Building a Numerical Error Budget

Why an Error Budget Is Needed

A sequential probability calculation can look precise while carrying several distinct numerical approximations. For example, a recursion may integrate a survivor density on a finite grid, interpolate that density at a stopping boundary, omit states outside a truncated range, and calibrate a boundary with a root solver. Each choice can change the reported crossing probability. Refining only one part does not establish that the final answer is accurate.

An error budget makes the accuracy claim explicit. It assigns an allowable contribution to each source, then checks whether their combined effect is below a prespecified tolerance. The budget applies to a particular computed quantity—such as a cumulative crossing probability, a stagewise rejection probability, or power—and to a specified model and numerical implementation.

This is numerical error, not sampling uncertainty in a trial estimate. Nor does an error budget assess whether the Gaussian model, information times, or stopping rules represent the intended design. Those are modeling and design questions. Here, take the probability model and target event as specified, and ask how closely the computation evaluates that target.

Four Contributions to Track

In a recursive calculation, distinguish the following sources. The separation is conceptual as well as practical: each source calls for a different diagnostic or remedy.

• Quadrature error: error from approximating an integral by a numerical integration rule. Adaptive integration can estimate local error, but its reported estimate is not automatically a rigorous bound.
• Grid error: error from representing a continuous state or density on a finite grid, including interpolation at a boundary. Compare refined grids or use a justified interpolation bound to assess it.
• Tail-truncation error: error from omitting state-space regions outside a finite integration range. Bound omitted probability mass under the model, rather than judging the range by visual inspection alone.
• Root-solving error: error from stopping boundary calibration before the root is exact. A solver’s residual in the calibration equation is not itself a bound on the resulting error in every reported probability.

These labels do not guarantee that the contributions are independent. In particular, a quadrature estimate obtained on a fixed grid does not measure grid error, and a root solver working with an approximate probability function inherits that function’s numerical error. Define the components operationally and avoid counting the same discrepancy twice.

Combining Component Bounds

Suppose a target probability is evaluated at fixed boundaries, and the implementation has four approximations. Write intermediate results so that each change introduces one approximation at a time. For example, start with the exact calculation, then replace exact integration with quadrature, then replace the continuous state representation with a grid, and finally truncate the state space. The difference between the first and last results is a telescoping sum of the successive changes.

Taking absolute values and applying the triangle inequality gives a bound by the sum of the absolute component differences. If those differences are bounded by \(\epsilon_{\mathrm{quad}},\epsilon_{\mathrm{grid}},\epsilon_{\mathrm{tail}},\epsilon_{\mathrm{root}}\), respectively, then

This result is conservative: errors may have opposite signs and partly cancel, but cancellation should not be relied on for a guaranteed tolerance. The decomposition also requires care. If a refinement difference already includes both quadrature and grid changes, it cannot be presented as a separate bound for each without additional evidence.

Tail Bounds and Root Sensitivity

For a simple conservative tail bound, suppose the state at look \(k\) has marginal distribution \(Z_k\sim N(\mu_k,\sigma_k^2)\), and the computation omits states outside \([-L_k,L_k]\). The probability of any omitted state over \(K\) looks is bounded by the union bound:

This bound does not require the looks to be independent. It can be conservative because the recursion may already have stopped on some paths, and because omitted events at different looks overlap. If the omitted region is one-sided or the states have a different specified distribution, use the corresponding tail probabilities. The bound must cover all omitted regions relevant to the target calculation.

For calibration, let \(g(b)=0\) define the exact boundary \(b^*\), and let \(\tilde g\) be the numerically evaluated calibration function. Suppose in a neighborhood of \(b^*\), \(|g'(b)|\geq m>0\). If a computed boundary \(\tilde b\) has exact-function residual \(|g(\tilde b)|\leq r\), the mean value theorem implies \(|\tilde b-b^*|\leq r/m\), provided the segment between the two boundaries lies in that neighborhood. If the reported probability \(h(b)\) satisfies \(|h'(b)|\leq M\) there, then

A solver often reports the residual \(|\tilde g(\tilde b)|\), not \(|g(\tilde b)|\). If numerical evaluation guarantees \(|\tilde g(b)-g(b)|\leq\epsilon_g\), then a reported solver residual at most \(\tau\) implies \(|g(\tilde b)|\leq\tau+\epsilon_g\). Thus root error must account for both the solver stopping tolerance and the numerical error in evaluating the calibration function.

Worked Examples

Allocating Tolerances and Diagnosing Failure

Start from a total absolute probability tolerance \(\epsilon\), chosen to be small enough for the intended use. Allocate it across the numerical sources so that

An equal allocation is easy to communicate, but not always efficient. A source that is cheap to reduce may receive a tighter allocation; a source that dominates runtime may receive a larger allocation if the remaining components can still meet the total requirement. For calibrated probabilities, include the error in evaluating the calibration function when converting solver tolerance into boundary error, as well as any effect of the boundary error on the final target.

Warning signs include probability changes larger than the allocated grid tolerance after refinement, quadrature estimates that fail to stabilize when integration settings change, tail bounds that exceed their allocation, and root boundaries that move materially when the calibration evaluation is made more accurate. A tiny root residual alone is not evidence of an accurate boundary if the calibration function is noisy or flat.

The triangle-inequality budget is safer than combining errors by a root-sum-of-squares formula. The latter requires a justified probabilistic error model and appropriate dependence assumptions; deterministic numerical errors do not ordinarily satisfy those assumptions. Conversely, simply refining every setting to an extreme value can waste computation without identifying the source of instability. Component-wise checks are more informative.

A budget also has limits. Refinement differences are diagnostics, not guaranteed bounds unless supported by a numerical analysis result. Derivative-based root bounds are local and require valid sensitivity bounds. The union bound may be loose. And a small numerical error cannot repair a misspecified model or an incorrect stopping rule. Report the total bound or assessment together with the component definitions and the tolerances used.

Common Pitfall