How Discounting Changes Posterior Precision

Precision and Borrowing in a Normal Model

When a trial borrows information from historical controls, a discount determines how strongly those observations contribute to the analysis. In a conjugate normal model, the discount has a particularly transparent effect: it adds a specified amount to posterior precision. That precision change is not the same as the change in the posterior mean. Precision describes the concentration of the posterior distribution; the mean describes its location.

We derive the result for a continuous control outcome with known sampling variance. The known-variance assumption is important: it makes the likelihood for the mean normal and allows the powered historical likelihood to combine exactly with a normal prior and current-data likelihood. All data are assumed to measure the same estimand, with the same outcome definition and sampling variance.

Model and Posterior Formula

Suppose current control outcomes \(X_{C1},\ldots,X_{Cn_C}\) and historical control outcomes \(X_{H1},\ldots,X_{Hn_H}\) are independent conditional on a common mean \(\theta\), with \(X_{Ci}\sim N(\theta,\sigma^2)\) and \(X_{Hj}\sim N(\theta,\sigma^2)\). The sampling variance \(\sigma^2\) is known and shared by both data sources. Put a proper normal prior on the mean, \(\theta\sim N(\mu_0,\tau_0^2)\), independent of the data. For this tutorial, \(a\) is fixed rather than estimated from these observations.

Write \(\bar{x}_C\) and \(\bar{x}_H\) for the observed current and historical sample means. Up to factors that do not depend on \(\theta\), the current likelihood is proportional to \(\exp\{-n_C(\theta-\bar{x}_C)^2/(2\sigma^2)\}\). The historical likelihood raised to \(a\) is proportional to \(\exp\{-a n_H(\theta-\bar{x}_H)^2/(2\sigma^2)\}\). Thus the powered historical data contribute precision \(a n_H/\sigma^2\), not \(n_H/\sigma^2\).

Completing the square in the product of the prior and both likelihood contributions gives the posterior variance \(V(a)\), precision \(P(a)\), and mean \(m(a)\):

The three terms in \(P(a)\) are prior precision, current-data precision, and discounted historical precision. Each has units of inverse squared outcome units. The posterior variance depends on sample sizes, the known sampling variance, the prior variance, and \(a\), but not on either observed sample mean. By contrast, the posterior mean depends on the observed means as well as on their precision weights.

Why the Precision Changes

The posterior expression follows directly from the quadratic terms in the log density. Ignoring terms independent of \(\theta\), twice the negative log posterior is

The coefficient of \(\theta^2\) in this expression is \(P(a)\). Completing the square therefore gives a normal posterior with variance \(1/P(a)\). This also justifies the interpretation of the historical contribution: multiplying the historical log likelihood by \(a\) multiplies its quadratic curvature, and hence its precision contribution, by \(a\).

Let \(P_0=1/\tau_0^2+n_C/\sigma^2\) be the precision without historical borrowing. Then

For fixed model inputs, increasing \(a\) therefore increases posterior precision and decreases posterior variance. Specifically,

This is a statement about the model's calculated uncertainty, not a guarantee that the model is appropriate. If historical observations are not exchangeable with current controls for the estimand being analyzed, the formula can produce a more concentrated posterior around a misleading value.

Precision Is Not the Posterior-Mean Shift

The mean is a precision-weighted combination of the prior mean and the two sample means. To see how its direction of change depends on the historical outcome, define \(q_H=n_H/\sigma^2\) and let \(B=\mu_0/\tau_0^2+n_C\bar{x}_C/\sigma^2\). Then \(m(a)=(B+a q_H\bar{x}_H)/(P_0+a q_H)\). Differentiating this ratio gives

Since \(q_H>0\) and \(P(a)>0\), the mean moves toward the historical mean as \(a\) increases: it moves upward when \(\bar{x}_H>m(a)\) and downward when \(\bar{x}_H<m(a)\). The posterior variance, in contrast, decreases with \(a\) regardless of the sample means under the stated model. The same precision gain can therefore accompany very different mean shifts in different data sets.

Boundary Cases and a Further Calculation

At \(a=0\), the historical likelihood contributes no precision or mean weight, so the analysis uses only the prior and current controls. At \(a=1\), the historical likelihood is included at its full model-based weight, and its precision contribution is \(n_H/\sigma^2\). These are algebraic limits for this model; full borrowing is not automatically justified by setting \(a=1\).

Assumptions, Diagnostics, and Alternatives

The formula is exact under the specified model, but its scientific use depends on substantive conditions. Current and historical controls must target a common \(\theta\); the outcome definition and measurement scale must be aligned; observations must be independent conditional on \(\theta\); and the common known \(\sigma^2\) must be appropriate. The normal likelihood also presumes that the modeled sampling distribution is adequately represented by the stated normal model. A fixed \(a\) scales the historical likelihood; it does not correct differences in estimands, outcome ascertainment, or sampling variance.

Before interpreting a precision gain, examine whether the historical controls are plausibly comparable with current controls. Useful checks include comparing the endpoint definition, eligibility and follow-up procedures, examining observed outcome distributions and variability, and considering calendar-time or care changes that could affect the control mean. Large discrepancies are warnings that the common-mean assumption may be implausible. Agreement does not prove exchangeability, particularly when historical sample sizes are small or relevant differences were not measured.

The principal alternatives answer different modeling questions. With no borrowing, set \(a=0\); this avoids historical influence but forgoes any useful information. With full borrowing, set \(a=1\); this uses the historical likelihood at its full assumed sampling precision and depends strongly on the common-mean and common-variance assumptions. A fixed intermediate discount reduces the historical contribution by a chosen amount, but still assumes the same underlying mean and variance structure. A model that explicitly allows study means or variances to differ can represent heterogeneity more directly, at the cost of additional assumptions and parameters. These approaches are not interchangeable: a smaller posterior variance from a fixed discount is not evidence that the historical and current data are compatible.

A frequent mistake is to say, “The historical mean is far from the current mean, so it must increase posterior uncertainty.” Under this known-variance normal model, that is incorrect: \(V(a)\) contains no sample means, so a fixed \(a\) produces the same posterior variance whether the historical and current means agree or disagree. The discrepancy can substantially shift \(m(a)\), and it can indicate model failure, but it does not change the formula's precision. Conversely, a smaller posterior variance should not be described as proof of greater accuracy if the shared-mean assumption is doubtful.