Reporting Posterior Means, Medians, and Quantiles

Why Posterior Summaries Need Context

A number such as “3.85” is not a complete treatment-effect result. Readers need to know what effect it summarizes, how the endpoint is measured, which patients are represented, and which feature of the posterior distribution the number describes. A posterior mean, a posterior median, and a posterior quantile answer different summary questions; they are not interchangeable labels for a single estimate.

In a clinical trial, the estimand states the treatment effect of interest. For example, it might be the treatment-minus-control difference in mean change from baseline at week 12 among randomized patients, under a specified approach to outcomes after treatment discontinuation. The endpoint scale states how that effect is expressed, such as millimetres of mercury or a log odds ratio. The analysis population identifies the patients to whom the summary pertains. State these alongside the numerical summaries so that readers can interpret the result without guessing.

For a continuous posterior, the \(p\)-quantile satisfies \(P(\theta\leq Q_p\mid D)=p\). For a discrete posterior, the cumulative probability can jump over \(p\); the definition using the infimum still specifies the quantile. A mean requires a posterior with a finite first moment. A median and quantiles can be defined even when the posterior mean does not exist.

Choosing the Summary

The mean uses the magnitude of all posterior values and is often useful when the estimand is an average effect or when decisions are evaluated using squared-error loss. It can, however, be pulled by a small amount of posterior mass far into a tail. The median is a central value defined by posterior probability on either side, and is less sensitive to extreme tail values. If the posterior is symmetric, has a unique center, and has a finite first moment, the mean and median coincide; otherwise, they may differ.

Selected quantiles describe locations in the posterior distribution. For example, \(Q_{0.05}\) and \(Q_{0.95}\) identify the 5th and 95th posterior percentiles. Reporting them gives readers information about spread and asymmetry that a single point summary cannot provide. The pair alone does not describe every feature of the posterior, and the posterior mean or median should not be used as a substitute for specifying which quantiles were calculated.

The choice of point summary should be prespecified or justified by the purpose of the analysis, not selected after comparing which number looks most favorable. In a regulatory report, consistency with the estimand, analysis plan, and other reported results matters. If both mean and median are informative, report both and name each explicitly.

A Brief Justification for Mean and Median

Their different roles can be seen through loss functions. Suppose a single reported value \(a\) is used to represent \(\theta\), and consider posterior expected squared error. For a posterior with finite variance,

The cross-product term vanishes because \(E[\theta-E(\theta\mid D)\mid D]=0\). The first term does not depend on \(a\), so the expected squared error is minimized at \(a=E(\theta\mid D)\), the posterior mean.

For absolute error, the expected loss is \(E[|\theta-a|\mid D]\). When the posterior has a continuous distribution with cumulative distribution function \(F\), its derivative with respect to \(a\), where defined, is \(2F(a)-1\). The loss decreases where \(F(a)<0.5\) and increases where \(F(a)>0.5\); consequently a median minimizes expected absolute error. This explains the loss-based distinction; it does not imply that every clinical report must select a summary solely by a formal loss function.

Summaries From Posterior Draws

In many analyses, posterior summaries are computed from draws rather than a closed-form distribution. For equally weighted draws, the mean is approximated by their arithmetic average. The median and quantiles are obtained by ordering the draws and applying a stated quantile convention. With weighted draws, the weights must be normalized to sum to one, and the cumulative normalized weights determine quantiles. Monte Carlo error means draw-based summaries are estimates of the corresponding posterior quantities, not exact values.

Report the Scale on Which the Summary Was Calculated

A treatment effect may be modeled on a transformed scale and reported on a more familiar scale. It is essential to distinguish “transform the posterior summary” from “summarize the transformed posterior.” These operations generally give different answers. For a continuous, strictly increasing transformation \(g\), a posterior quantile transforms directly: the \(p\)-quantile of \(g(\theta)\) is \(g(Q_p)\). The mean generally does not: \(E[g(\theta)\mid D]\) is not usually equal to \(g(E[\theta\mid D])\).

Practical Checks and Common Reporting Errors

Before finalizing a report, check that the posterior being summarized corresponds to the intended estimand, model, prior, and analysis population. Confirm that treatment-effect direction is stated, that units or the dimensionless ratio scale are clear, and that each number is labelled as a mean, median, or specified quantile. When posterior draws are used, check that the draws and weights are handled as intended and that the quantile algorithm is documented when reproducibility requires it. Summaries calculated from poorly mixing or otherwise inadequate draws may not reliably represent the posterior; numerical precision in a table cannot repair that problem.

A common mistake is to report the exponential of a posterior mean log odds ratio as “the posterior mean odds ratio.” In the example above, \(\exp(0.20)=1.221\) is the posterior median odds ratio, not the posterior mean odds ratio; the latter is approximately 1.278. The correction is to calculate the target summary on the scale being reported, or label the transformed summary accurately. Another frequent error is to give quantiles without naming their probabilities, or to omit the population and endpoint. A list of numbers without those labels is ambiguous even when the arithmetic is correct.

Mean, median, and quantiles are compact summaries, not complete representations of a posterior distribution. Two distributions can have the same mean and quantiles but differ in other ways. If summaries appear inconsistent with plots or other model outputs, investigate the model, computation, transformations, and population definitions rather than choosing whichever summary supports a preferred conclusion. The full posterior distribution may be needed to understand features that a few selected numbers conceal.