Worked Binary-Response Generalized Urn

From an Urn Rule to Exact Probabilities

A generalized urn rule becomes concrete when its starting composition, response model, and replacement rule are all specified. The resulting calculations answer two different questions: what urn composition can follow the next assignment, and how likely is each treatment to be assigned after that update? Keeping those questions separate is essential. A state probability describes the urn; an allocation probability describes a draw from the urn.

Suppose there are \(K\) treatments. Just before patient \(n\) is assigned, let \(X_n=(X_{n1},\ldots,X_{nK})\) be the numbers of balls of the respective treatment types, and let \(S_n=\sum_{j=1}^K X_{nj}\) be the total. The patient receives treatment \(j\) with probability \(X_{nj}/S_n\). Write \(A_n=j\) when this occurs. The response \(R_n\) is binary, with \(R_n=1\) for success and \(R_n=0\) for failure. For each treatment \(j\), assume the response probability is \(p_j\), so that \(P(R_n=1\mid A_n=j,X_n)=p_j\).

For this tutorial, the response probability does not otherwise depend on the current urn composition or previous patients’ outcomes. Once treatment and response are known, the replacement increment is fixed by the rule. These assumptions define a simple one-step model; they are not claims that response probabilities are known in an actual trial.

All compositions that can be reached must have nonnegative entries and a positive total, so that a subsequent draw is defined. Increments may in principle contain negative entries, but feasibility must be checked on every reachable path. The calculations below use fixed increments and feasible states.

One-Step State Probabilities

Given a current composition \(x=(x_1,\ldots,x_K)\) with total \(S=\sum_j x_j>0\), the probability of drawing treatment \(j\) is \(x_j/S\). Conditional on that draw, success has probability \(p_j\) and failure has probability \(1-p_j\). Therefore, the joint probabilities of the two possible draw-response events are

Each event produces a composition \(x+d_{j,r}\). If different events produce the same composition, their probabilities must be added. For any candidate next state \(y\), the transition probability is

Here \(\mathbf{1}\{\cdot\}\) equals 1 when its condition is true and 0 otherwise. This expression accounts for all draw-response events leading to \(y\), including ties in the resulting state. Summing the transition probabilities over all distinct next states gives 1: for each treatment, the success and failure probabilities add to 1, and the treatment-draw probabilities add to 1.

Allocation Probabilities After the Update

After the urn changes to state \(y\), the next patient’s probability of receiving treatment \(k\), conditional on that state, is \(y_k/\sum_{\ell=1}^K y_\ell\). This is a conditional allocation probability. Before the updated state is observed, the probability averages over its possible values:

This is an application of the law of total probability: the next assignment can be calculated within each possible updated state and then averaged using that state’s probability. The state total can differ across states when replacement increments add or remove different numbers of balls, so the conditional allocation fraction must be calculated separately for each state.

Why the Calculation Works—and How to Check It

The transition formula follows directly by partitioning the next state according to the current draw and response. These events are mutually exclusive and exhaustive: exactly one treatment is drawn, and its response is either success or failure. For a fixed treatment \(j\), the probabilities of the two events sum to

Summing this identity over treatments gives \(\sum_j x_j/S=1\). Each event contributes its probability to exactly one resulting state, although several events may contribute to the same state. Therefore the probabilities of all distinct next states sum to 1. For the next allocation, the conditional treatment fractions in each possible state also sum to 1. Averaging those fractions over states consequently gives allocation probabilities that sum to 1 as well.

A practical calculation can be organized as a short sequence: list each treatment-response event, calculate its joint probability, apply its increment, aggregate probabilities for any duplicate states, and only then calculate the next allocation probabilities. This order keeps state transitions distinct from subsequent urn draws.

Diagnostics, Comparisons, and Limitations

Several checks catch common errors. The current treatment-draw probabilities must sum to one. For every treatment, its success and failure probabilities must sum to one. The event probabilities must therefore sum to one, and after aggregating paths, the distinct state probabilities must still sum to one. Each updated composition must have nonnegative counts and a positive total. Finally, the allocation fractions within every state must sum to one.

A frequent mistake is to report \(x_j/S\) as the probability of the next allocation even after an update has occurred. That is the draw probability from the current state. The next patient’s probability conditional on updated state \(y\) is \(y_j/\sum_k y_k\); if \(y\) is not known, average this fraction over the possible updated states. Another mistake is to treat an increment as a final count. For example, adding \(d_{j,r}\) means \(x+d_{j,r}\), not replacing \(x\) with \(d_{j,r}\).

This calculation is more explicit than simply stating a generalized urn rule: it gives exact finite-step probabilities for a specified state and response model. A rule with a different replacement scheme changes the increments and therefore the transition probabilities; a state-dependent scheme requires using the increment appropriate to the current state as well as the draw and response. A fixed allocation plan, by contrast, does not update treatment probabilities through urn replacement. These approaches answer related design questions but are not interchangeable calculations.

The one-step calculation is exact under the stated model, but it does not establish that the assumed response probabilities are correct, that outcomes are observed immediately, or that a particular replacement rule is desirable. If responses are delayed, the update may occur after a different number of assignments; if response probabilities vary over time or depend on history, the constant \(p_j\) model needs modification. Exact arithmetic cannot compensate for an incorrectly specified rule or response model.