Bayes Factor Stopping Regions

• Bayes factor stopping regions are defined by thresholds at which a sequential Bayesian test halts, providing an operational criterion for early stopping.
• They integrate decision-theoretic risk minimization and controlled Type I error rates by interpreting evidence in posterior odds.
• Their applications span clinical trials and experimental designs, with extensions to soft classification and continuous-time drift testing.

A Bayes factor stopping region is the set of all sample paths or statistic values for which the Bayes factor crosses pre-specified thresholds, at which point a sequential Bayesian test halts sampling and renders a terminal inference. This construction lays the mathematical and operational basis for Bayesian sequential analysis, providing an evidence-based criterion for early stopping that is interpretable in terms of posterior odds, decision-theoretic risk minimization, and (in special cases) frequentist error control. The theoretical and practical ramifications of these stopping regions have been the focus of significant analysis in sequential testing theory, optimal stopping, group-invariant models, clinical trial design, and the study of optional stopping in Bayesian inference.

1. Mathematical Definition and Construction

A Bayes factor stopping region is determined by the evolution of the (possibly sequentially updated) Bayes factor $\mathrm{BF}_n$, which, at each sample size $n$, compares the marginal likelihoods of all data observed so far under two competing hypotheses: $$\mathrm{BF}_n = \frac{m_1(X_{1:n})}{m_0(X_{1:n})}, \quad m_i(X_{1:n}) = \int p_i(X_{1:n}|\theta_i)\pi_i(\theta_i)\,d\theta_i,$$ where %%%%2%%%% is the model under $H_i$, and $\pi_i$ is its prior (Heide et al., 2017, Hendriksen et al., 2018).

A Bayes factor stopping rule is any rule that stops sampling the first time $n=\tau$ such that

$$\mathrm{BF}_\tau \geq c_1 \quad\text{or}\quad \mathrm{BF}_\tau \leq c_0,$$

for constants $c_1>1>c_0$ (or, by symmetry, $c_0=1/c_1$). The corresponding Bayes factor stopping region $R$ is the set: $$R = \bigcup_{n} \left\{(X_{1:n}) : \mathrm{BF}_n \geq c_1\ \text{or}\ \mathrm{BF}_n \leq c_0\right\},$$ with the continuation region being the complement $\{\mathrm{BF}_n \in (c_0, c_1)\}$ (Heide et al., 2017, Hendriksen et al., 2018). At stopping, the decision is made in favor of $H_1$ if $\mathrm{BF}_\tau\geq c_1$ or $H_0$ if $\mathrm{BF}_\tau\leq c_0$.

This structure generalizes to composite hypotheses with nuisance parameters and to the continuous-time setting, where (e.g. in the drift testing of Brownian motion) the Bayes factor is constructed from filtered posteriors over the unknown parameter, defining boundaries in the space of posterior probabilities or Bayes factors (Ekström et al., 2015, Campbell et al., 20 Jan 2025).

2. Bayes Factor Stopping Regions in Sequential Drift Testing

In the canonical Bayesian sequential test of a Brownian motion's drift,

$$dX_t = \theta\,dt + dW_t,\quad H_0: \theta < 0,\quad H_1: \theta \geq 0,$$

the stopping region can be recast as an optimal stopping problem for the posterior probability process $\pi_t = \mathbb{P}(\theta\geq0 | \mathcal{F}_t)$: $$\tau^* = \inf\{t \geq 0: \pi_t\notin (b_1(t), b_2(t))\},$$ with $0 < b_1(t) < 1/2 < b_2(t) < 1$. In Bayes factor terms, this is equivalent to monitoring the process $B_t = \pi_t/(1-\pi_t) \cdot \Lambda_0$, where $\Lambda_0$ are the prior odds. The stopping boundaries $(b_1(t), b_2(t))$ (or equivalently, in the Bayes factor scale) are characterized by systems of integral equations in both finite- and infinite-horizon settings (Ekström et al., 2015).

The table below summarizes the principal components of Bayes factor stopping regions in Brownian motion drift testing:

Element	Description	Reference
Signal process	$dX_t = \theta\,dt + dW_t$	(Ekström et al., 2015)
Statistic	Posterior $\pi_t$; Bayes factor $B_t$	(Ekström et al., 2015)
Stopping rule	Exit from interval $(b_1(t),b_2(t))$	(Ekström et al., 2015)
Boundary equations	System of integral equations (finite/infinite)	(Ekström et al., 2015)

These boundaries are monotonic in time (non-decreasing for $b_1$, non-increasing for $b_2$) and are continuous functions of $t$, with long-term asymptotics determined by the prior mass near the decision boundary (Ekström et al., 2015).

3. Theoretical Guarantees Under Optional Stopping

Bayes factor stopping regions possess several distinct mathematical guarantees in the context of optional stopping:

• Stopping-rule independence ($\tau$-independence): For any stopping time $\tau$, the posterior odds at stopping depend only on the observed data, not on the manner in which the stopping time is chosen, provided the prior is fixed before data collection (Heide et al., 2017, Hendriksen et al., 2018).
• Posterior calibration: The empirical frequency with which certain Bayes factor values arise under $H_1$ versus $H_0$ matches the nominal Bayes factor; for unit prior odds, $$\mathrm{BF}=b \implies \mathbb{P}(H_1|BF=b)/\mathbb{P}(H_0|BF=b)=b$$ (Heide et al., 2017, Hendriksen et al., 2018).
• Semi-frequentist Type I error control: Markov's inequality yields that, for any $\tau$,

$$\mathbb{P}_0(\exists\,n:\,\mathrm{BF}_n\ge 1/\alpha) \leq \alpha,$$

so that exceeding the threshold $c_1=1/\alpha$ at any time yields a maximal Type I error of $\alpha$ (Hendriksen et al., 2018).

However, these guarantees can weaken if "default" or design-dependent priors are used, or when Bayesian updates are not those of a subjective Bayesian (see Section 4).

4. Impact of Prior Choice and Calibration Results

The efficacy and interpretation of Bayes factor stopping regions under optional stopping are sensitive to the nature of the employed priors (Heide et al., 2017). Three principal types are identified:

• Type 0 (Group-invariant/Haar) priors: For nuisance parameters with group structure (e.g., scale in normal families), right-Haar priors guarantee strong calibration under all stopping rules.
• Type I (Proper, non-invariant) priors: Default priors not tied to a group invariance (e.g., Cauchy on effect size) only ensure prior-based calibration; strong calibration may fail, and under optional stopping, significant miscalibration can arise at fixed parameter values.
• Type II (Design-dependent) priors: Priors depending on sample size, design, or sampling scheme (e.g., g-prior in regression) render calibration ill-defined under optional stopping.

This classification guides recommendations: optional stopping is valid for fully subjective Bayesians or for group-invariant nuisance calibration. Otherwise, the Bayes factor can severely misrepresent evidence when stopping is dictated by accumulating data, especially for parameters of scientific interest (Heide et al., 2017).

5. Explicit Boundary Characterizations and Algorithmic Implementation

In practical Bayesian sequential design—especially in clinical or experimental trials—the Bayes factor stopping region is often implemented as z-statistic boundaries across pre-scheduled (group-sequential) interim analyses. For common models, the log Bayes factor is a monotone function of the cumulative z-statistic at each look, allowing direct computation of critical values via root-finding:

$$z_{i,\text{crit}}(k) = \frac{\mu^2/\sigma_i^2 - 2 \ln k}{2(\mu/\sigma_i)}$$

For a fixed schedule of $J$ interim looks, one computes stopping ($z_i \geq u_i$ or $z_i \leq \ell_i$) and continuation regions at each look $i$, translating Bayes factor boundaries into operational criteria for practice (Pawel et al., 6 Jan 2026). Key operating characteristics—power, expected sample size, type I/II error rates—are computed via multivariate normal integrals over these stopping regions, avoiding simulation (Pawel et al., 6 Jan 2026).

6. Extensions: Soft Classification and Free-Boundary Problems

A recent generalization replaces the classical hard 0-1 loss in sequential drift testing with a soft classification penalty $g(\pi)$, inducing expanded continuation regions whose boundaries are characterized by free-boundary PDEs and analytic two-tangent conditions on convex envelopes (Campbell et al., 20 Jan 2025). The optimality region in posterior or Bayes factor space is determined by the shape of $g$, the signal-to-noise ratio, and the observation cost, seamlessly extending optimal stopping theory to soft-decision and continuous-risk settings.

For instance, with $g(\pi)=2\pi(1-\pi)$ (symmetric $L_1$ loss), the boundaries $b_1(K),b_2(K)$ satisfy implicit equations tied to the joint effect of loss and information, demonstrating that as the signal strength increases, the region where continued sampling is optimal covers almost all posterior mass—matching the classical sequential probability ratio test in the limit (Campbell et al., 20 Jan 2025).

7. Practical and Conceptual Implications

Bayes factor stopping regions provide a principled, interpretable, and implementable criterion for sequential inference in both exploratory and confirmatory research. They connect Bayesian evidence monitoring with decision-theoretic risk minimization, permit frequentist-style error rate control in well-specified settings, and allow rapid design optimization in sequential and group-sequential trials. Nonetheless, the properties of Bayes factor stopping regions rest crucially on the choice and interpretation of the prior, the invariance structure of the model, and the measurability of stopping rules. Full calibration and error control require strict adherence to group-invariant prior constructions and stopping rules dependent only on maximal invariants (or the Bayes factor itself) (Heide et al., 2017, Hendriksen et al., 2018).

A plausible implication is that, in routine scientific use with default or pragmatic priors, Bayesian sequential designs using Bayes factor stopping rules may only provide reliable evidence quantification and error control in nuisance or exploratory settings, and care must be taken to avoid evidence overstatement in confirmatory inference with parameters of interest.

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References:

(Ekström et al., 2015): Bayesian sequential testing of the drift of a Brownian motion (Heide et al., 2017): Why optional stopping can be a problem for Bayesians (Hendriksen et al., 2018): Optional Stopping with Bayes Factors: a categorization and extension of folklore results, with an application to invariant situations (Campbell et al., 20 Jan 2025): A Bayesian sequential soft classification problem for a Brownian motion's drift (Pawel et al., 6 Jan 2026): Bayes Factor Group Sequential Designs