Two Ways to Express a Posterior Decision
A clinical decision rule may ask directly how much posterior probability lies in a favorable region, or it may ask whether a credible interval lies entirely on the favorable side of a clinically relevant boundary. These summaries can express the same evidence, but only when their probability cutoffs and interval conventions are matched. In particular, “the credible interval excludes the boundary” is incomplete unless the interval type and the direction of exclusion are specified.
Let \(\theta\) be the treatment-effect estimand, with larger values defined as more favorable, and let \(c\) be the prespecified boundary for a clinically meaningful effect. Given trial data \(D\), define \(F(c)=P(\theta\le c\mid D)\). If the posterior distribution is continuous, \(P(\theta=c\mid D)=0\), so the posterior probability of exceeding the boundary is \(q=P(\theta>c\mid D)=1-F(c)\). The same comparison applies to other directions of benefit after reversing the inequalities.
The probability is conditional on the specified model, prior, and observed data. It is not the probability that a null hypothesis is true, nor is it a repeated-sampling error rate. The boundary \(c\) and the decision cutoff should be chosen for the clinical decision at hand; they need not be zero or any particular universal probability.
Why an Equal-Tail Interval Can Encode a Tail Rule
For a continuous posterior with a strictly increasing CDF near the relevant quantiles, the equal-tail credible interval with posterior content \(1-\alpha\) is \([F^{-1}(\alpha/2),F^{-1}(1-\alpha/2)]\). Its lower endpoint is \(L=F^{-1}(\alpha/2)\). A strict favorable exclusion means \(c<L\). Because \(F\) is increasing, this is equivalent to \(F(c)<\alpha/2\), and hence to \(q>1-\alpha/2\). Thus a central 95% interval excludes \(c\) strictly on its favorable side exactly when the posterior tail probability is greater than 0.975, subject to the stated regularity and boundary conventions.
A one-sided lower credible bound uses a different convention. Its lower endpoint is \(L_1=F^{-1}(\alpha)\), so strict exclusion, \(c<L_1\), corresponds to \(F(c)<\alpha\), or \(q>1-\alpha\). Therefore, a one-sided 95% lower bound aligns with a tail cutoff of 0.95; a central 95% interval aligns with a tail cutoff of 0.975. Calling both “95% credible” does not make their favorable decision thresholds the same.
These equivalences concern the posterior probability and quantiles of the same estimand under the same posterior. With discrete posterior mass, a CDF can jump at a boundary, and quantile conventions can affect the relationship. For continuous distributions that are not strictly increasing around the relevant value, quantiles may also fail to be unique. In either case, define the interval and the operational decision rule directly rather than assuming the equivalence holds unchanged.
Worked Example: Strong Tail Evidence and a Central 95% Interval
Suppose a hypothetical trial estimates mean reduction in systolic blood pressure, where larger reductions are better. Let \(\theta\) be the reduction in mmHg relative to control, let \(c=0\) mmHg, and suppose the posterior is \(N(3,1.2^2)\). Assume this is a proper, continuous posterior and that the direction of benefit is indeed \(\theta>0\).
Standardizing the boundary gives \(z=(0-3)/1.2=-2.5\). Therefore,
The central 95% equal-tail interval is \(3\pm1.96(1.2)\) mmHg, or approximately \([0.648,5.352]\) mmHg. Its lower endpoint exceeds 0, so it excludes the boundary favorably. The tail probability exceeds the central-interval-equivalent cutoff of 0.975, and both rules give a favorable decision. The interval describes posterior uncertainty in the mean reduction; it does not imply that every future patient will have a reduction in that range.
Interval Conventions Can Change the Decision
A one-sided interval and a central equal-tail interval allocate posterior probability differently. A highest-density interval (HDI) is different again: it is a region containing the specified posterior probability whose points have density at least as high as points excluded, when such a region is well-defined. An HDI can be asymmetric or disconnected. Consequently, an HDI’s exclusion of \(c\) is not generally equivalent to a fixed posterior tail-probability cutoff. For multimodal posteriors, a single reported interval may also conceal meaningful low-density gaps or multiple regions.
The following example uses a deliberately separated continuous posterior to make the interval-convention issue visible. Although this shape is unusual, it shows why a posterior tail probability and a central interval answer distinct questions when the distribution is irregular.
Worked Example: A Central Interval Spans a Low-Density Gap
Suppose the hypothetical posterior for a treatment effect \(\theta\), in mmHg of additional blood-pressure reduction, assigns probability 0.04 uniformly over \([-100,-99]\) and probability 0.96 uniformly over \([1,2]\). The unfavorable and favorable regions are separated by a gap, and the clinical boundary is \(c=0\). The posterior is continuous and has no point mass at 0.
All the positive component is above 0 and all the negative component is below it, so \(q=P(\theta>0\mid D)=0.96\). A one-sided 95% lower bound is the 0.05 quantile. The first 0.04 of probability lies in the negative component; the next 0.01 is a fraction \(0.01/0.96\) of the positive component’s unit-length interval. Thus the lower bound is \(1+0.01/0.96\approx1.010\) mmHg. This one-sided interval excludes 0 favorably, consistent with \(q>0.95\).
For the central 95% interval, the lower endpoint is the 0.025 quantile. It lies \(0.025/0.04=0.625\) of the way through the negative component, at \(-100+0.625=-99.375\) mmHg. The upper endpoint is the 0.975 quantile: after the negative component, the positive component must supply \(0.975-0.04=0.935\) probability, or \(0.935/0.96\) of its unit length. This gives \(1+0.935/0.96\approx1.974\) mmHg. The central interval \([-99.375,1.974]\) includes 0, despite the posterior tail probability being 0.96. There is no contradiction: the central 95% rule corresponds to a 0.975 tail cutoff, which 0.96 does not reach. The gap also makes a single interval a poor summary of the posterior’s shape.
A posterior tail rule directly answers how much probability is on one side of a specified clinical boundary. A credible interval gives a range summary, and its decision implication depends on whether it is one-sided, equal-tail, or highest-density. When a rule is intended to operationalize a clinical action, specify the estimand, direction, boundary, posterior cutoff, interval convention if one is used, and whether equality counts as meeting the rule.
Boundary Equality, Computation, and Diagnostics
Strict versus non-strict inequalities matter at the decision cutoff. Continuity implies \(P(\theta=c\mid D)=0\); it does not imply that a posterior tail probability cannot equal its decision cutoff. For example, a continuous posterior can have \(F(c)=\alpha/2\), so that \(q=1-\alpha/2\) exactly. If the tail rule is \(q\ge\gamma\), equality passes; if it is \(q>\gamma\), equality does not. An interval rule using \(c<L\) also differs at equality from a rule using \(c\le L\). State the convention in advance and use it consistently.
Worked Example: Exact Equality at the Central 95% Cutoff
Let the hypothetical posterior for a treatment effect in mmHg be \(N(0,1)\), and let the favorable region be \(\theta>c\), where \(c=-\Phi^{-1}(0.975)\) mmHg. This is a continuous posterior, and its central 95% lower endpoint is exactly \(c\).
The tail probability is \(q=P(\theta>c\mid D)=\Phi(\Phi^{-1}(0.975))=0.975\). Thus a rule \(q\ge0.975\) passes, while a rule \(q>0.975\) does not. The strict interval-exclusion rule \(c<L\) also does not pass because \(L=\Phi^{-1}(0.025)=-\Phi^{-1}(0.975)=c\); a convention that counts equality as exclusion would pass. The probability of the single value \(\theta=c\) is zero, but that fact does not prevent the tail probability from equalling the decision cutoff. This example is why the operational treatment of ties must be explicit.
In practice, posterior probabilities and quantiles may be estimated from MCMC draws. With \(M\) retained draws \(\theta^{(1)},\ldots,\theta^{(M)}\), estimate the tail probability by the fraction above \(c\). Estimate interval endpoints from the relevant empirical quantiles, using a documented quantile convention. If the estimate is near the decision cutoff, Monte Carlo error can make a nominal pass or fail unstable. Check chain mixing and effective sample size for the tail indicator \(I(\theta^{(m)}>c)\), not only for the parameter mean; the indicator can be poorly estimated even when a trace plot of \(\theta\) looks acceptable. Where feasible, report the numerical estimate and its Monte Carlo uncertainty, or continue sampling when that uncertainty could change the decision.
Common Pitfalls and Limitations
- Matching the nominal interval percentage to the same tail cutoff. A central 95% interval does not correspond to a 0.95 favorable tail cutoff; under the continuous equal-tail setup it corresponds to 0.975.
- Using “excludes” without specifying direction or interval type. State whether the interval is one-sided, central equal-tail, or an HDI, and which side of the clinical boundary is favorable.
- Assuming a continuous posterior prevents cutoff ties. It prevents posterior point mass at the boundary, not equality between a tail probability and a decision threshold.
- Ignoring posterior shape. An interval may be misleading for skewed or multimodal distributions. Inspect the posterior and, when relevant, report the direct tail probability alongside an interval.
- Treating a favorable posterior summary as a universal clinical decision. The decision still depends on the chosen estimand, margin, harms, benefits, and prespecified decision criterion.
The direct tail probability is often the clearest summary when the action is explicitly defined by a boundary. An interval is useful for communicating uncertainty over a range, but its decision meaning must be tied to its construction. Neither summary repairs a poorly chosen estimand, a misspecified model, or inadequate posterior computation.
Name \(\theta\), the clinically relevant boundary \(c\), and the favorable direction in clinical units.
Specify a posterior tail cutoff, or define the credible-interval type, content, and strict or inclusive boundary convention.
For a continuous equal-tail interval, verify the corresponding tail cutoff; assess posterior shape and Monte Carlo precision near the decision boundary.
Check Your Understanding
Answer each question, then compare with the solutions below.
- For a continuous posterior, what favorable tail-probability cutoff corresponds to strict exclusion of \(c\) by a central 90% equal-tail interval?
- A hypothetical posterior has \(P(\theta>c\mid D)=0.96\). Does this meet a one-sided 95% tail rule? Does it necessarily meet the rule implied by a central 95% interval?
- Why does \(P(\theta=c\mid D)=0\) not settle whether a decision rule passes when its tail probability equals the cutoff?
- What should be examined if an MCMC estimate of the posterior tail probability is close to its decision threshold?
Solutions. (1) Here \(\alpha=0.10\), so the central-interval equivalence is \(q>1-\alpha/2=0.95\) for strict exclusion; the specified operational rule must settle equality. (2) It meets a one-sided 95% rule if the rule is \(q\ge0.95\), and also if it is \(q>0.95\). It does not meet the strict central 95% interval-equivalent cutoff \(q>0.975\). (3) Zero posterior mass at the boundary concerns the event \(\theta=c\); the posterior probability \(q\) can still equal a chosen cutoff. The rule must specify whether equality passes. (4) Check mixing and effective sample size for the indicator \(I(\theta>c)\), and assess Monte Carlo uncertainty. If that uncertainty could change the decision, additional computation may be needed.