Normal Posterior Updates With Known Variance

Why a Normal Posterior Update Is Useful

A continuous trial endpoint may be summarized by an arm-specific mean, such as average change in a symptom score or average reduction in blood pressure. When the sampling variance is treated as known and both the sampling model and prior are normal, the posterior distribution for that mean can be calculated exactly. The update combines the prior information and the trial estimate, giving more weight to whichever is more precise.

This is a model-based calculation, not a rule for averaging any two numbers. The sampling variance, the prior variance, and the data must match the assumptions below. In particular, a variance estimated from the same small trial is not literally known just because it has been inserted into the formula.

From Individual Outcomes to the Trial Estimate

One common model starts with individual outcomes \(X_1,\ldots,X_n\) in one trial arm. Conditional on the arm mean \(\theta\), assume that the observations are independent and identically distributed as \(X_i\mid\theta\sim N(\theta,\sigma^2)\), where the sampling variance \(\sigma^2\) is known. The sample mean \(\bar{x}\) then satisfies \(\bar{X}\mid\theta\sim N(\theta,\sigma^2/n)\). Thus the update uses \(y=\bar{x}\) and \(v=\sigma^2/n\).

The normal model is exact under these assumptions. The variance \(\sigma^2\) describes the variation of individual outcomes around the arm mean; the variance of their mean is smaller by a factor of \(n\). Alternatively, a trial may report an estimate \(y\) and a known standard error \(SE\). In that case, use \(v=SE^2\), provided the estimate is modeled as unbiased and normally distributed around \(\theta\) with that variance.

The assumptions matter: observations must be independent under the specified sampling model, the mean must be the parameter being estimated, and the variance used in the likelihood must be known and appropriate. If participants are clustered, repeated measurements are dependent, or the variance changes in a way the model ignores, then \(\sigma^2/n\) may not be the correct variance for \(\bar{x}\).

Deriving the Precision-Weighted Update

The posterior density is proportional to the likelihood times the prior density. For an estimate \(y\), the parts of those densities that depend on \(\theta\) are

Write \(\lambda_0=1/\tau_0^2\) and \(\lambda_y=1/v\). Expanding the expression inside the brackets gives a quadratic in \(\theta\):

To complete the square, set \(m=(\lambda_0\mu_0+\lambda_y y)/(\lambda_0+\lambda_y)\). The \(\theta\)-dependent terms then equal \((\lambda_0+\lambda_y)(\theta-m)^2\), plus a constant that does not depend on \(\theta\). Consequently the posterior density is proportional to a normal density with mean \(m\) and variance \(1/(\lambda_0+\lambda_y)\). This establishes both the posterior mean and variance in the definition.

The variance decreases because the posterior precision is the sum of prior and data precision. The posterior mean is a weighted average: its weights are the two precisions divided by their sum. Since both precisions are positive, the mean must lie between \(\mu_0\) and \(y\). A more precise source pulls it closer.

Using a Reported Estimate and Updating in Stages

The same formula applies when the trial provides an estimate and its known standard error rather than individual observations. It also permits sequential updating: after one data set has been used, its posterior can serve as the prior for new, conditionally independent data. The new likelihood must contain only the new data. Reusing an estimate that already includes earlier observations would count those observations twice.

Checking the Model and Choosing an Approach

The update is exact only for the specified normal likelihood and normal prior. Before relying on it, check whether the endpoint and trial design make a normal mean model reasonable, whether the reported standard error reflects the actual design, and whether the prior mean and spread are scientifically defensible. Earlier material on prior predictive checks is useful for examining the implications of prior choices in terms of possible outcomes; it does not make a misspecified sampling model correct.

Potential warning signs include strong skewness or heavy tails in individual outcomes, influential extreme observations, outliers inconsistent with the normal sampling model, dependence between participants, and substantial differences in variability across groups that the model does not represent. For the arm mean, also verify that the variance used is a variance of the estimate, not the individual-outcome variance mistakenly used without dividing by \(n\). When only an aggregate estimate is available, the normal known-variance assumption and the appropriateness of its reported standard error must be justified from the study design and analysis.

A conjugate normal update is attractive when its assumptions are credible: it gives an exact posterior with a simple calculation. If the outcome distribution or sampling process is materially nonnormal, a likelihood appropriate to that endpoint may be needed, and computation may no longer have this closed form. If the variance is not known but must be estimated jointly with the mean, this fixed-variance update is not the appropriate full model; that setting requires a different posterior calculation. These alternatives address different modeling assumptions rather than being interchangeable computational shortcuts.

A normal prior also has support over the entire real line. That can be reasonable for an unconstrained mean, but may assign material prior probability to impossible values when the endpoint has hard bounds. Check the prior’s implications in context. A posterior calculation can be numerically correct and still be unhelpful if the prior or likelihood represents implausible values or an unsuitable data-generating process.