Why the Object Being Discounted Matters
A power weight is often described informally as giving historical data “less influence.” To implement that idea, it matters what is being weighted. In a power prior, the weight is applied to the historical likelihood, which represents how the observed historical outcomes depend on the model parameter. A parameter density, by contrast, is a density relative to a particular parameter scale. Treating those two objects as interchangeable can change the analysis.
For a binary control outcome, let \(Y_0\) be the number of events among \(n_0\) historical controls, and let \(p\) be their event probability. The binomial likelihood is proportional to \(p^{Y_0}(1-p)^{n_0-Y_0}\). Applying a weight \(a\) to this likelihood discounts the historical counts’ contribution to the likelihood. Re-expressing the same probability as a logit does not change the likelihood contribution; however, a density on the logit scale is not the same density as one on the probability scale.
This tutorial considers a fixed \(a\), not a weight estimated from data. In the binary example, assume a common event probability for the historical and current control observations, independent binomial sampling conditional on that probability, and a proper baseline prior. Applying the model to a real trial also requires the endpoint and relevant patient populations to be sufficiently comparable for a common control probability to be defensible. A power weight does not repair a mismatch in those assumptions.
Likelihood Weighting in the Probability Parameterization
Write \(y_0\) for the observed historical event count and \(n_0-y_0\) for the non-event count. Ignoring the binomial coefficient, which does not depend on \(p\), the likelihood is \(L(p)\propto p^{y_0}(1-p)^{n_0-y_0}\). Under a baseline prior \(p\sim\operatorname{Beta}(\alpha,\beta)\), applying the power prior gives
Collecting the powers of \(p\) and \(1-p\) gives a Beta distribution with shape parameters \(\alpha+a y_0\) and \(\beta+a(n_0-y_0)\). Thus, in this model, the historical successes and non-events each contribute in proportion to \(a\), while the baseline prior is not itself multiplied by \(a\). With current data \(y_1\) events among \(n_1\) controls, the posterior shapes become \(\alpha+a y_0+y_1\) and \(\beta+a(n_0-y_0)+n_1-y_1\).
Worked Example: Discounting Historical Binary Outcomes
Suppose, hypothetically, that 4 of 20 historical controls had the event. Use a \(\operatorname{Beta}(1,1)\) baseline prior and fixed weight \(a=0.5\). Find the resulting distribution for \(p\) and its mean.
The likelihood is proportional to \(p^4(1-p)^{16}\). Raising it to \(0.5\) gives \(p^2(1-p)^8\). Multiplying by the uniform baseline density contributes no additional powers, so
The mean is \(3/(3+9)=0.2500\). This is the mean of the power-prior distribution for \(p\), not a statement that there is a 25% probability that the historical event rate equals a particular value. The fractional contributions—2 historical events and 8 non-events—are a convenient description of the likelihood’s exponents, not new observed patients.
Re-expressing the Same Model on the Logit Scale
Now define \(\theta=\operatorname{logit}(p)=\log\{p/(1-p)\}\), so that \(p=e^\theta/(1+e^\theta)\). The likelihood as a function of \(\theta\) is
The power prior applies \(a\) to this likelihood too: \(L(\theta)^a\propto e^{a y_0\theta}/(1+e^\theta)^{a n_0}\). It is the same historical likelihood contribution expressed in a different coordinate. But the prior density must also be expressed with respect to the new coordinate. Since \(dp/d\theta=p(1-p)\), a density \(\pi_p(p)\) on the probability scale corresponds to \(\pi_\theta(\theta)=\pi_p(p)p(1-p)\) on the logit scale.
This Jacobian factor is not an extra historical-data weight. It is part of expressing a probability-scale prior as a density over \(\theta\). For example, a uniform prior on \(p\) is not a uniform prior on \(\theta\): its logit-scale density is proportional to \(p(1-p)\). Using a flat logit-scale prior instead would specify a different baseline prior, and therefore a different analysis.
Worked Example: Checking the Same Power Prior on the Logit Scale
Use the hypothetical data from the preceding example: \(y_0=4\), \(n_0=20\), \(a=0.5\), and a uniform baseline prior on \(p\). Verify that expressing the model on the logit scale produces the same distribution for \(p\).
The powered likelihood on the logit scale is proportional to \(e^{2\theta}/(1+e^\theta)^{10}\). The uniform prior on \(p\), transformed to \(\theta\), contributes \(p(1-p)=e^\theta/(1+e^\theta)^2\). Their product is therefore
The last expression is a density kernel with respect to \(d\theta\). To identify the corresponding density with respect to \(dp\), use \(d\theta=dp/\{p(1-p)\}\). The resulting kernel in \(p\) is \(p^2(1-p)^8\), which is \(\operatorname{Beta}(3,9)\). It matches the probability-scale calculation. The agreement comes from keeping the same likelihood and transforming the same prior correctly.
Why Powering a Parameter Density Is Different
A common alternative is to take a historical posterior density and raise that density to \(a\). That procedure is not generally the power prior defined above. If the historical posterior is proportional to \(L(D_0\mid\theta)\pi_0(\theta)\), raising it to \(a\) gives a kernel proportional to \(L(D_0\mid\theta)^a\pi_0(\theta)^a\). The power prior instead uses \(L(D_0\mid\theta)^a\pi_0(\theta)\). The baseline prior has been weighted in the first construction but not the second.
There is an additional issue: a parameter density is defined relative to a coordinate. If a density is powered on the probability scale, then transformed to the logit scale, that is generally not the same operation as first transforming the density to the logit scale and then powering it. The density’s Jacobian factor is also raised in the latter operation. This dependence on the scale is why “discount the density” does not specify a unique, parameterization-independent data-weighting rule.
Worked Example: Powering the Historical Posterior Density
For the same hypothetical 4 events among 20 controls and a \(\operatorname{Beta}(1,1)\) baseline prior, the historical posterior on the probability scale is \(\operatorname{Beta}(5,17)\). Instead of powering the likelihood, suppose we raise the historical posterior density on the logit scale to \(a=0.5\), then express the result as a density on \(p\). Compare its mean with the power-prior result.
The logit-scale historical posterior density is proportional to \(p^5(1-p)^{17}\), because the probability-scale \(\operatorname{Beta}(5,17)\) density gains a factor \(p(1-p)\) when transformed. Raising this logit-scale density to \(0.5\) gives the kernel \(p^{2.5}(1-p)^{8.5}\) with respect to \(d\theta\). Since \(d\theta=dp/\{p(1-p)\}\), its kernel with respect to \(dp\) is \(p^{1.5}(1-p)^{7.5}\). Thus the resulting probability-scale distribution is
The likelihood power prior gave \(\operatorname{Beta}(3,9)\), with mean \(0.2500\). The difference is not a rounding effect: powering the logit-scale posterior density also powers the prior contribution and its scale-dependent density factor. The two constructions answer different questions.
Adding Current Controls and Checking the Model
Worked Example: Combining Discounted Historical and Current Controls
Suppose, hypothetically, the 4 events among 20 historical controls receive weight \(a=0.5\), and 3 events are observed among 10 current controls. Retain the \(\operatorname{Beta}(1,1)\) baseline prior and assume both groups have the same event probability. Calculate the posterior mean.
The weighted historical contributions are \(0.5(4)=2\) events and \(0.5(16)=8\) non-events. The current counts contribute 3 events and \(10-3=7\) non-events without discounting. Therefore,
The posterior mean is \(6/(6+16)=0.2727\), or approximately 27.3% on the event-probability scale. The calculation relies on a common-probability model; it does not establish that historical and current controls are comparable. If the endpoint definition, eligibility criteria, or care environment differ in ways that affect the event rate, the pooled model may be inappropriate even when \(a\) is small.
Useful checks include confirming that \(a=0\) removes historical counts from the posterior shapes, \(a=1\) includes them at full weight, and intermediate \(a\) scales both event and non-event contributions. Also check that the likelihood is written for the observed data and that any prior has the intended meaning on its stated scale. A surprising result after switching from \(p\) to \(\theta\) often signals a changed prior or a missing Jacobian, rather than a change in the binomial likelihood.
Approaches, Limitations, and a Common Pitfall
| Approach | What is weighted? | Interpretation and caution |
|---|---|---|
| Power prior | Historical likelihood \(L(D_0\mid\theta)^a\) | Weights historical likelihood information while retaining the baseline prior as specified. |
| Full historical borrowing | Historical likelihood with \(a=1\) | Uses the historical likelihood at full weight under the model. |
| No historical borrowing | Historical likelihood with \(a=0\) | Leaves the baseline prior and current data to determine the posterior. |
| Power a parameter density | A density such as the historical posterior density | Also changes prior contributions and depends on the parameter scale; it is not generally the same power-prior construction. |
A power prior’s weight is a modeling choice, not a guarantee against bias from historical-current differences. In the binomial-conjugate case, the weighted counts make the arithmetic transparent, but they should not be mistaken for literal additional participants or a complete account of uncertainty about between-study differences. A common probability model can fail, and a fixed \(a\) cannot make incompatible data exchangeable.
- Wrong reasoning: “I have a historical posterior density, so I can discount it by raising it to \(a\); this is the power prior.” Correction: The power prior raises the historical likelihood and multiplies by the baseline prior. Powering a posterior density also changes the prior contribution and depends on the density’s parameterization.
- Wrong reasoning: “A flat prior on the probability scale is flat on the logit scale.” Correction: A uniform density in \(p\) transforms to a density proportional to \(p(1-p)\) in \(\theta\). A flat prior on \(\theta\) is a different prior.
- Wrong reasoning: “Using a small \(a\) makes historical data safe to borrow regardless of differences.” Correction: A smaller weight reduces the historical likelihood contribution; it does not validate the common endpoint, population, or event-probability assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- With \(y_0=6\), \(n_0=30\), a \(\operatorname{Beta}(1,1)\) baseline prior, and \(a=0.25\), what is the power-prior distribution for \(p\)?
- For a probability-scale density \(\pi_p(p)\), what factor converts it to a density on \(\theta=\operatorname{logit}(p)\)?
- Why does raising a historical posterior density to \(a\) generally differ from raising the historical likelihood to \(a\)?
- For the data in question 1, what happens to the historical contribution at \(a=0\) and \(a=1\)?
Solutions. (1) There are 24 non-events, so the shapes are \(1+0.25(6)=2.5\) and \(1+0.25(24)=7\): \(\operatorname{Beta}(2.5,7)\). (2) Multiply by \(dp/d\theta=p(1-p)\), so \(\pi_\theta(\theta)=\pi_p(p)p(1-p)\). (3) The historical posterior includes both likelihood and baseline prior; powering it raises both contributions, and a density also depends on its parameter scale. (4) At \(a=0\), neither historical count contributes; at \(a=1\), all 6 events and 24 non-events contribute at full weight.