Truncating the Gaussian State Space

Why a Gaussian State Space Is Truncated

A Gaussian test statistic has support over the entire real line, but numerical integration must use a finite range or a finite collection of states. Choosing that range is not merely a computational convenience: probability outside it is omitted from the calculation. The key question is therefore not whether the range is finite, but whether the omitted probability is small enough for the intended accuracy.

Consider a group-sequential design with looks at information levels \(I_1<\cdots<I_K\). Write \(X_j\) for the standardized statistic at look \(j\). In the canonical Gaussian model with constant drift \(\theta\), assume the increments of the underlying information-scale process are independent Gaussian increments. Then \(X_j\sim N(\theta\sqrt{I_j},1)\), and, for \(j>1\),

In particular, under \(H_0\), set \(\theta=0\). The transition variance is positive because information increases between looks. The marginal distribution at look \(j\) has mean \(\mu_j=\theta\sqrt{I_j}\) and variance 1. The bounds below use these marginal distributions; dependence between looks is allowed and is not ignored.

Intervals need not be centered at zero. Under a nonzero drift, the marginal states are shifted, so limits chosen for a null calculation may omit substantially more probability under an alternative. Limits can also differ from look to look. What matters for the error guarantee is the omitted probability under the parameter value whose crossing probability is being calculated.

A Bound on the Effect of Truncation

Let \(A\) be the event of crossing a specified sequential boundary by the final look. Define \(T\) as the event that at least one state lies outside its integration interval at a look used in the calculation. The truncated calculation retains the crossing probability from paths that never leave the chosen intervals, \(P(A\cap T^c)\). Therefore its absolute error is bounded as follows.

The last inequality is the union bound. It does not require independent looks; sequential statistics are generally dependent. It can be conservative because a path may be outside the range at more than one look, and not every omitted path would have crossed a boundary. Nevertheless, it gives a transparent guarantee: if the sum of marginal tail probabilities is at most \(\varepsilon\), truncation changes the crossing probability by at most \(\varepsilon\).

This is a bound on truncation error, not on every source of numerical error. Numerical quadrature and discretization can introduce additional error. If the calculation also approximates integrals, its total discrepancy from the target is bounded by the truncation bound plus a bound for those other errors, when such bounds are available. In practice, report the chosen limits and the tail-probability bound rather than treating a finely computed integral as automatically accurate.

Choosing Limits for a Probability Tolerance

A practical way to select limits is to allocate a total tail-probability budget across looks. Choose nonnegative values \(\varepsilon_j\) whose sum is no greater than the desired tolerance \(\varepsilon\). Then select \(l_j\) and \(u_j\) so that \(q_j\le\varepsilon_j\). A symmetric interval around the marginal mean is convenient when tails are balanced, but asymmetric limits can be more efficient when the mean is shifted or one side of the state range is more important.

For a unit-variance marginal distribution, a symmetric interval \([\mu_j-c_j,\mu_j+c_j]\) has omitted probability \(2\Phi(-c_j)\). Thus a sufficient condition for the total truncation error to be at most \(\varepsilon\) is $$ \sum_{j=1}^{K}2\Phi(-c_j)\le\varepsilon. $$

Equal allocation uses the same tail probability at every look, but it is not required. For example, more of the budget may be allocated to looks with wider integration ranges or to looks where the boundary geometry makes omitted states especially relevant. The union-bound guarantee still holds as long as the allocated probabilities sum to no more than the tolerance.

How to Apply the Bound in a Recursive Calculation

The survivor-density recursion carries probability forward only for paths that have not crossed a boundary. Restricting each integration to a finite state interval discards additional mass. The following procedure makes that omission explicit before interpreting the resulting crossing estimate.

For a boundary-crossing calculation, limits should include the boundary locations at each look. If a boundary lies outside the interval, crossings in that excluded region are omitted; the tail bound still controls the probability of such omissions, but a range that cuts through a region of substantial boundary-crossing probability is a warning to widen the interval.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to compare the calculated tail bound with the precision needed for the scientific or design decision. A bound of \(0.001\) may be negligible for a rough power assessment but unacceptable when distinguishing crossing probabilities that differ by \(0.0002\). Also inspect the separate \(q_j\) values: a small total can conceal one look that contributes most of the error. Under a nonzero drift, check both tails at every look rather than assuming symmetry around zero.

State-space truncation is different from the main alternatives for Gaussian integration. A Gaussian transition quadrature can approximate an integral over an unbounded transition distribution without imposing the same state interval, though a finite numerical rule still has its own approximation error. Conversely, restricting a state grid gives direct control over where the recursion is evaluated, but it introduces a tail omission that must be budgeted. Direct multivariate Gaussian probability calculations offer another way to evaluate some crossing events, but they do not make tail or numerical accuracy checks unnecessary.

The union bound is intentionally general and can be conservative. It adds marginal tail probabilities even when the same rare path contributes to several looks, and it counts tail paths that would not contribute to the event of interest. A tighter event-specific bound may be possible, but the simple sum is dependable and does not require modeling dependence beyond the marginal distributions. Its limitations are equally important: it does not quantify quadrature error, grid discretization error, or errors from an incorrect Gaussian model.

Common Mistakes

• Using a null tail bound for an alternative calculation. Under drift, the marginal means shift. Recalculate \(q_j\) under the parameter value used for the crossing probability.
• Assuming look independence. The union bound applies to dependent looks; multiplying one tail probability by the number of looks is valid only when those probabilities are equal, not because the looks are independent.
• Rounding before summing. Retain adequate precision in each tail probability, sum first, and round the final bound. For example, \(3\{\Phi(-4.5)+\Phi(-3.5)\}=0.000708080\ldots\), not a value obtained by multiplying a prematurely rounded tail sum.
• Treating the tail bound as total numerical error. It bounds only the error from state-space truncation. Quadrature and discretization need separate assessment.
• Calling the bound the actual error. The actual error can be much smaller, because not all omitted paths cross a boundary and the union bound can count a path more than once.