Bayesian Credible Intervals and Stopping-Rule Ignorability

Why Stopping Can Matter Differently to Bayesians and Frequentists

A group sequential design can stop data collection when interim results cross a prespecified boundary. The stopping decision changes which data are observed and can change the repeated-sampling distribution of an estimator. Yet, under specific conditions, the stopping decision does not add a new parameter-dependent factor to the Bayesian likelihood once the full observed data path is known.

That likelihood result is not a claim that stopping is irrelevant to every inferential question. In particular, a Bayesian credible interval describes posterior probability conditional on the observed data and prior, while a frequentist confidence interval is evaluated by its coverage over repeated trials. These are different properties. A posterior can be unchanged by the stopping rule and still fail to have its stated frequentist coverage.

Definition: A stopping rule is ignorable for Bayesian updating from the full stopped data when, under the stated sampling model, including the prespecified rule in the likelihood multiplies the ordinary likelihood for the observed data by a factor that does not depend on the unknown parameter. The posterior is then the same as the posterior obtained from the ordinary likelihood and the same prior.

The Likelihood of a Stopped Data Path

Let \(Y_1,Y_2,\ldots\) denote sequential observations, and let \(\tau\) be the stopping time: the number of observations collected before the rule stops the trial. Suppose the observed data are the complete path \(y_1,\ldots,y_n\), with \(\tau=n\). Under a model in which the observations are independent with density or probability function \(f_\theta\), the likelihood for this stopped path is

L_{\mathrm{stop}}(\theta;y_1,\ldots,y_n) = \left\{\prod_{i=1}^{n} f_\theta(y_i)\right\} \mathbf{1}\{\tau(y_1,\ldots,y_n)=n\}.

Here the indicator is one if this data path satisfies the stopping rule at \(n\), and zero otherwise. For the observed path it equals one, and it is determined by the observed data and the prespecified rule, not by \(\theta\). Thus it does not change the likelihood as a function of \(\theta\). With prior density \(\pi(\theta)\), Bayes' rule gives

\pi(\theta\mid y_1,\ldots,y_n,\tau=n) \propto \pi(\theta)L_{\mathrm{stop}}(\theta;y_1,\ldots,y_n) \propto \pi(\theta)\prod_{i=1}^{n} f_\theta(y_i).

This derivation assumes that the sampling model is correctly specified, the rule was set independently of the unknown parameter, and the full data used by the rule are available for the likelihood. A stopping decision may depend on earlier observations, but once those observations are part of the observed path, the deterministic rule indicator adds no parameter-dependent information.

A different calculation arises if the analyst observes only that the trial stopped at stage \(n\), rather than the data that led to that decision. The likelihood for that coarsened observation is \(P_\theta(\tau=n)\), which generally depends on \(\theta\). More generally, if the analysis conditions on a stopping event while omitting relevant data, it must account for the probability of that event under each parameter value. Ignoring that dependence can yield an incorrect posterior.

Worked Example: A Normal Observation That Crosses an Interim Boundary

Suppose \(X\mid\theta\sim N(\theta,1)\), and the prior is \(\theta\sim N(0,1)\). The prespecified rule stops after this observation if \(|X|\geq 1\). In a hypothetical trial, \(X=1.2\), so the trial stops at the first look. Find the posterior distribution and a 95% equal-tailed credible interval.

The observed value satisfies the stopping rule because \(|1.2|\geq1\). Its stopped likelihood is

L_{\mathrm{stop}}(\theta;1.2) = \frac{1}{\sqrt{2\pi}} \exp\left[-\frac{(1.2-\theta)^2}{2}\right] \mathbf{1}\{|1.2|\geq1\}.

The indicator equals one for this observation and contains no \(\theta\). Combining the normal prior and likelihood gives posterior variance \(1/(1+1)=0.5\) and posterior mean \((0+1.2)/(1+1)=0.6\). Therefore

\theta\mid X=1.2,\tau=1\sim N(0.6,0.5).

The equal-tailed 95% interval is \(0.6\pm1.96\sqrt{0.5}\), or approximately \([-0.786,1.986]\). Under this model and prior, the posterior probability that \(\theta\) lies in this interval is 0.95. This statement is conditional on the observed data and prior; it is not a claim that this interval procedure covers the fixed true parameter in 95% of repeated stopped trials.

When the Stopping Event Is the Information

The distinction between a complete path and a stopping-stage label is important in practice. If only a coarse record is retained, the event that caused stopping can be informative about \(\theta\), even when a fully observed path would make the stopping indicator redundant. The following simple calculation shows the difference.

Worked Example: Updating From a Stop Flag Alone

Suppose \(\theta\) is either 0 or 1, with prior probabilities \(P(\theta=0)=P(\theta=1)=0.5\). A hypothetical monitoring procedure records only a stop flag \(S\). Its model gives \(P(S=1\mid\theta=0)=0.2\) and \(P(S=1\mid\theta=1)=0.8\). Find the posterior probability of \(\theta=1\) after observing \(S=1\).

Because the underlying data are not available, the likelihood is the probability of the observed flag, not a likelihood for the full data path. Bayes' rule gives

P(\theta=1\mid S=1) = \frac{P(S=1\mid\theta=1)P(\theta=1)} {P(S=1\mid\theta=0)P(\theta=0)+P(S=1\mid\theta=1)P(\theta=1)} = \frac{0.8(0.5)}{0.2(0.5)+0.8(0.5)} =0.8.

The stop flag changes the prior probability from 0.5 to 0.8 because its probability depends on \(\theta\). The ignorability result for a complete observed path does not permit replacing this likelihood with a constant. The data here have been deliberately coarsened to the flag alone.

Credible Probability Is Not Frequentist Coverage

A \(100(1-\alpha)\%\) Bayesian credible interval \(C(Y)\) is a set satisfying \(P(\theta\in C(Y)\mid Y)=1-\alpha\), for the specified prior and model (up to any stated rounding). Frequentist coverage instead evaluates \(P_\theta\{\theta\in C(Y)\}\) over repeated samples generated at a fixed \(\theta\). With sequential sampling, that probability must be calculated under the actual stopping design, with the chosen conditional or unconditional coverage target made explicit.

The next calculation uses a normal model to show why a posterior interval's nominal probability does not itself establish its frequentist coverage. It also illustrates a practical check: the posterior variance must be transformed correctly when computing the sampling distribution of a rescaled statistic.

Worked Example: A Credible Interval With Different Frequentist Coverage

Suppose \(X\mid\theta\sim N(\theta,1)\) and the prior is \(\theta\sim N(0,1)\). In a hypothetical observation, \(X=6\). Find the 95% posterior interval, then calculate the repeated-sampling coverage of this interval formula at the fixed value \(\theta=3\).

The conjugate posterior has mean \(X/2\) and variance \(1/2\). At \(X=6\), it is \(N(3,0.5)\), so the 95% equal-tailed posterior interval is

\frac{6}{2}\pm1.96\sqrt{0.5} = 3\pm1.386 \approx[1.614,4.386].

For repeated sampling at \(\theta=3\), \(X\sim N(3,1)\). The interval formula for a general observation \(X\) is \(X/2\pm1.96\sqrt{0.5}\). Since division by 2 divides the variance by \(2^2\),

X/2\mid\theta=3\sim N(1.5,0.25), \qquad \operatorname{SD}(X/2\mid\theta=3)=0.5.

The interval covers the fixed value 3 when \(1.614\leq X/2\leq4.386\), using the rounded endpoints. Therefore its repeated-sampling coverage at \(\theta=3\) is

P(1.614\leq X/2\leq4.386\mid\theta=3) = \Phi\left(\frac{4.386-1.5}{0.5}\right) - \Phi\left(\frac{1.614-1.5}{0.5}\right) \approx \Phi(5.772)-\Phi(0.228) \approx0.4098.

Thus the interval has 95% posterior probability under the stated prior and observed value, but the interval formula has about 41.0% fixed-sample frequentist coverage at \(\theta=3\). This is a fixed-sample calculation for that formula, not a conditional coverage calculation among trials selected to stop at a particular look. For an actual sequential design, its stopping rule and the intended coverage target must be included in the repeated-sampling calculation.

Checking the Assumptions and Choosing an Interval

Stopping-rule ignorability is a likelihood statement, not a guarantee that every analysis choice is harmless. Before relying on it, check what information the analysis conditions on and whether it matches the data actually available.

The rule is fixed and parameter-independent. A prespecified boundary based on interim data meets this condition. A rule changed after inspecting results, or selected using information about the unknown parameter not represented in the model, needs further modelling.
The relevant data path is retained. If only the stopping stage, a threshold-crossing flag, or a summary that omits rule-driving data is available, the probability of that retained event may depend on \(\theta\).
The sampling model and prior are appropriate. Stopping-rule cancellation does not correct a misspecified likelihood, a prior chosen in response to the observed results without accounting for that choice, or unrecorded changes to the design.
The reported interval answers the intended question. A posterior credible interval answers a posterior-probability question. A stopping-aware frequentist interval is designed to meet a repeated-sampling coverage criterion, often conditional on a stage or unconditional across stages.

These approaches serve different purposes. A posterior interval updates a prior using the likelihood and can remain unchanged by a prespecified rule when the full data are used. A stopping-aware frequentist interval accounts for the stopped sampling law to achieve a specified coverage property. A conventional fixed-sample interval may be easy to compute, but its nominal coverage need not survive sequential stopping. No one of these statements substitutes for the others.

Approach	What is calibrated or summarized?	Key caution
Bayesian credible interval	Posterior probability given the model, prior, and observed data	Nominal posterior probability does not ensure frequentist coverage
Stopping-aware frequentist interval	Repeated-sampling coverage under the specified stopping design	State whether coverage is conditional on stage or unconditional
Naive fixed-sample interval	Coverage under its fixed-sample derivation	Sequential selection can alter its coverage

Common Mistake

A common mistake is to reason: “The stopping indicator cancels from the posterior, so the posterior interval must have its nominal frequentist coverage.” The first clause can be correct when the complete data path and the stated assumptions are used. The conclusion does not follow: posterior probability and repeated-sampling coverage average over different sources of uncertainty. To assess coverage, define the interval procedure for repeated trials and calculate its probability under the actual stopping design.

Another calculation error is to divide a normal random variable by 2 while leaving its variance unchanged. If \(X\sim N(\mu,\sigma^2)\), then \(X/2\sim N(\mu/2,\sigma^2/4)\). In the preceding example the variance is \(0.25\), and the standard deviation is \(0.5\); using either as the wrong quantity changes the coverage calculation.

Key takeaway: With a prespecified parameter-independent rule and the complete stopped data, the rule often contributes only a parameter-free likelihood factor, leaving the Bayesian posterior unchanged. That result does not confer frequentist coverage on a credible interval; coverage must be evaluated under the chosen repeated-sampling design and target.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A prespecified rule stops when the first observation exceeds a fixed boundary. Why does the stop indicator cancel from the posterior when the exact observation is retained?
In the binary example, what posterior probability results from observing only the stop flag, and why is it not the prior probability?
For \(X\sim N(3,1)\), what are the mean and variance of \(X/2\)?
Does a 95% posterior credible interval necessarily have 95% frequentist coverage at every fixed \(\theta\)? Explain.
Name two conditions needed to use the stopping-rule ignorability argument.

Solutions. (1) For the observed path, the rule indicator is one and does not depend on \(\theta\), so it is a constant factor in the likelihood. (2) The posterior probability is 0.8; the stop-flag likelihoods, 0.2 and 0.8, differ by parameter value. (3) The mean is 1.5 and the variance is \(1/4=0.25\). (4) No. Credible probability is conditional on the model, prior, and data; coverage is a repeated-sampling property. (5) Examples include a prespecified parameter-independent rule and retention of the full data path used by the rule; a correctly specified sampling model is also required.