From Clinical Expectations to a Beta Prior
For a binary treatment response, a clinician may be able to describe an expected response rate and how uncertain that expectation is before the trial begins. A Beta prior turns those beliefs into a probability distribution for the unknown response probability. Elicitation is not just a matter of choosing two parameters: the resulting distribution should also be checked against plausible response rates and tail probabilities.
Suppose each patient has a binary response, and let \(p\) denote the response probability for the patient population relevant to the trial. The standard Beta-Bernoulli model assumes responses are conditionally independent given \(p\), with the same response probability \(p\) for each patient. The prior describes uncertainty about this common probability; it is not a model for patient-to-patient variation in response probabilities.
The ESS is a useful summary, but it is not a universal measure of prior information across all models or elicitation methods. In the Beta-Bernoulli model, \(a\) and \(b\) behave algebraically like prior response and nonresponse counts, which motivates the convention. They need not be integers and should not automatically be interpreted as actual patients previously observed.
Deriving the Conversion
Let the clinician’s expected response probability be \(m\), with \(0<m<1\), and the desired prior ESS be \(\kappa>0\). By definition, the Beta mean is \(a/(a+b)\), while the ESS is \(a+b\). Thus the two elicited quantities impose the equations below.
Substituting \(a+b=\kappa\) into the mean equation gives \(a/\kappa=m\), so \(a=m\kappa\). Subtracting this value from the total gives \(b=\kappa-a=(1-m)\kappa\). Both parameters are positive when \(0<m<1\) and \(\kappa>0\), as required for a proper Beta distribution.
This conversion matches the specified mean and concentration, but it does not guarantee that every other clinical belief is represented well. For example, the elicited pair also determines the prior variance:
Consequently, once the mean and ESS are fixed, the variance and all quantiles are fixed too. If clinicians also give a plausible interval or a probability for an important tail event, that information is a check on the proposed prior—not a third quantity that can generally be matched by the same two parameters.
Worked Example: A Low Expected Response Rate
Suppose a clinician expects a response probability of \(m=0.10\) and considers the prior information equivalent, by the Beta-Bernoulli convention, to \(\kappa=10\) patients. Find the Beta parameters and the central 90% prior quantiles.
The conversion gives \(a=m\kappa=0.10(10)=1\) and \(b=(1-m)\kappa=0.90(10)=9\). The proposed prior is \(p\sim\operatorname{Beta}(1,9)\), with mean \(1/(1+9)=0.10\).
For \(\operatorname{Beta}(1,9)\), the cumulative distribution function is \(F(x)=1-(1-x)^9\). The quantile at probability \(u\) solves \(F(x)=u\), so \(x=1-(1-u)^{1/9}\). Therefore:
The central 90% prior interval is approximately 0.0057 to 0.2831, in response-probability units. This interval is asymmetric around 0.10, which is unsurprising for a prior concentrated near the lower boundary of the probability scale. Before using it, the clinician should judge whether response probabilities in this range are plausible. For example, a belief that values above 0.20 are almost impossible would conflict with this prior’s upper tail and should prompt discussion.
Checking Quantiles and Tail Beliefs
A quantile check makes the consequences of the elicited mean and ESS visible. For a specified probability \(u\), the \(u\)-quantile \(q_u\) is the value satisfying \(P(p\le q_u)=u\). The central 90% prior interval is \([q_{0.05},q_{0.95}]\); it is a range containing 90% of the prior probability, not a confidence interval for an estimated response rate.
Quantiles are especially helpful when a clinician can describe clinically meaningful thresholds more readily than a standard deviation. Questions such as “How plausible is a response rate below 10%?” or “What rate would be unusually high?” can be translated into prior tail probabilities or quantiles. A numerical Beta CDF or quantile calculation is generally used for arbitrary \(a\) and \(b\); the resulting values should be reported with enough precision for the clinical comparison, not as if the inputs were exact measurements.
Worked Example: Testing a Lower-Tail Belief
A clinician gives \(m=0.40\) and \(\kappa=5\), and separately states that the chance of a response probability below 0.10 should be about 1%. Convert the mean and ESS, then check that tail belief.
The parameters are \(a=0.40(5)=2\) and \(b=0.60(5)=3\), giving \(p\sim\operatorname{Beta}(2,3)\). For this prior, the cumulative distribution function is \(F(x)=6x^2-8x^3+3x^4\). Evaluating at the clinical threshold gives:
The implied lower-tail probability is 5.23%, not about 1%. The three statements—mean 0.40, ESS 5, and a 1% chance below 0.10—are therefore not all represented by this Beta prior. The discrepancy is not fixed by rounding. The elicitation needs clarification: the clinician might revise the mean, the ESS, or the tail belief, or the modeling team might consider a different prior family if the beliefs are genuinely important and cannot be reconciled by a Beta distribution.
What the ESS Controls—and What It Does Not
At a fixed mean \(m\), increasing \(\kappa\) increases both Beta parameters proportionally and reduces the prior variance \(m(1-m)/(\kappa+1)\). The prior becomes more concentrated around its mean. Decreasing \(\kappa\) makes it more diffuse. This gives the ESS an interpretable role in elicitation, but the same ESS does not imply the same prior shape when the mean changes.
Fractional parameters are valid. For instance, \(m=0.60\) and \(\kappa=8\) give \(a=4.8\) and \(b=3.2\). These values define a proper Beta distribution even though neither is an integer. They are parameters, not a claim that 4.8 patients responded and 3.2 did not.
Worked Example: Comparing the Quantiles With a Clinical Range
Suppose the elicited mean is \(m=0.50\) and the ESS is \(\kappa=4\). The conversion gives \(a=2\) and \(b=2\), so \(p\sim\operatorname{Beta}(2,2)\). Its CDF is \(F(x)=3x^2-2x^3\). Numerical inversion gives a central 90% prior interval of approximately \([0.1354,0.8646]\).
A clinician says that the prior should place 90% of its probability between response probabilities 0.20 and 0.80. Check the proposed prior’s mass in that interval:
The proposed prior assigns 79.2% probability to that range, rather than 90%. Its central 90% interval is wider than the clinician’s stated range. The mean is consistent with the stated midpoint, but the ESS of 4 does not express the desired degree of concentration. If the mean and interval belief are firm, the ESS should be revisited and the resulting prior checked again.
A Practical Elicitation and Review Sequence
A disciplined sequence separates the clinical judgments from the mathematical conversion. It also makes disagreements visible before the prior is used in an adaptive trial.
Specify the binary endpoint, assessment time, and patient population to which \(p\) refers. The elicited mean is not meaningful if these are left ambiguous.
Record the expected response probability \(m\) and the conventional Beta-Bernoulli ESS \(\kappa\), including how the clinician interprets that strength of belief.
Calculate \(a=m\kappa\) and \(b=(1-m)\kappa\), and verify that both are positive and sum to \(\kappa\).
Compute prior quantiles and probabilities on both sides of meaningful response thresholds. Compare those implications with the original clinical beliefs.
If the summaries conflict, discuss which elicited quantities are most reliable; do not silently alter a parameter or describe the resulting prior as matching all the beliefs.
This procedure assumes a single response probability is a reasonable model for the target patients and that a Beta distribution is an acceptable summary of prior uncertainty. If response probabilities vary substantially across sites or patient subgroups, a single \(p\) may be inadequate. Similarly, if the clinician’s beliefs are strongly multimodal or have unusual tails, matching a mean and ESS may conceal important structure.
Alternatives, Diagnostics, and Common Mistakes
Moment-based elicitation is convenient when the clinician can state an expected rate and an information strength. Quantile-based elicitation instead begins with one or more plausible ranges or tail probabilities and finds a prior compatible with them. With only two Beta parameters, two independent constraints can generally be used to determine them, but additional beliefs become checks. A pseudo-count description is another communication aid: \(a\) and \(b\) resemble prior response and nonresponse counts in the conjugate model. It is an interpretation of the parameterization, not evidence that those outcomes were observed.
Useful diagnostics include checking that \(a+b=\kappa\), verifying that the computed mean is \(m\), inspecting central quantiles, and evaluating probabilities around clinically meaningful thresholds. A prior can pass the mean check and still put implausible mass near zero or one. Conversely, an unexpected-looking asymmetric interval may be a natural consequence of a mean near a boundary rather than a calculation error.
- Confusing the expected rate with an observed sample rate. The elicited \(m\) is the prior mean for \(p\), not necessarily the response fraction in any historical data set.
- Treating ESS as an actual patient count. The concentration \(a+b\) is an effective-information convention. It can be fractional and does not establish that prior data exist.
- Assuming the mean determines uncertainty. Many Beta priors share a mean but have different concentrations and quantiles. The ESS is needed to set the spread under this parameterization.
- Ignoring contradictory clinical statements. If a prior’s quantiles disagree with a stated plausibility range, reporting only its mean does not resolve the disagreement.
- Using a finite Beta prior at an exact boundary mean. For \(m=0\) or \(m=1\), the conversion gives one shape parameter equal to zero, which is not a proper Beta distribution. Exact certainty at a boundary requires a different modeling decision; it should not be represented by an ordinary finite-parameter Beta prior.
Check Your Understanding
Work each item, then compare with the solutions below.
- A clinician specifies a prior mean response probability of 0.25 and an ESS of 12. Find \(a\) and \(b\), and calculate the prior variance.
- For a mean of 0.60 and an ESS of 8, are fractional Beta parameters valid? Give the parameters and explain what they do not mean.
- For \(p\sim\operatorname{Beta}(1,4)\), calculate \(P(p>0.50)\). What kind of elicited belief could this help check?
- A clinician gives \(m=0.20\) and \(\kappa=10\), but also says the prior should be centered at 0.30. Can the conversion satisfy both mean statements? Explain.
Solutions. (1) \(a=0.25(12)=3\) and \(b=0.75(12)=9\). The variance is \(ab/[(a+b)^2(a+b+1)]=27/[12^2(13)]=0.01442\). (2) Yes. The parameters are \(a=0.60(8)=4.8\) and \(b=0.40(8)=3.2\), both positive, and their sum is 8. They are not literal counts of observed responders and nonresponders. (3) For a \(\operatorname{Beta}(1,4)\) prior, \(P(p>x)=(1-x)^4\), so \(P(p>0.50)=0.50^4=0.0625\). This checks beliefs about the chance that the response probability exceeds a clinically relevant threshold. (4) No. The conversion uses \(m=0.20\), so the prior mean is 0.20; it cannot also be centered at 0.30. The statements need clarification before the prior is finalized.