From a Final Decision Rule to a Future-Data Event
A trial may end with a rule such as “go if the final posterior probability that the response rate exceeds 50% is at least 0.95.” Before all outcomes have been observed, that rule is not yet a decision: it is a condition that the final data will either satisfy or fail to satisfy. The predictive question is, under the current posterior and the planned data collection, what is the probability that the eventual data will satisfy this exact final rule?
The calculation has two distinct parts. First, translate the final posterior criterion into a region of possible future data. Then find the probability of that region using the current posterior predictive distribution. This makes the final decision rule operational without replacing it with a different rule, such as a cutoff on the current posterior mean.
In a discrete-outcome trial, \(G\) may be a set of counts, such as “at least eight of the remaining ten participants respond.” In a continuous-outcome trial, it may be an interval or half-line of possible future sample means. The boundary depends on the posterior decision rule, the model, the information already observed, and the planned remaining sample size.
Binary Outcomes: Find the Pass Boundary First
Suppose the response probability is \(p\), and the current posterior distribution is \(\operatorname{Beta}(a,b)\). If \(K\) successes are observed among \(m\) future participants, then the final posterior is \(\operatorname{Beta}(a+K,b+m-K)\), assuming conditionally independent Bernoulli outcomes with common probability \(p\). A final rule based on \(P(p>p_0\mid D,K)\) can therefore be checked for each possible value of \(K\).
When \(p_0\) is fixed and the rule is a lower-tail superiority criterion, this posterior probability increases with \(K\). Consequently, the set of passing counts has a boundary: there is a smallest count \(k_{\min}\) that passes, and the final-go event is \(K\geq k_{\min}\). The boundary is found by evaluating the final posterior criterion, not by choosing a predictive-probability cutoff.
For a beta posterior, a useful identity is \(P(p>p_0)=P\{B\leq a-1\}\), where \(B\sim\operatorname{Binomial}(a+b-1,p_0)\). This gives an exact way to evaluate the posterior tail probability after each possible final count. Once the pass boundary is known, the predictive distribution of \(K\) under the current \(\operatorname{Beta}(a,b)\) posterior is beta-binomial:
Here \(B(\cdot,\cdot)\) is the beta function. The predictive probability of final go is the sum of these probabilities over the passing counts. The derivation is direct: condition on \(p\), use \(P(K=k\mid p)=\binom{m}{k}p^k(1-p)^{m-k}\), and average over the current posterior density of \(p\). Integration gives the displayed beta-binomial probability.
Worked Example: A Beta-Binomial Final-Go Boundary
Hypothetical trial. At an interim analysis, the posterior response probability is \(p\mid D\sim\operatorname{Beta}(13,9)\). Ten participants remain. At the final analysis, the rule is go if \(P(p>0.50\mid\text{all data})\geq0.95\). Find the predictive probability of meeting that rule.
If \(K=k\) of the ten future participants respond, the final posterior is \(\operatorname{Beta}(13+k,19-k)\). For this posterior, the beta-binomial identity gives \(P(p>0.50\mid D,K=k)=P\{\operatorname{Binomial}(31,0.50)\leq12+k\}\). At \(k=7\), this is \(P\{\operatorname{Binomial}(31,0.50)\leq19\}=0.9252\), below 0.95. At \(k=8\), it is \(P\{\operatorname{Binomial}(31,0.50)\leq20\}=0.9646\), above 0.95. Since this probability increases with \(k\), the final rule passes exactly when \(K\geq8\).
The current posterior predictive distribution is beta-binomial with \(a=13\), \(b=9\), and \(m=10\). To make the tail calculation transparent, set the unnormalized weight at \(k=0\) to 1. Successive weights obey \(\frac{w_{k+1}}{w_k}=\frac{10-k}{k+1}\frac{13+k}{18-k}\). Applying this ratio gives weights for \(k=0,\ldots,10\) of approximately \(1,\ 7.222,\ 26.765,\ 66.912,\ 124.902,\ 182.000,\ 210.000,\ 190.000,\ 129.545,\ 60.455,\ 14.778\). Their total is 1013.578.
The weights for passing counts \(k=8,9,10\) sum to \(129.545+60.455+14.778=204.778\). Therefore,
Under the stated model and planned sample size, the probability that the final posterior go criterion will be met is about 20.2%. This is a probability of future data falling in the passing set, not a posterior probability that \(p>0.50\).
Continuous Outcomes: Translate the Rule Into a Cutoff
The same logic applies when future data are continuous. Under a normal likelihood with known variance and a normal current posterior, the final posterior mean is a function of the future sample mean. A posterior-tail rule can therefore be rearranged into a cutoff for that future sample mean. The predictive probability is then a normal tail probability, provided the assumed normal model and fixed future sample size are appropriate.
Worked Example: A Normal-Normal Future-Mean Boundary
Hypothetical trial. Let \(\theta\) be a treatment effect in units of a clinical score, with current posterior \(\theta\mid D\sim N(0.30,0.20^2)\). Sixteen future independent observations will have distribution \(N(\theta,1.2^2)\). The final rule is go if \(P(\theta>0\mid D,\bar{Y})\geq0.95\). Find the predictive probability of go.
With \(m=16\) future observations, the final posterior variance and mean are \(v_1=(1/0.20^2+16/1.2^2)^{-1}=0.02769\) and \(\mu_1=v_1(0.30/0.20^2+16\bar{Y}/1.2^2)=0.2077+0.3077\bar{Y}\). A normal posterior has \(P(\theta>0)\geq0.95\) exactly when its mean is at least \(1.6449\) posterior standard deviations above zero. Here the required mean is \(1.6449\sqrt{0.02769}=0.2737\). Thus the final rule is equivalent to \(\bar{Y}\geq(0.2737-0.2077)/0.3077\approx0.2146\) score units.
Under the current posterior, the future sample mean has predictive distribution \(N(0.30,0.20^2+1.2^2/16)=N(0.30,0.13)\). Its standard deviation is \(\sqrt{0.13}=0.3606\). Therefore,
The predictive probability of meeting the final rule is about 59.4%. This calculation assumes the stated known outcome variance, independent future observations, and no change to the planned sample size or analysis model.
Discrete Boundaries and the Role of the Model
A final posterior threshold need not correspond to a single smooth boundary. With binary outcomes, the possible values of \(K\) are integers, so the posterior probability can jump across the required threshold. It is entirely possible for one count to fall short and the next to pass by a visible margin. The criterion is still applied exactly as specified: identify which possible counts satisfy it, then sum their predictive probabilities.
Worked Example: A Small Trial With Only One Passing Count
Hypothetical trial. Suppose the current posterior is \(\operatorname{Beta}(2,2)\), four participants remain, and final go requires \(P(p>0.50\mid\text{final data})\geq0.90\). If \(K=k\), the final posterior is \(\operatorname{Beta}(2+k,6-k)\). The posterior tail probability equals \(P\{\operatorname{Binomial}(7,0.50)\leq1+k\}\).
For \(k=3\), this probability is \(P(B\leq4)=99/128=0.7734\), which does not pass. For \(k=4\), it is \(P(B\leq5)=120/128=0.9375\), which does pass. Thus only \(K=4\) meets the final rule. Under the current beta-binomial predictive distribution, \(P(K=4\mid D)=\binom44 B(6,2)/B(2,2)=(1/42)/(1/6)=1/7\approx0.1429\). The predictive probability of final go is therefore 14.3%.
These examples rely on a specified likelihood, current posterior, future sample plan, and final decision rule. If any of these changes, the passing set or its predictive probability can change. In particular, a predictive probability is conditional on the assumed model; it does not by itself account for model misspecification, changes in eligibility or outcome measurement, or unplanned changes to enrollment.
Diagnostics, Alternatives, and Common Mistakes
A practical calculation should be checked at both stages. First verify that the final posterior used for each hypothetical future outcome includes the interim information exactly once. Then verify the pass set by checking outcomes on both sides of the proposed boundary. In a binary example, confirm that the final posterior probability is below threshold at \(k_{\min}-1\) and at or above threshold at \(k_{\min}\); if monotonicity is not established, evaluate all possible counts rather than assuming a single cutoff. Finally, check that predictive probabilities over all possible future outcomes sum to one and that the passing-event probability lies between zero and one.
For conjugate models, analytic posterior updates and exact predictive sums are often convenient. Numerical integration or direct calculation over a discrete outcome space can also be used when the predictive distribution is not available in closed form. These approaches should target the same quantity, \(P(Y\in G\mid D)\), and use the same final rule. They differ in computational method, not in the event being evaluated.
- Confusing the two probabilities. \(P(\theta\in A\mid D)\) describes current posterior support for the favorable parameter region. \(P(Y\in G\mid D)\) describes the chance that future data will make the final posterior rule pass. They are not interchangeable.
- Using a current threshold as the final rule. A cutoff on the current posterior mean or current posterior tail probability is not automatically equivalent to the specified final criterion. Derive the set of future outcomes from the final posterior calculation.
- Ignoring discreteness. With a fixed number of binary outcomes, the posterior probability may leap from below to above \(\gamma\). Do not round a passing boundary into a fractional number of responses.
- Changing the analysis while predicting. The future likelihood, prior/posterior updating convention, estimand, and final threshold must match the planned final analysis. Otherwise the calculated event is not the event the trial will actually use.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In one sentence, distinguish the final-go event from its predictive probability.
- A binary trial’s posterior go criterion passes at \(K=6\) and every larger count, but fails at \(K=5\). What is the final-go event?
- Why is it incorrect to interpret a predictive probability of 0.20 as a 20% posterior probability that the treatment is effective?
- In the small beta-binomial example, why does the predictive probability of go equal the probability of exactly four responses?
Solutions. (1) The final-go event is the set of future datasets that produce a final posterior satisfying the go threshold; its predictive probability is the current posterior predictive probability that future data fall in that set. (2) The event is \(K\geq6\). (3) The 0.20 concerns whether future data will make a specified decision rule pass, while a posterior probability about effectiveness concerns the parameter conditional on available data. (4) Four is the only possible future count that meets the specified final criterion, so the passing set contains only \(K=4\), and its predictive probability is \(P(K=4\mid D)\).