Choosing Priors for Treatment Response Rates

Why Prior Choice Matters

In Thompson sampling for a clinical trial, each treatment arm has an uncertain response rate. Before any outcomes are observed, a prior distribution represents the uncertainty assigned to that rate. The prior then combines with trial outcomes to form a posterior distribution, from which Thompson sampling draws candidate response rates. The probability that an arm’s draw is highest determines its allocation probability.

Consequently, the prior does more than set a starting estimate. It also determines how much uncertainty the algorithm represents before data arrive, and may influence which arm is more likely to receive the first participants. A prior should be chosen to express defensible assumptions about the response rate, not simply because it is mathematically convenient.

Definition: For treatment arm \(i\), let \(\theta_i\) be its binary response probability. A Beta prior, written \(\theta_i\sim\operatorname{Beta}(a_i,b_i)\) for \(a_i,b_i>0\), has density proportional to \(\theta_i^{a_i-1}(1-\theta_i)^{b_i-1}\) on \(0<\theta_i<1\). Its prior mean is \(a_i/(a_i+b_i)\), and its prior variance is \(a_i b_i/\{(a_i+b_i)^2(a_i+b_i+1)\}\).

The comparisons here assume that, conditional on \(\theta_i\), responses in arm \(i\) are Bernoulli outcomes with probability \(\theta_i\), and that the response definition and follow-up are consistent with the clinical question. A prior distribution is a model for uncertainty about this response probability; it is not a claim that the probability itself changes randomly from patient to patient. For multiple arms, priors may be specified separately. Unless a model explicitly links arms, separate priors do not borrow information across arms.

Three Common Choices

The uniform prior, \(\operatorname{Beta}(1,1)\), has constant density over response probabilities from zero to one. Its mean is \(0.5\). This is a convenient neutral starting point on the probability scale, but it is not assumption-free: it gives equal density to equal-length intervals on that scale, and so encodes a particular notion of what “flat” means.

The Jeffreys prior for a Bernoulli probability is \(\operatorname{Beta}(1/2,1/2)\). Its density is proportional to \(\theta^{-1/2}(1-\theta)^{-1/2}\), so it is larger near zero and one than in the middle. It has mean \(0.5\), but more prior variance than the uniform prior. Its standard justification is invariance under smooth reparameterization: the prior transforms consistently when the parameter is expressed on another smooth scale. That property does not make it universally appropriate for every clinical setting.

A clinically informed Beta prior, \(\operatorname{Beta}(a,b)\), uses parameters chosen to represent a clinically defensible expectation and degree of uncertainty. For example, \(\operatorname{Beta}(4,6)\) has mean \(0.4\) and is more concentrated than either of the preceding priors. The sum \(a+b\) is often called the prior effective sample size: algebraically, it determines the weight of the prior mean in the posterior mean. It is a useful measure of concentration, not a guarantee that the prior is literally equivalent to observing \(a+b\) patients.

Prior	Mean	Variance	Key implication
\(\operatorname{Beta}(1,1)\), uniform	0.5	1/12	Equal density across the probability scale
\(\operatorname{Beta}(1/2,1/2)\), Jeffreys	0.5	1/8	More mass near the boundaries than the uniform prior
\(\operatorname{Beta}(a,b)\), clinically informed	\(a/(a+b)\)	\(ab/\{(a+b)^2(a+b+1)\}\)	Can represent a context-specific centre and degree of uncertainty

The Jeffreys density is unbounded at the endpoints, but its total probability is finite. It does not assign positive probability to exactly zero or exactly one: those individual points have probability zero under any continuous Beta prior. A high density near a boundary is therefore not the same as a point mass at that boundary.

Worked Example: Uniform and Jeffreys Priors After One Response

Suppose, hypothetically, a trial has observed one responder in an arm and no other outcomes. Compare the posterior means under a uniform prior and a Jeffreys prior. Assume the response is Bernoulli with a common response probability for this arm.

For a Beta prior and \(x\) responses among \(n\) participants, the posterior is \(\operatorname{Beta}(a+x,b+n-x)\). With one response, \(x=1\) and \(n=1\). The uniform prior gives \(\operatorname{Beta}(1,1)\), hence

\theta\mid \text{data}\sim\operatorname{Beta}(1+1,1+0) =\operatorname{Beta}(2,1), \qquad E(\theta\mid\text{data})=\frac{2}{3}\approx 0.667.

The Jeffreys prior gives \(\operatorname{Beta}(1/2,1/2)\), hence

\theta\mid \text{data}\sim\operatorname{Beta}(1.5,0.5), \qquad E(\theta\mid\text{data})=\frac{1.5}{1.5+0.5}=0.750.

Both priors have mean \(0.5\) before the outcome, but they do not respond identically to a single success. The Jeffreys prior’s greater concentration near the boundaries affects its update. These posterior means describe uncertainty about the arm’s response probability; they are not, by themselves, the Thompson sampling allocation probabilities.

Updating and the Initial Allocation

The Beta family is conjugate to the Bernoulli likelihood. If \(x\) responses are observed among \(n\) participants in arm \(i\), the likelihood is proportional to \(\theta_i^x(1-\theta_i)^{n-x}\). Multiplying it by the Beta prior density gives

\theta_i\mid\text{data}\sim \operatorname{Beta}(a_i+x,\ b_i+n-x).

This update gives the posterior mean

E(\theta_i\mid\text{data}) =\frac{a_i+x}{a_i+b_i+n} =\frac{a_i+b_i}{a_i+b_i+n}\frac{a_i}{a_i+b_i} +\frac{n}{a_i+b_i+n}\frac{x}{n},

for \(n>0\). The posterior mean is a weighted average of the prior mean and the observed response proportion, with weights determined by prior concentration and sample size. This identity helps explain why a more concentrated prior typically moves less after a fixed amount of data. It does not imply that every feature of the posterior, including its tail probabilities, can be understood from the mean alone.

At the start of a Thompson sampling trial, there are no observed outcomes, so the allocation probabilities are determined by the prior distributions. For \(K\) arms, draw one response probability from each arm’s prior and allocate to the arm with the largest draw. Thus arm \(i\)’s initial allocation probability is the probability that its prior draw is largest. With independent, identical, continuous priors across arms, symmetry gives each arm probability \(1/K\): ties have probability zero, and no arm is favored by the joint distribution. Different prior distributions need not produce equal initial allocation probabilities.

Worked Example: Initial Allocation Under Two Different Priors

Consider two hypothetical treatments. For illustration, suppose clinical knowledge leads to independent priors \(\theta_A\sim\operatorname{Beta}(2,1)\) and \(\theta_B\sim\operatorname{Beta}(1,2)\). These priors have means \(2/3\) and \(1/3\), respectively. Assume the Bernoulli response model is appropriate and that the priors represent the available pretrial information.

The density of \(\theta_A\) is \(2u\), and the cumulative distribution function of \(\theta_B\) is \(F_B(u)=2u-u^2\) for \(0<u<1\). Since the draws are independent, the probability that arm A is selected initially is

P(\theta_A>\theta_B) =\int_0^1 2u(2u-u^2)\,du =\int_0^1(4u^2-2u^3)\,du =\frac{4}{3}-\frac{1}{2} =\frac{5}{6}\approx 0.833.

Arm B’s probability is \(1-5/6=1/6\approx0.167\), because the continuous draws tie with probability zero. These values are the initial Thompson sampling allocation probabilities under the specified priors—not probabilities that either treatment is truly superior. The example also illustrates why clinically informed priors must be justified carefully: when prior beliefs differ substantially, they can strongly influence allocation before trial outcomes are observed.

Worked Example: Prior Strength and a Short Response History

Suppose, hypothetically, an arm has a clinically informed prior \(\operatorname{Beta}(4,6)\), with mean \(4/10=0.400\). After 10 participants, 3 have responded. Assume outcomes satisfy the Bernoulli model and the prior applies to the same response definition, patient population, and treatment regimen.

The posterior is \(\operatorname{Beta}(4+3,6+7)=\operatorname{Beta}(7,13)\), so its mean is

E(\theta\mid\text{data})=\frac{7}{7+13} =\frac{7}{20}=0.350.

For comparison, a uniform prior with the same data gives \(\operatorname{Beta}(4,8)\), whose mean is \(4/12\approx0.333\). The observed proportion is \(3/10=0.300\). The clinically informed prior pulls the posterior mean toward \(0.400\) more strongly because its prior concentration is \(10\), compared with \(2\) for the uniform prior. The difference in posterior means is not evidence that one prior is correct; it shows that a prior choice can matter with a modest sample.

Choosing and Checking a Prior

The uniform and Jeffreys priors are useful reference choices, but neither automatically reflects the clinical setting. A clinically informed prior can incorporate relevant historical evidence or expert knowledge, provided that the information is sufficiently comparable to the planned trial. Differences in eligibility criteria, outcome definitions, follow-up, background care, or treatment delivery can make apparently relevant historical response rates misleading.

Before using a prior in an adaptive trial, examine its implications on the response-rate scale. Check its mean and spread, and inspect whether it assigns substantial probability to response rates that clinical knowledge regards as implausible. Also compare the posterior and allocation probabilities under reasonable alternatives. Large changes signal that the design is prior-sensitive and should be reported and considered in planning.

A prior-predictive check asks what outcomes the prior implies before the trial data are observed. For a single future Bernoulli response, the prior predictive response probability is \(E(\theta)=a/(a+b)\). For multiple future responses, the Beta-Bernoulli model implies a beta-binomial predictive distribution. These checks can reveal implausible implications, but they do not prove that a prior is valid. A poor fit between historical evidence and the planned trial remains a substantive concern even when the prior’s numerical summaries look reasonable.

These checks complement, rather than replace, the clinical rationale for a prior. In particular, a prior’s effective sample size is a summary of mathematical concentration. It should not be treated as a literal count of exchangeable patients unless the evidence and model support that interpretation.

Common Mistakes and Limitations

Calling the uniform prior uninformative. It is flat on the response-probability scale, which is a choice of scale and assumption—not the absence of assumptions.
Reading the Jeffreys density as endpoint certainty. Its density rises near zero and one, but it assigns probability zero to either exact endpoint.
Equating prior effective sample size with real participants. The sum \(a+b\) controls the algebraic weight of the prior mean; its clinical interpretation depends on how the prior was constructed.
Assuming equal prior means guarantee equal initial allocation. Thompson sampling compares draws, so the entire prior distributions matter. Equal means alone do not establish equal probabilities of being largest.
Ignoring transportability. Historical response information may not apply if the target patients, treatment, outcome definition, or follow-up differ.
Reporting only the posterior mean. Allocation depends on posterior probabilities of being best, which can differ even when posterior means are similar.

No single prior is best in every trial. Uniform and Jeffreys priors offer standard reference assumptions; clinically informed priors can encode context-specific knowledge, but require justification and sensitivity checks. The appropriate choice depends on the scientific setting, the information available before enrollment, and the consequences of allowing those beliefs to affect early allocation.

Check Your Understanding

Answer each question, then compare with the solutions below.

What are the mean and variance of a \(\operatorname{Beta}(1,1)\) prior? How do its assumptions differ from those of a Jeffreys prior?
An arm has prior \(\operatorname{Beta}(3,7)\) and then records 4 responses among 10 participants. Find the posterior distribution and posterior mean.
Two arms have independent, identical continuous priors. What is each arm’s initial Thompson sampling allocation probability, and why?
Why does a prior effective sample size not automatically equal the number of real historical patients represented?

Solutions. (1) The mean is \(1/(1+1)=0.5\), and the variance is \(1\cdot1/\{(2)^2(3)\}=1/12\). The uniform prior has constant density on the probability scale; Jeffreys’ \(\operatorname{Beta}(1/2,1/2)\) has more density near the boundaries and is justified by a reparameterization-invariance property. (2) The posterior is \(\operatorname{Beta}(3+4,7+6)=\operatorname{Beta}(7,13)\), with mean \(7/20=0.350\). (3) Each arm’s probability is \(1/K\) when there are \(K\) arms. Identical continuous priors make the arms exchangeable, and ties occur with probability zero, so symmetry gives equal probabilities. (4) The sum of the Beta parameters determines algebraic concentration, but interpreting it as actual historical patients requires those data to be relevant and suitably modeled as comparable to the trial population and outcome.