Credible Intervals for a Continuous Endpoint Mean

What a Credible Interval Says

A point estimate such as a posterior mean summarizes a distribution, but it does not show how uncertain the endpoint mean remains after observing trial data. A posterior credible interval gives a range of parameter values with a specified probability under the posterior distribution. Its interpretation is conditional on the observed data, the likelihood, and the prior.

For a continuous endpoint, let \(X_1,\ldots,X_n\) be the observed values for \(n\) trial participants, and let \(\mu\) be their population mean under the model. Values might be changes in systolic blood pressure, measured in mmHg. The intervals below concern \(\mu\), not the range containing a specified proportion of individual participants.

The phrase “equal-tailed” means the posterior probability outside the interval is split equally between its two tails. Other interval constructions are possible, but the equal-tailed form is especially direct when the posterior distribution is symmetric. A credible interval is not automatically a frequentist confidence interval: the two can have the same endpoints in particular models, while their probability statements remain different.

Known Variance: Normal Posterior Quantiles

Suppose the endpoint observations are conditionally independent and normally distributed, \(X_i\mid\mu\sim N(\mu,\sigma^2)\), where the variance \(\sigma^2\) is known. With a normal prior \(\mu\sim N(\mu_0,\tau_0^2)\), the posterior is normal. Its variance \(V_n\) and mean \(m_n\) are

The posterior variance combines prior precision \(1/\tau_0^2\) and data precision \(n/\sigma^2\). For a central \(100(1-\alpha)\%\) interval, let \(z_{1-\alpha/2}\) be the \(1-\alpha/2\) quantile of the standard normal distribution. The interval is \(m_n\pm z_{1-\alpha/2}\sqrt{V_n}\). This calculation assumes that the normal sampling model is adequate, that the variance really is treated as known, and that the stated prior represents the intended information about \(\mu\).

Unknown Variance: A Student t Posterior

In many endpoint analyses the variance is not known. A common reference-prior analysis uses the normal sampling model \(X_i\mid\mu,\sigma^2\sim N(\mu,\sigma^2)\), independently, with prior density proportional to \(1/\sigma^2\) for \(\mu\in\mathbb{R}\) and \(\sigma^2>0\). This prior is improper, so it is essential to check that the resulting posterior is proper before interpreting posterior probabilities.

Write \(SSE=\sum_{i=1}^n(X_i-\bar{x})^2=(n-1)s^2\). Under this model and prior, when \(n\geq2\) and \(s^2>0\), the marginal posterior for \(\mu\) is a Student t distribution with \(n-1\) degrees of freedom, location \(\bar{x}\), and scale \(s/\sqrt{n}\). In this tutorial, that statement means \((\mu-\bar{x})/(s/\sqrt n)\) has a standard \(t_{n-1}\) posterior distribution.

The scale \(s/\sqrt n\) is the posterior scale parameter here; it is not the standard deviation of the Student t distribution. When \(n>3\), the latter is \(s/\sqrt n\sqrt{(n-1)/(n-3)}\). The interval quantiles must use the t distribution, not a normal distribution, particularly when the sample is small.

For a central \(100(1-\alpha)\%\) interval, let \(t_{1-\alpha/2,n-1}\) be the corresponding standard Student t quantile. Then

The interval is centered at the sample mean because the marginal posterior is symmetric about \(\bar{x}\). Its width reflects both the observed variability and the sample size. As \(n\) increases, the scale \(s/\sqrt n\) tends to decrease, though the actual interval also depends on the observed \(s\) and the appropriate t quantile.

Why the Unknown-Variance Result Holds

A short derivation shows both where the Student t distribution enters and why zero sample variance is a serious boundary case. The normal likelihood can be factored using \(\sum_i(X_i-\mu)^2=SSE+n(\mu-\bar{x})^2\). Multiplying by the prior gives a joint posterior kernel proportional to

Integrating over \(\mu\) contributes a factor proportional to \(\sqrt{\sigma^2/n}\). Thus the marginal kernel for \(\sigma^2\) is proportional to

If \(SSE>0\), the exponential term controls the behavior near zero, and the kernel is integrable at both zero and infinity for \(n\geq2\). If \(SSE=0\), the exponential term is 1, leaving \((\sigma^2)^{-(n+1)/2}\), which is not integrable near zero. Thus an all-identical sample gives an improper posterior, even if it contains at least two observations.

To obtain the marginal distribution of \(\mu\), integrate the joint kernel over \(\sigma^2\). For \(SSE>0\), the resulting kernel in \(\mu\) is proportional to \([SSE+n(\mu-\bar{x})^2]^{-n/2}\). Since \(SSE=(n-1)s^2\), this is proportional to \(\left[1+\{(\mu-\bar{x})/(s/\sqrt n)\}^2/(n-1)\right]^{-n/2}\), the kernel of a \(t_{n-1}\) distribution with the stated location and scale. This derivation also identifies the conditions needed before using its quantiles.

Posterior Probability and Frequentist Coverage

The unknown-variance interval above has the same algebraic form as the usual two-sided t confidence interval for a normal mean. Under independent normal sampling, its repeated-sampling coverage is exactly \(1-\alpha\): the statistic \((\bar{x}-\mu)/(s/\sqrt n)\) has a \(t_{n-1}\) distribution when the true \(\mu\) is fixed. Therefore, over repeated samples generated with that fixed mean, the interval procedure contains \(\mu\) in a proportion \(1-\alpha\) of samples.

That coverage statement is not the interpretation of the Bayesian interval. In the Bayesian analysis, after observing these data, the parameter \(\mu\) has posterior probability \(1-\alpha\) of falling in the particular interval computed from them. In the frequentist interpretation, \(\mu\) is fixed and the interval varies across repeated samples. Identical endpoints do not make these statements interchangeable. With an informative proper prior, the Bayesian interval will generally have different endpoints and need not have exact frequentist coverage.

A central equal-tailed interval is also not the only possible Bayesian interval. For a symmetric normal or Student t posterior, the equal-tailed interval is centered at the posterior location and is also a highest posterior density interval. For an asymmetric posterior, equal-tailed and highest posterior density intervals may differ. The intended construction should therefore be stated, rather than simply calling an interval “95%” without explaining its convention.

Diagnostics, Pitfalls, and Limitations

Before reporting an interval, check whether its model and calculation are coherent. For the unknown-variance reference-prior result, verify that there are at least two observations and that \(s^2>0\). Inspect the endpoint distribution for marked skewness, outliers, or features incompatible with a normal model; with small samples, departures can materially affect the t-based posterior. Confirm that participants can reasonably be treated as independent observations. For the known-variance analysis, also ask whether treating \(\sigma^2\) as known is defensible rather than merely convenient.

• Using a normal quantile with small \(n\). Under the unknown-variance reference-prior model, the posterior is Student t with \(n-1\) degrees of freedom. Replacing its quantile with 1.96 generally understates tail uncertainty for small samples.
• Ignoring posterior propriety. If all observations are identical, \(s^2=0\); the stated improper prior does not yield a proper posterior. A t interval cannot be rescued by inserting zero into its formula.
• Calling the interval a probability about repeated samples. The 95% Bayesian statement concerns the parameter conditional on the observed data and model. Repeated-sampling coverage is a separate property of an interval procedure.
• Confusing the mean with individual outcomes. The interval quantifies uncertainty in \(\mu\), not a range expected to contain 95% of participants' endpoint values.

A Bayesian interval is sensitive to the likelihood and prior. The flat prior on \(\mu\) and \(1/\sigma^2\) prior on variance are reference choices, not universal defaults that erase modeling assumptions. This tutorial addresses a single normal endpoint mean; it does not establish how to compare two treatment groups or how to model a variance as the scientific target. When normality, independence, prior choice, or the handling of missing endpoint values is questionable, the interval should be accompanied by suitable sensitivity analyses and limitations rather than treated as self-validating.