Why the Precision Prior Matters
A commensurate prior links a current treatment effect to a historical effect while allowing them to differ. The precision in that link controls how tightly they are expected to agree: low precision allows a wide discrepancy, whereas high precision concentrates the discrepancy near zero. Choosing a prior for this precision is therefore choosing how much room the model gives the current effect to depart from the historical one.
There is no universal precision value that means “strong borrowing.” Its interpretation depends on the units of the treatment effect and on how uncertain the historical estimate is. A prior should be judged by the discrepancies and borrowing it makes plausible in the intended clinical setting, not only by its algebraic form or a single summary such as its mean.
The model says that, conditional on a given precision, the effects are centered on one another. It does not say they must be equal: for every finite \(\tau\), the discrepancy remains uncertain. A continuous prior on positive \(\tau\) assigns probability zero to the exact value \(\tau=0\), but may put substantial probability near zero. That distinction matters: near-zero precision allows a very broad discrepancy; it is not an exact “no borrowing” point mass.
How Precision Controls Potential Borrowing
To make the link concrete, suppose the current trial provides an approximately normal estimate \(\hat{\theta}_C\) with sampling variance \(s_C^2\), and the historical effect is represented by an independent normal distribution \(\theta_H\sim N(m_H,V_H)\). This summary is appropriate only when the historical information is relevant to the current estimand and the uncertainty \(V_H\) is carried forward without reusing current data. Conditional on \(\tau\), integrating over \(\theta_H\) gives \(\theta_C\sim N(m_H,V_H+1/\tau)\).
Combining that conditional prior with the current normal likelihood gives the following posterior mean. Its coefficient on the historical mean is a useful measure of the borrowing available at a particular value of \(\tau\).
Here \(B(\tau)\) is the conditional weight on the historical mean. As \(\tau\) approaches zero, \(1/\tau\) grows and \(B(\tau)\) approaches zero: the historical link becomes weak. As \(\tau\) grows without bound, \(B(\tau)\) approaches \(s_C^2/(s_C^2+V_H)\). If the historical effect is treated as known, \(V_H=0\), this upper limit is one; if it is uncertain, its uncertainty limits the maximum weight.
This calculation is conditional on \(\tau\). When \(\tau\) is assigned a prior and estimated jointly, the posterior distribution averages over possible precision values; the prior alone does not determine the realized borrowing. Still, the prior determines which strengths of link are available before seeing the current data, and therefore deserves direct examination.
Worked Example: Translating an Exponential Precision Prior
Suppose effects are measured in points and the prior is exponential with rate \(0.04\) point\(^{2}\). Its density is \(\pi(\tau)=0.04e^{-0.04\tau}\), for \(\tau>0\). The rate has units point\(^{2}\), so that the exponent is dimensionless and \(\tau\) has units point\(^{-2}\).
The mean precision is \(1/0.04=25\) point\(^{-2}\), while the median solves \(1-e^{-0.04\tau}=0.5\): \(\operatorname{median}(\tau)=\log(2)/0.04\approx17.33\) point\(^{-2}\). The discrepancy standard deviation conditional on a particular \(\tau\) is \(\sigma_\Delta=1/\sqrt{\tau}\). Because this is a monotone decreasing transformation, its median under this prior is also about \(1/\sqrt{17.33}=0.240\) point.
The probability of precision below \(1\) point\(^{-2}\) is \(P(\tau<1)=1-e^{-0.04}\approx0.03921\). The probability above \(50\) point\(^{-2}\) is \(P(\tau>50)=e^{-0.04(50)}=e^{-2}\approx0.1353\). These thresholds describe prior mass in a broad-link region and a tighter-link region, respectively. Whether those regions are clinically meaningful must be assessed in points, through their implied discrepancies.
Comparing Prior Shapes on Precision
Different prior families can have similar central values but markedly different probabilities near zero or in the upper tail. Those differences are especially important because near-zero precision permits weak borrowing, while large precision makes the prior discrepancy small. For a lognormal prior, \(\log(\tau)\) is normal; this guarantees positive precision and can represent a broad range of values multiplicatively. An exponential prior also has positive support but has a particular shape: its density is largest near zero and decreases steadily as precision increases.
Worked Example: A Lognormal Prior and Its Tails
Let \(\log(\tau)\sim N(\log(25),1)\), with \(\tau\) measured in point\(^{-2}\). The median precision is \(25\) point\(^{-2}\), so the median of the monotone transformation \(\sigma_\Delta=1/\sqrt{\tau}\) is \(0.200\) point.
For the threshold \(1\) point\(^{-2}\), \(P(\tau<1)=\Phi((\log(1)-\log(25))/1)=\Phi(-\log 25)=\Phi(-3.2189)\approx0.0006435\). For \(100\) point\(^{-2}\), \(P(\tau>100)=1-\Phi((\log(100)-\log(25))/1)=1-\Phi(\log 4)=1-\Phi(1.3863)\approx0.08283\).
At these thresholds, the lognormal prior assigns much less mass below \(1\) than the exponential prior in the previous example, and about \(8.283\%\) above \(100\). The comparison is not a ranking of priors: it shows why a median alone is insufficient. The priors’ probabilities over scientifically interpretable regions differ, and those differences affect the discrepancy scales available for borrowing.
A useful comparison begins with the same thresholds for every candidate prior. For example, define a broad-discrepancy region by \(\tau<\tau_L\) and a tight-link region by \(\tau>\tau_U\), where the thresholds are selected from effect-scale considerations. Then calculate the prior probabilities of both regions. Also inspect intermediate values: a prior can put little mass in either extreme while concentrating at moderate precision. A table of a few quantiles of \(\tau\), together with the corresponding \(1/\sqrt{\tau}\), can make the interpretation clearer.
| Prior feature | Implication for the discrepancy | Question to assess |
|---|---|---|
| Substantial mass near zero precision | Allows large discrepancy standard deviations and weak historical influence | Does the model permit enough departure if the historical effect is not transportable? |
| Substantial mass at large precision | Allows a very small discrepancy standard deviation and a tight historical link | Could this encourage excessive agreement despite meaningful clinical differences? |
| Concentration over a narrow range | Restricts the plausible discrepancy scales | Are those scales credible before the current trial results are observed? |
Scale Choices and Induced Priors
A prior may be specified directly on \(\tau\), or indirectly by first specifying a distribution for the discrepancy standard deviation \(\sigma_\Delta=1/\sqrt{\tau}\). These choices are not interchangeable by name: a seemingly simple prior on \(\sigma_\Delta\) induces a particular density on precision. The change of variables makes that implication explicit.
Worked Example: A Half-Normal Prior on Discrepancy Scale
Suppose \(\sigma_\Delta\) has a half-normal prior with scale \(A>0\), measured in effect units. Its density for \(\sigma_\Delta>0\) is \(f(\sigma_\Delta)=\sqrt{2/\pi}\,A^{-1}\exp[-\sigma_\Delta^2/(2A^2)]\). Since \(\sigma_\Delta=\tau^{-1/2}\), the absolute derivative is \(\left|d\sigma_\Delta/d\tau\right|=\tfrac12\tau^{-3/2}\).
Applying the change-of-variables formula gives the precision density \[ \pi(\tau) = \frac{\sqrt{2/\pi}}{2A}\tau^{-3/2} \exp\!\left(-\frac{1}{2A^2\tau}\right), \qquad \tau>0. \]
As \(\tau\) approaches zero, the exponential factor suppresses the density; as \(\tau\) becomes large, the density has a \(\tau^{-3/2}\) tail. Thus this prior can allow very large precision values even though it was specified on a discrepancy scale. In fact, its mean precision is infinite: for large \(\tau\), the integrand \(\tau\pi(\tau)\) is proportional to \(\tau^{-1/2}\), whose integral to infinity diverges. This is a reason to inspect tail probabilities and implied discrepancy scales rather than relying on the prior’s name.
Checking Prior Implications
A practical assessment has two parts. First, calculate prior probabilities for interpretable precision regions and translate representative values into discrepancy standard deviations in the endpoint’s units. Second, propagate the prior through the model to see what differences between current and historical effects it predicts before current data are observed. Under the conditional normal model, this means drawing \(\tau\) from its prior and then drawing \(\Delta=\theta_C-\theta_H\) from \(N(0,1/\tau)\). The resulting distribution is a mixture of normal discrepancies; its tails may be much wider than those suggested by a single median precision.
The same prior draws can be used to calculate the conditional historical weight \(B(\tau)\) for plausible values of \(s_C^2\) and \(V_H\). This reveals whether the prior places meaningful probability on near-zero borrowing, near-maximum borrowing, or mainly intermediate values. Such checks should reflect the current trial’s design and the uncertainty in the historical estimate; a precision prior does not have a context-free borrowing interpretation.
State whether the treatment effect is measured in points, a log-odds unit, or another scale. Precision is in the inverse square of that unit.
Translate clinically meaningful differences into plausible standard deviations, then into corresponding precision thresholds using \(\tau=1/\sigma_\Delta^2\).
Calculate probability near zero and in the upper tail, and inspect quantiles rather than comparing only means or medians.
Simulate discrepancies and conditional historical weights under plausible trial variances. Revise the prior if these implications do not match the intended protection or borrowing behavior.
Common Pitfalls and Limitations
- Reversing the precision direction. A larger \(\tau\) means a smaller conditional discrepancy variance \(1/\tau\), not a wider link.
- Reading precision without units. A value such as \(25\) has little meaning until its units are stated. Changing the effect scale changes the numerical precision values.
- Using only a central summary. Equal medians do not imply equal probabilities of weak or tight borrowing. Inspect regions and tails that matter clinically.
- Calling near-zero mass an exact no-borrowing component. A continuous prior can allow very weak links but assigns zero probability to the single point \(\tau=0\).
- Confusing prior permission with posterior behavior. Large precision values make strong borrowing possible, but the posterior behavior also depends on the current data, the historical uncertainty, and the model assumptions.
No precision prior can compensate for a poorly defined common estimand or an inappropriate historical data summary. The normal link also imposes a symmetric, zero-centered discrepancy conditional on precision. If the scale, transportability assumptions, or discrepancy shape are unsuitable, comparing precision priors alone will not repair the model. Prior predictive checks expose some consequences, but they do not establish that the historical and current effects are exchangeable.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If precision is measured in point\(^{-2}\), what is the conditional discrepancy standard deviation at \(\tau=16\) point\(^{-2}\)?
- For an exponential prior with rate \(0.04\) point\(^{2}\), calculate \(P(\tau>50)\).
- With \(s_C^2=0.04\) point\(^{2}\) and \(V_H=0.01\) point\(^{2}\), calculate \(B(\tau)\) at \(\tau=25\) point\(^{-2}\).
- Why can two priors with the same median precision still permit different amounts of weak borrowing?
- What does a continuous prior on positive precision imply about the probability of exactly \(\tau=0\)?
Solutions. (1) The standard deviation is \(1/\sqrt{16}=0.25\) point. (2) \(P(\tau>50)=\exp(-0.04\times50)=e^{-2}\approx0.1353\). (3) \(B(25)=0.04/(0.04+0.01+1/25)=0.04/0.09\approx0.4444\); at that precision, the historical mean receives about \(44.44\%\) of the conditional posterior mean weight. (4) Their probabilities near zero and across other tail regions can differ, even if their medians agree, so they can assign different probabilities to weak links. (5) The probability of the single value \(\tau=0\) is zero; the prior may nevertheless place probability near zero.