Evidence and allocation answer different questions
In an adaptive clinical trial, accumulating response data can change which treatment appears most promising. That change may influence how the next participant is randomized, but the evidence about treatment effects and the randomization decision are not the same thing. Confusing them can make an allocation rule appear more certain—or more evidentially justified—than it is.
The distinction is between a probability about an unknown treatment parameter, conditional on the observed data, and a probability generated by a specified assignment procedure. The first summarizes the fitted Bayesian model. The second describes the trial’s randomization for an eligible participant at a particular decision time. An allocation probability is not itself a posterior probability that the assigned arm is best.
When the posterior distributions are continuous, exact ties for best ordinarily have probability zero, so the \(\pi_j(D_t)\) values sum to one across the eligible arms. If a model has point masses or ties are possible, the analysis must specify how ties are handled when calculating the probability of being best. Assignment probabilities also sum to one across the arms eligible for that participant. But the fact that both vectors sum to one does not make them interchangeable.
How an allocation rule transforms evidence
A rule takes current information—possibly posterior probabilities of being best—and produces assignment probabilities. The mapping is a design choice, not a consequence forced by the posterior. For example, a rule may allocate in proportion to \(\pi_j\), use a power of \(\pi_j\), or reserve some probability for exploration of every eligible arm. Rules that use other summaries, such as posterior expected outcomes, need not use \(\pi_j\) at all.
One common family assigns weights \(w_j=\pi_j^\gamma\), where \(\gamma>0\), then normalizes those weights. The normalization ensures a valid randomization distribution, but does not turn the resulting probabilities back into posterior probabilities of being best.
A short derivation shows how the tuning parameter changes allocation. For any two arms \(i\) and \(j\) with positive posterior-best probabilities, the rule gives
Thus \(\gamma=1\) gives \(q_j=\pi_j\), while \(\gamma>1\) exaggerates the relative allocation to arms with larger \(\pi_j\), and \(0<\gamma<1\) makes the allocation distribution flatter. The exponent changes the assignment policy; it does not change the posterior evidence.
A rule may also set aside a fraction \(\varepsilon\) for uniform exploration across \(K\) eligible arms. If the remaining fraction follows the powered weights, the assignment probability is
This mixture still sums to one: the uniform terms sum to \(\varepsilon\), and the normalized powered terms sum to \(1-\varepsilon\). It can give an arm a positive assignment probability even when its posterior probability of being best is very small. Conversely, constraints or a different rule could result in no assignments to an arm despite some posterior probability that it is best.
Worked Example: A Square-Root Rule Is Not the Posterior
Suppose a hypothetical analysis of three eligible arms uses 10,000 posterior draws. In 5,000 draws arm A has the largest treatment parameter, in 3,000 arm B does, and in 2,000 arm C does. The estimated posterior-best probabilities are therefore \((0.50,0.30,0.20)\). Consider the square-root allocation rule, \(\gamma=0.5\), without an exploration mixture.
The weights are \(w_A=\sqrt{0.50}\approx0.7071\), \(w_B=\sqrt{0.30}\approx0.5477\), and \(w_C=\sqrt{0.20}\approx0.4472\). Their sum is approximately \(1.7020\). Normalizing gives
The values \(0.50, 0.30, 0.20\) summarize posterior evidence. The values \(0.415, 0.322, 0.263\) are the rule’s probabilities for assigning the next participant. For example, \(0.415\) does not mean the posterior probability that A is best is \(0.415\); that posterior probability is \(0.50\). If 60 future assignments were made under unchanged probabilities, the expected counts would be approximately \(60(0.415)=24.9\), \(60(0.322)=19.3\), and \(60(0.263)=15.8\). These are expectations, not guaranteed realized counts.
Worked counterexamples
Worked Example: A High Probability of Being Best, but a Different Assignment Probability
Suppose a hypothetical posterior analysis gives \((\pi_A,\pi_B,\pi_C)=(0.80,0.15,0.05)\). The trial uses a squared-probability rule, \(\gamma=2\), with an exploration fraction \(\varepsilon=0.15\) spread equally across the three arms.
First calculate the powered weights: \(0.80^2=0.64\), \(0.15^2=0.0225\), and \(0.05^2=0.0025\). Their sum is \(0.665\), so the normalized weights are approximately \(0.9624\), \(0.03383\), and \(0.003759\). The uniform exploration contribution to each arm is \(0.15/3=0.05\). Therefore,
The assignment probabilities sum to approximately one, allowing for rounding. Arm A has posterior probability \(0.80\) of being best, but its assignment probability is approximately \(0.868\). Arms B and C each receive assignment probabilities greater than their posterior-best probabilities because the uniform exploration component adds probability to them. Neither difference indicates an arithmetic inconsistency: the posterior vector and allocation vector answer different questions.
Worked Example: Same Evidence, Three Allocation Policies
Suppose three arms have posterior-best probabilities \((0.70,0.20,0.10)\). Compare three rules: direct proportional allocation; a square-root rule; and a rule reserving 10% of assignment probability for equal exploration, with the other 90% allocated in proportion to posterior-best probability.
Under direct proportional allocation, \(q=(0.70,0.20,0.10)\). This is the special case in which assignment probabilities equal posterior-best probabilities. Under the square-root rule, the weights are \(\sqrt{0.70}\approx0.8367\), \(\sqrt{0.20}\approx0.4472\), and \(\sqrt{0.10}\approx0.3162\), with sum approximately \(1.6001\). The resulting allocation probabilities are approximately \((0.5229,0.2795,0.1976)\).
For the exploration-mixture rule, each arm receives \(0.10/3\approx0.03333\) from exploration, and the remaining probabilities are \(0.90(0.70,0.20,0.10)\). Thus
All three rules use the same posterior evidence, but they produce different assignment probabilities. Under the mixture rule, for example, the probability of assignment to arm A is about \(0.6633\), not \(0.70\). The rule deliberately allocates some probability to the other arms. With 30 assignments and unchanged probabilities, the expected counts under this rule are approximately \(19.9\), \(6.40\), and \(3.70\), respectively.
Assumptions, interpretation, and checks
The posterior-best probabilities are meaningful only relative to the specified outcome model, prior, data, and definition of “best.” The treatment parameter must be oriented consistently—for example, larger response probability may be better, whereas a smaller event rate may be better after defining the parameter accordingly. The posterior calculation also requires the response data and model to be handled as specified, including the timing of responses used at the decision. The assignment probabilities, in turn, require a fully specified randomization rule and a clear set of arms eligible at that moment.
To audit a decision, record the posterior summary used by the rule, the rule’s formula and tuning parameters, any exploration or operational constraints, and the resulting \(q_j(D_t)\). Then verify that each probability is between zero and one and that the probabilities across eligible arms sum to one. If the rule uses posterior draws directly rather than estimated \(\pi_j\) values, document how the draws are converted into assignment probabilities. Posterior simulation error can affect the estimates of \(\pi_j\), and hence the resulting \(q_j\); it does not erase the conceptual distinction between the quantities.
An adaptive assignment probability is conditional on the information available at a particular decision. If the data or eligibility set changes, the probability for a later participant may change too. Nor does a probability for one assignment guarantee the same fraction of participants will be assigned to each arm over a finite trial. Randomization produces a realized sequence of assignments, which can differ from its probabilities.
| Quantity | Question answered | What determines it? |
|---|---|---|
| \(\pi_j(D_t)\) | Given the model and current data, how probable is it that arm \(j\) has the best parameter? | Posterior distribution and definition of best |
| \(q_j(D_t)\) | Under the rule in force, what is the chance the next participant is assigned to arm \(j\)? | Randomization rule, current information, and design constraints |
| Observed allocation fraction | What fraction of participants was actually assigned to arm \(j\)? | Realized assignments over the specified period |
These quantities can coincide under a particular rule, but coincidence is not their definition. Direct proportional allocation makes \(q_j=\pi_j\) by choice. The square-root and exploration examples show that other rules can yield different values with exactly the same posterior evidence. Fixed allocation is another alternative: its assignment probabilities are set by design and need not change with accumulating outcomes. A posterior expected outcome is also a different summary from the probability of being best; a rule based on expected outcomes may therefore produce yet another allocation vector.
Common mistake and practical limits
- Wrong reasoning: “The posterior probability that A is best is 80%, so the next participant has an 80% chance of receiving A.” This conclusion does not follow unless the allocation rule specifically sets \(q_A=\pi_A\).
- Correction: Calculate the posterior probability of being best from the posterior model, then apply the documented randomization rule to obtain the assignment probability. Label and report the two quantities separately.
- Wrong reasoning: “A realized assignment fraction of 40% proves the algorithm assigned that arm with probability 40%.” A finite sequence of random assignments can differ from the probabilities used at each decision, especially if those probabilities changed over time.
- Correction: Report the rule’s conditional probabilities at each decision, or a clearly defined summary of them, separately from realized counts or fractions.
A correct distinction does not establish that a rule is ethically preferable, statistically efficient, or well calibrated. Those are design questions requiring their own justification. The practical point here is narrower: evidence is summarized by the posterior, while assignment is governed by a policy. Communicating one as if it were the other can mislead investigators, clinicians, and participants about what the trial’s randomization means.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- An analysis gives posterior-best probabilities \((0.60,0.30,0.10)\). Under direct proportional allocation, what are the assignment probabilities?
- For the same posterior vector, is an assignment probability of \(0.60\) required by Bayesian inference? Explain.
- Two arms have posterior-best probabilities \(0.40\) and \(0.10\). Under a powered rule with \(\gamma=2\), what is their assignment-probability ratio?
- Why can an arm have positive assignment probability even if its posterior probability of being best is zero under the current model?
Solutions. (1) The direct proportional rule sets \(q_j=\pi_j\), so the assignment probabilities are \((0.60,0.30,0.10)\). (2) No. That equality follows from this chosen rule; another rule, such as a square-root rule or an exploration mixture, can produce a different assignment probability from the same posterior. (3) The powered rule gives \(q_1/q_2=(0.40/0.10)^2=4^2=16\). Normalizing across all arms affects the individual probabilities but not this pairwise ratio. (4) A rule may include an exploration component that assigns positive probability independently of the posterior-best weight, as in a uniform mixture. That is a property of the allocation policy, not evidence that the arm has positive posterior probability of being best.