equally likely under a common success-chance mixture. The count in a chosen batch of size $n$ therefore has a hypergeometric distribution, with mass $\left(\genfrac{}{}{0ex}{}{s}{y}\right)\left(\genfrac{}{}{0ex}{}{m+n-s}{n-y}\right)/\left(\genfrac{}{}{0ex}{}{m+n}{n}\right)$. Under the uniform-count predictive property, $s$ is uniform on $0,\dots ,m+n$. Dividing the joint pooled-and-batch probability by the first batch's uniform marginal gives the posterior predictive distribution, with factor $(m+1)/(m+n+1)$. Substituting $s=x+y$ in the hypergeometric expression does not give a single normalized hypergeometric mass as $y$ varies, since its population success total then varies too.

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.

Two exchangeable random variables satisfy $(X,Y)\stackrel{d}{=}(Y,X)$: their joint probability distribution is unchanged by swapping the coordinates. Thus for every measurable set $A$,Exchangeability implies identical marginal distributions, but does not imply independence.

For each auxiliary count $x$, the correctly normalized joint expression above is symmetric in $p_M,p_T$. Summing it over $x=0,\dots ,n$ preserves that symmetry, so these are exchangeable random variables. Their marginal distributions are therefore identical. Since $p_M$ was generated from a beta distribution,The Beta-binomial exchangeable coupling changes the dependence, while leaving both prior distributions unchanged.

Calling the effects exchangeable random variables means that their joint prior distribution is unchanged by permuting study labels. In a hierarchical Bayesian model, conditional independent draws ${\beta }_j\mid \mu ,\tau \sim N(\mu ,{\tau }^2)$ achieve this, and integrating shared hyperparameters induces dependence between studies. This permits partial pooling without asserting that all effects are identical.

The assumption is reasonable when the studies concern comparable treatments, populations, outcomes and follow-up, and no known study characteristic gives one effect a systematically different prior center. Relevant differences can instead enter a linear regression for study effects, after which the residual effects may be exchangeable. The numerical table counts deaths, so its event probabilities are mortality probabilities and ${\beta }_j<0$ indicates a lower mortality odds ratio. Interpreting those counts as beneficial responses would reverse the clinical meaning.

Under the intended additional assumption of mutually independent, continuously calibrated null p-values, each test fails to reject with probability $0.95$. The familywise error rate is thereforeThere is a qualification to the printed use of “not correlated”: zero pairwise correlation coefficients do not generally imply mutual independence. A multivariate normal distribution with zero off-diagonal covariances does give independence, but an arbitrary biomarker distribution need not. The formula above requires independent rejection events, not just uncorrelated measurements.

The usual intuition for positively dependent tests is that they reject together, reducing the effective number of opportunities for a false positive. A rigorous sufficient assumption is positive association of random variables for the rejection indicators $I_1,\dots ,I_{10}$. Products of their non-rejection indicators are decreasing functions; association and induction giveThus under this stronger positive-dependence assumption the probability is no larger, and perfect dependence gives $0.05$ instead of $0.4013$. Strict inequality is not forced by every form of positive dependence.

Pairwise positive correlation coefficients alone, as printed, is insufficient. For a counterexample choose a vector of ten exchangeable random variables that are rejection indicators as follows: with probabilities $0.545,0.450,0.005$, respectively, reject none, reject one uniformly selected test, or reject all ten. Thenbut the familywise error rate is $0.455>1-0.95^{10}$. This can be realized with valid null p-values: conditional on the indicators, independently draw $p_i$ uniformly on $(0,0.05)$ if $I_i=1$ and on $(0.05,1)$ otherwise. Every $p_i$ is uniform on $(0,1)$, and $\mathrm{Cov}(p_i,p_j)=0.25\mathrm{Cov}(I_i,I_j)>0$. Taking biomarker statistics $X_i=1-p_i$ gives pairwise positively correlated null statistics with exactly these one-sided tests. Therefore the unconditional larger-or-smaller claim needs a specified dependence model.

Let ${\mathcal{G}}_n=\sigma (S_n,S_{n+1},\dots )$. Since $X_{n+j}=S_{n+j}-S_{n+j-1}$,Permuting $X_1,\dots ,X_n$ preserves their joint probability distribution and leaves every generator of ${\mathcal{G}}_n$ unchanged. Therefore, for each bounded ${\mathcal{G}}_n$-measurable $H$,This is the symmetry of exchangeable random variables. By the defining identity of conditional expectation, all $\mathbb{E}[X_j\mid {\mathcal{G}}_n]$ are equal almost surely. Summing these conditional expectations and using that $S_n$ is ${\mathcal{G}}_n$-measurable givesHence the conditional expectation of a summand given future partial sums isThe integrability of each $X_j$ justifies every conditional expectation above.