Fixed Versus Random Discount Parameters

Why the Discount Can Be Fixed or Random

A power-prior discount controls how much historical control data contribute to inference about a current trial’s control rate. Fixing the discount makes that contribution a prespecified fraction of the historical likelihood. Assigning a prior to the discount instead represents uncertainty about how much to borrow, and the current data can update that uncertainty. The resulting posterior is generally a mixture of posteriors, not the posterior obtained by substituting one average discount.

Consider binary outcomes. Let \(y_0\) of \(n_0\) historical control participants experience the event, and let \(y\) of \(n\) current control participants experience it. Assume a common event probability \(\theta\) for historical and current controls when borrowing is allowed. Let the baseline prior be \(\theta\sim\text{Beta}(a,b)\), where \(a,b>0\), and let \(\delta\in[0,1]\) be the power-prior discount. The binomial likelihoods are \(L_0(\theta)=\binom{n_0}{y_0}\theta^{y_0}(1-\theta)^{n_0-y_0}\) and \(L(\theta)=\binom{n}{y}\theta^y(1-\theta)^{n-y}\).

These definitions describe how the historical likelihood is used; they do not establish that the two studies truly share the same event probability. The shared-\(\theta\) assumption is the borrowing model’s substantive premise. A discount can reduce the influence of historical data, but it does not by itself model every possible difference between study populations, outcome definitions, or care settings.

Fixed Discounts: A Direct Beta Update

For a fixed value of \(\delta\), the posterior density is proportional to the baseline prior times the current likelihood times the discounted historical likelihood. Ignoring factors that do not depend on \(\theta\), the kernel is

This is the kernel of a beta distribution. Therefore,

The algebra works because exponentiating the historical binomial likelihood by \(\delta\) multiplies both historical event and non-event counts by \(\delta\). It does not mean that fractional participants were observed. It is a likelihood-weighting device. The beta posterior mean is the first shape parameter divided by the sum of the two shape parameters, and its variance is the product of the shape parameters divided by the squared shape sum times one plus that sum.

Random Discounts: Posterior Mixtures

To define a random-discount analysis precisely, first choose a joint-model convention. For the conditional power prior, the historical-data-adjusted prior for \(\theta\), conditional on \(\delta\), is normalized separately at each discount:

where \(D_0\) denotes the historical data and \(\pi_0\) is the \(\text{Beta}(a,b)\) baseline prior. A prior \(\pi(\delta)\) is then assigned to the discount. After observing current data, the joint posterior is proportional to \(L(\theta)\pi(\theta\mid\delta,D_0)\pi(\delta)\). Integrating out \(\delta\) gives the posterior for \(\theta\) as a mixture:

For a continuous discount, the integral averages the conditional posterior distributions using the posterior density of \(\delta\). For a discrete discount distribution, it is a weighted sum. The discount’s posterior density under this conditional formulation is proportional to the prior density \(\pi(\delta)\) times the current-data predictive probability under \(\pi(\theta\mid\delta,D_0)\):

This expression explains how uncertainty in borrowing enters the posterior. The current likelihood updates the relative support for discount values through their predictive fit to the current observations. The posterior for \(\theta\) then averages across the remaining uncertainty in \(\delta\), rather than treating one chosen discount as known.

Why the Normalization Convention Matters

A different construction, often called a joint power prior, uses a joint density proportional to \(L_0(\theta)^\delta\pi_0(\theta)\pi(\delta)\), with one overall normalizing constant rather than dividing by \(C(\delta)\) at each discount. These are different models for the random discount. In the conditional construction, the specified \(\pi(\delta)\) remains the discount distribution before seeing the current data, conditional on the historical data. In the joint construction, integrating over \(\theta\) weights each discount by \(C(\delta)\). As a result, even before the current data are observed, the induced distribution of \(\delta\) need not equal the nominal \(\pi(\delta)\).

Derivation: Uncertainty Adds a Between-Discount Component

A mixture’s variance includes variation within its component posteriors and variation between their means. Let \(m(\delta)=E(\theta\mid y,D_0,\delta)\) and \(v(\delta)=\operatorname{Var}(\theta\mid y,D_0,\delta)\). The law of total variance gives

To justify the identity, write \(\theta-E(\theta\mid y,D_0)\) as \([\theta-m(\delta)]+[m(\delta)-E(\theta\mid y,D_0)]\), square, and take the posterior expectation. The cross-product has expectation zero because \(E[\theta-m(\delta)\mid\delta,y,D_0]=0\). The first squared term gives the average conditional variance, and the second gives the variance of the conditional means.

The second term is nonnegative. Unless the component means are identical or the posterior puts all its mass on one discount value, uncertainty about \(\delta\) adds posterior variance beyond the average within-discount variance. This does not mean every credible interval must be wider than every fixed-discount interval: the fixed analysis may use a different component, and mixture quantiles depend on the full distribution. It does mean that reporting only a fixed-discount posterior can omit a source of uncertainty present in the random-discount model.

For the two-point example, the mixture mean is approximately 0.3614, but the mixture is not a beta distribution. A credible interval should therefore be calculated from the mixture cumulative distribution, \(0.3267F_{\text{Beta}(4,8)}(t)+0.6733F_{\text{Beta}(12,20)}(t)\), rather than from a beta distribution formed by averaging shape parameters. Likewise, the posterior probability of a clinically relevant event-rate range should be averaged using the mixture distribution.

Model Checks, Alternatives, and Limitations

A fixed discount is transparent and easy to communicate, but conclusions can depend strongly on the chosen value. A random discount propagates uncertainty about that choice, but it does not remove the need to specify the discount prior and the normalization convention. Neither approach automatically protects against all forms of historical-current conflict. In particular, a random discount can respond to conflict only through the way conflict affects the likelihood and the chosen model; the posterior may still borrow substantially when the current sample is small.

Useful checks include calculating results at \(\delta=0\) and \(\delta=1\) as boundary cases, inspecting the posterior distribution of \(\delta\), and comparing the conditional component posteriors over discounts with materially different posterior weight. Confirm that the likelihood, the baseline prior, and the discount convention used in computation match the model described. With a continuous prior on \(\delta\), numerical integration or simulation should adequately represent regions where the posterior discount density is concentrated. A computed posterior that changes materially under finer numerical resolution is a warning about inadequate computation, not a clinical finding.

The main alternatives answer different questions. A fixed-discount analysis conditions on one chosen borrowing level. A conditional random-discount analysis averages over borrowing levels while preserving the specified discount distribution before current data. A joint power-prior analysis uses a different joint distribution and can induce different discount weights. These analyses should not be treated as interchangeable sensitivity settings unless their prior and normalization choices are aligned. The model also relies on the binary binomial likelihood and a common \(\theta\); extensions to other outcomes or more complex historical-data structures require their own likelihood specifications.