Robust D-Optimality Under Parameter Uncertainty

Three Ways to Aggregate Uncertain Information

A locally D-optimal design maximizes the determinant of its Fisher information matrix at a specified parameter value. When parameters are uncertain, however, a design produces a different information matrix at different plausible values. There is no single automatic way to summarize those matrices: averaging their log-determinants, taking the log-determinant of their average, and protecting against the worst case answer different design questions.

Let \(d\) denote a design, let \(\theta\) be the uncertain model-parameter vector, and let \(I(d,\theta)\) be the \(q\times q\) Fisher information matrix for the \(q\) parameters of interest. Assume the matrices are symmetric and positive semidefinite. Let \(\pi\) denote the specified probability distribution for \(\theta\), when a probability distribution is used. D-optimality scores the information through \(\log\det I\), which is the log of the product of the information eigenvalues.

The first criterion averages the D score over uncertainty, so outcomes are weighted by their probabilities under \(\pi\). The second first averages the matrices and then scores that average matrix; it is not the average of the scenario-specific scores. The third uses a set of plausible values rather than their probabilities: even a low-probability point can determine the score if it belongs to \(\Theta\). If uncertainty is expressed by a distribution rather than an explicit set, a worst-case score may use the essential infimum over its support; the intended set or support must be stated.

The usual geometric interpretation helps explain the criteria. Under the local large-sample normal approximation, the covariance of an estimator is approximately \(I(d,\theta)^{-1}\). When \(I\) is positive definite, the volume of a fixed-level joint confidence ellipsoid is proportional to \(\det(I)^{-1/2}\). Maximizing \(\log\det I\) therefore favors smaller joint uncertainty volume. This interpretation is local and relies on the information matrix and approximation being appropriate; it does not make the three aggregation rules interchangeable.

Conditions and the Relationship Between the Expectation Criteria

For a finite log-determinant at a particular parameter value, \(I(d,\theta)\) must be positive definite. If it is singular, its determinant is zero and its pointwise log-determinant is \(-\infty\), using the extended-value convention. The expected log-determinant therefore needs care: it is an expectation of an extended-valued quantity, not automatically a finite number. In particular, if singular matrices occur with positive probability and the positive part of the log-determinant has finite expectation, the expected score is \(-\infty\). If both the positive and negative parts have infinite expectation, the expectation is undefined without additional integrability conditions. Finite scenario distributions with finite matrix entries make these checks especially direct.

For \(J_{\mathrm{logE}}\), the expected matrix must exist with finite entries and be positive definite. Individual scenario matrices may be singular while their average is positive definite. Thus \(J_{\mathrm{logE}}\) can be finite when \(J_{\mathrm{Elog}}\) is \(-\infty\). For a finite worst-case score, every matrix in the specified set must be positive definite, and the log-determinant must be bounded below there. If any included scenario is singular, the worst-case score is \(-\infty\).

When the expectations and scores are well-defined, concavity of \(\log\det\) gives a useful comparison:

A short derivation establishes the relevant concavity. For a positive definite matrix \(A\) and a symmetric direction \(H\), the first derivative of \(\log\det(A+tH)\) at \(t=0\) is \(\operatorname{tr}(A^{-1}H)\), and its second derivative is

The inequality follows because \(A^{-1/2}HA^{-1/2}\) is symmetric, so the trace of its square is the sum of squared eigenvalues and is nonnegative. Thus \(\log\det\) is concave on the positive definite matrices, and Jensen's inequality yields the displayed result when the expectation conditions hold. Strict inequality generally occurs when the information matrix varies across scenarios in a nontrivial way. This is a relation between scores for the same design, not a guarantee that the two criteria rank competing designs identically.

Worked Comparison of Competing Designs

The criteria can prefer different designs, not merely report different numerical scores. The following hypothetical example uses three candidate designs and a two-scenario distribution. Matrices are expressed in common parameter coordinates; the scenarios and probabilities are the same for all candidates.

Uncertainty Distributions, Diagnostics, and Limitations

In practice, inspect the information matrices across the uncertainty range before trusting a criterion. Check for nonpositive eigenvalues, near-singularity, abrupt score changes, and scenarios in which one parameter direction receives almost no information. A numerical log-determinant should generally be computed using a stable factorization, such as a Cholesky factorization when the matrix is positive definite, rather than by directly multiplying eigenvalues that may have very different magnitudes. A near-zero eigenvalue can make the score extremely sensitive to rounding or approximation error.

Monte Carlo estimates of an expectation also need diagnostics: increase simulation size, examine variability of the estimated score, and verify that rare parameter regions are not being missed. For expected log-determinant, unstable or extremely negative contributions can indicate near-singular matrices or inadequate tail behavior. For log-determinant of expected information, verify that the estimated mean matrix is positive definite and that its eigenstructure is stable. For worst-case D, test boundary points and regions where the smallest eigenvalue becomes small; a coarse grid can miss a narrow minimum.

A common mistake is to calculate the average information matrix and call its log-determinant the “expected D score.” It is instead \(J_{\mathrm{logE}}\); the expected D score is \(J_{\mathrm{Elog}}\). Jensen's inequality shows that the former is at least as large as the latter for a fixed design under suitable conditions, but it does not imply that it is always the more appropriate objective. Another mistake is to interpret a worst-case score as probability weighted: by definition, the least favorable point in the specified set controls it, regardless of that point's probability.

These criteria also have limitations. Expected criteria depend on the chosen uncertainty distribution and can conceal poor performance in low-probability regions. Worst-case D depends strongly on the uncertainty set and can be dominated by an extreme point that is implausible or poorly specified. Log-determinant summarizes joint information volume, not uncertainty in a particular parameter or a guarantee of good estimation under model misspecification. Moreover, matrix singularity and missing integrability cannot be repaired by silently replacing a zero eigenvalue with an arbitrary small positive number; such regularization changes the criterion and must be justified explicitly.