Why Posterior Shape Matters
A treatment-effect summary should help readers understand not only a central value, but also the range and structure of effects still compatible with the data and model. A posterior may be strongly skewed, or it may have several distinct modes. In either case, a mean and one interval can conceal features that affect clinical interpretation: a long tail, a substantial chance of harm, or separate regions corresponding to different plausible effect sizes.
The posterior distribution is conditional on the observed data and the specified model and prior. For reporting, define the treatment-effect parameter and its scale before summarizing it. In the examples below, a negative effect on the difference scale means benefit, while a ratio below one means benefit. The direction must be stated for the actual endpoint; it is not universal across clinical outcomes.
The posterior mean is the expectation of the effect under its posterior distribution. A posterior median \(m\) is any value satisfying both \(P(\theta\leq m\mid\text{data})\geq 0.5\) and \(P(\theta\geq m\mid\text{data})\geq 0.5\). These inequalities are the appropriate definition when the posterior has point masses; for a continuous distribution without an atom at the median, each probability is 0.5. A mean can lie in a low-density region between modes, while the median can fall at a boundary between concentrations. Neither summary, by itself, describes the whole shape.
Reporting a Skewed Posterior
For a skewed posterior, report a central summary together with quantiles or a distributional display. Quantiles are especially useful because they describe positions in the posterior distribution without requiring symmetry. For a posterior cumulative distribution function \(F\), define the generalized \(p\)-quantile as \(q_p=\inf\{x:F(x)\geq p\}\). An equal-tail interval at level \(1-\alpha\) has endpoints \(q_{\alpha/2}\) and \(q_{1-\alpha/2}\). With a continuous posterior, each tail has probability \(\alpha/2\). With atoms, the probability strictly below or above an endpoint need not equal \(\alpha/2\), and the interval's posterior probability can exceed \(1-\alpha\).
The original effect scale matters. If \(\theta\) is a log ratio and \(R=\exp(\theta)\) is the ratio, exponentiating changes the scale and can change the posterior shape. In particular, a normal posterior for \(\theta\) gives a lognormal posterior for \(R\), which is right-skewed when its variance is positive. Symmetry of an arbitrary posterior for \(\theta\), however, does not by itself guarantee that the transformed posterior is right-skewed.
Worked Example: Reporting a Lognormal Treatment Ratio
Suppose a hypothetical trial estimates a treatment-to-control risk ratio \(R\), with \(R<1\) indicating lower risk under treatment. The posterior for the log risk ratio \(\theta=\log R\) is \(N(-0.20,0.18^2)\). This is an assumed posterior model for illustration, not a real study. Because the model is normal on the log scale, \(R=\exp(\theta)\) has a lognormal posterior.
The median ratio is the exponential of the normal mean: \( \operatorname{median}(R\mid\text{data})=\exp(-0.20)=0.8187 \). The posterior mean ratio is different. For a normal variable with mean \(\mu\) and variance \(\sigma^2\), completing the square in the normal-density integral gives \(E[\exp(\theta)]=\exp(\mu+\sigma^2/2)\). Substituting the values gives \(E[R\mid\text{data}]=\exp(-0.20+0.18^2/2)=\exp(-0.1838)=0.8321\).
A central 95% equal-tail interval for \(\theta\) is \(-0.20\pm1.96(0.18)=[-0.5528,0.1528]\). Since the exponential function is strictly increasing, the corresponding interval for \(R\) is \([\exp(-0.5528),\exp(0.1528)]=[0.5753,1.165]\). It includes 1, the value of no relative difference. The posterior probability of benefit is \(P(R<1\mid\text{data})=P(\theta<0\mid\text{data})=\Phi(0.20/0.18)\approx0.8667\).
A suitable report would give the median ratio, the equal-tail interval, and the probability of a ratio below 1, while identifying the ratio and its direction. The mean ratio of 0.8321 is not the same as the median, and neither alone conveys the long right tail of the lognormal posterior. Here the normal-on-log-scale assumption justifies that right-skew description; it should not be generalized to every symmetric log-effect posterior.
Multimodality and Clinically Relevant Probabilities
For a multimodal posterior, one mean can sit between concentrations of probability and describe an effect that is not itself a plausible region. A single interval may include low-density values between modes, while reporting only the mode with the highest peak can ignore substantial probability elsewhere. Report the locations and relative importance of the modes, and pair them with probabilities tied to clinically interpretable thresholds.
A density plot or histogram can reveal separate concentrations; a table of posterior probabilities for benefit, clinically important benefit, and harm can quantify their relevance. A useful display marks the null and clinical thresholds on the effect scale. A posterior mean and a central interval may still be included for comparability, but should not be presented as a complete description when the distribution is clearly multimodal.
Worked Example: A Two-Mode Posterior for a Continuous Outcome
Consider a hypothetical posterior for the treatment-minus-control difference in systolic blood pressure, \(\theta\), measured in mmHg. Negative values favor treatment. Suppose \(\theta\mid\text{data}\sim0.65N(-5,1^2)+0.35N(1,1^2)\). This means the posterior is a mixture: it assigns probability 0.65 to the first normal component and 0.35 to the second. Its well-separated concentrations are near \(-5\) and \(1\) mmHg.
The posterior mean combines the component means using their weights: \(E(\theta\mid\text{data})=0.65(-5)+0.35(1)=-2.90\) mmHg. That mean is between the two concentrations. It is a valid average, but does not show that much of the posterior probability is concentrated near either \(-5\) or \(1\) mmHg.
The posterior probability of any reduction is \(P(\theta<0\mid\text{data})=0.65\Phi(5)+0.35\Phi(-1)\) \(=0.65(0.9999997)+0.35(0.1587)=0.7055\). If a reduction of at least 3 mmHg is defined as clinically important, then \(P(\theta\leq-3\mid\text{data})=0.65\Phi(2)+0.35\Phi(-4)\) \(=0.65(0.97725)+0.35(0.0000317)=0.6352\). The posterior probability of an increase is \(1-0.7055=0.2945\), because this mixture is continuous and has no probability mass exactly at zero.
These results support a report that states the two concentrations, the posterior mean of \(-2.90\) mmHg, and the probabilities for benefit, clinically important benefit, and an increase. They do not support describing the mean as the single most plausible effect. The mixture is an assumed posterior distribution; in an actual analysis, model adequacy and computation would also need to be checked.
Intervals, Transformations, and Discrete Posterior Summaries
Equal-tail intervals are defined by quantiles and are straightforward to compute and compare across analyses. A highest-posterior-density (HPD) region instead comprises values with the greatest posterior density or mass, chosen to contain a target posterior probability. For a continuous unimodal posterior, an HPD interval is often compact around the mode. For a multimodal posterior, the HPD region may consist of disconnected pieces. HPD regions depend on the parameter scale: under a nonlinear transformation, density heights change, so the HPD region on the transformed scale need not be the transformed HPD region on the original scale. Equal-tail quantiles transform as \(g(q_p)\) under a continuous, strictly increasing transformation.
To see why, let \(Y=g(\theta)\), where \(g\) is strictly increasing, and let \(q_p\) be the generalized \(p\)-quantile of \(\theta\). Then \(F_Y(y)=F_\theta(g^{-1}(y))\) on the range of \(g\). If \(g\) is continuous and strictly increasing, the transformed \(p\)-quantile is \(g(q_p)\). The probability \(P(\theta\leq q_p\mid\text{data})\) is generally at least \(p\), not necessarily exactly \(p\); it equals \(p\) under suitable continuity conditions. This distinction matters for posterior distributions with atoms.
Worked Example: Quantiles When the Posterior Has Atoms
Suppose a hypothetical posterior distribution for a treatment effect \(\theta\), in mmHg, assigns probabilities 0.10, 0.40, 0.35, and 0.15 to \(-4,-1,3,\) and \(5\), respectively. This discrete distribution can represent, for example, a posterior summarized by rounded draws. Its cumulative probabilities at those values are 0.10, 0.50, 0.85, and 1.00.
The mean is \(0.10(-4)+0.40(-1)+0.35(3)+0.15(5)=-0.40-0.40+1.05+0.75=1.00\) mmHg. A median is \(-1\): \(P(\theta\leq-1\mid\text{data})=0.50\) and \(P(\theta\geq-1\mid\text{data})=0.90\), so both defining inequalities hold. The median does not require the two probabilities to be exactly equal.
Using \(q_p=\inf\{x:F(x)\geq p\}\), the 2.5th percentile is \(-4\), because the cumulative probability there is 0.10, and the 97.5th percentile is \(5\), because the cumulative probability first reaches 0.975 at 5. Thus the 95% equal-tail interval is \([-4,5]\) mmHg. Its posterior probability is 1.00 in this example, not exactly 0.95: both endpoints are included and all the posterior mass lies within them. The probability strictly below the lower endpoint and strictly above the upper endpoint is zero, not 0.025 each. Reporting the quantile definition and the interval's actual posterior probability avoids falsely claiming exact tail probabilities.
Diagnostics, Pitfalls, and Limitations
Before choosing a summary, inspect posterior draws or a density estimate for asymmetry, long tails, multiple concentrations, and isolated extreme values. Check whether a different effect scale changes the apparent shape, and verify that any transformation preserves the direction of clinical benefit. For a mixture, report component weights or clinically meaningful probability regions rather than relying on a single central number. If computation is based on simulation, check that apparent modes and tail probabilities are stable across chains and are not artifacts of poor mixing or too few effective draws.
- Pitfall: treating the mean as the typical or most plausible effect. A mean between two modes may be in a low-density region. Report the distribution's separate concentrations and relevant posterior probabilities alongside the mean.
- Pitfall: claiming exact equal tail probabilities for an empirical or discrete posterior. Atoms can make the cumulative probability jump across a quantile. State the generalized quantile convention and, when relevant, the actual posterior mass in the interval.
- Pitfall: assuming an effect's shape is unchanged by rescaling. A nonlinear transformation can alter skewness and density-based summaries. State the scale and transform quantile endpoints using the correct monotonicity rule.
- Pitfall: presenting an interval as the whole story. A central interval can span a low-density gap between modes or conceal a clinically important tail probability. Pair it with a distributional display or threshold probabilities when those features matter.
No summary method removes dependence on the model, prior, endpoint definition, or scale. A density display can also be misleading if smoothing hides a mode or exaggerates noise, especially with limited draws. The practical goal is not to report every possible summary, but to give enough information for readers to see the posterior structure relevant to the decision. The introductory material on posterior means, medians, and quantiles provides the basic summaries; here the key additional requirement is to check whether those summaries faithfully represent the shape.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A log-effect posterior is normal with mean \(-0.10\) and standard deviation \(0.20\). The ratio is the exponential of the log effect. What are the ratio median and posterior mean, and why can they differ?
- A posterior has probabilities 0.20 at \(-2\) and 0.80 at \(1\). Is \(-2\) a median under the two-inequality definition? What is the generalized 50th percentile?
- A multimodal posterior has a mean of \(-1\) mmHg, but two concentrations near \(-6\) and \(2\) mmHg. Name two additions that would make the report more informative.
- Why should an HPD region on a log-ratio scale not automatically be exponentiated and described as an HPD region for the ratio?
Solutions. (1) The ratio median is \(\exp(-0.10)=0.9048\). The posterior mean is \(\exp(-0.10+0.20^2/2)=\exp(-0.08)=0.9231\). They differ because exponentiation is nonlinear and the lognormal posterior is skewed. (2) No: \(P(\theta\leq-2)=0.20<0.5\), so \(-2\) is not a median. The generalized 50th percentile is \(1\), since the cumulative probability first reaches 0.50 there. (3) Describe or display both concentrations and report probabilities for clinically relevant thresholds, such as benefit and harm. These reveal structure the mean alone misses. (4) HPD regions depend on density heights, which change under a nonlinear transformation. Exponentiated quantiles give the corresponding quantiles for a strictly increasing transformation, but HPD regions are not generally invariant in that way.