Equivalence — the series · Normal Distribution

The variance touches p and never crosses it. d(x) = f(x)′M⁻¹f(x) along the diagonal of a square region, for the D-optimal measure. The line at 6 is the number of parameters in the model. Kiefer and Wolfowitz's theorem says a design is D-optimal exactly when the largest d anywhere in the region is p — not approximately, equals — so the optimal curve is tangent to that line at its support points and below it everywhere else. Here the largest value anywhere on a 41×41 grid is 6.000000000.

The theorem that says when to stop

A search that maximises the volume of the information has no way of knowing it has finished, because nothing tells it what the maximum is. Kiefer and Wolfowitz's equality does — a design is D-optimal exactly when the worst prediction anywhere in the region equals the number of parameters, which is 6.000000000059 here, gated at machine precision.

part 1 · optimality

What visiting fewer settings costs, 6 parameters. Carathéodory's bound puts the support of an optimal measure between 6 and 21. 6 settings: D-efficiency 88.90%, G-efficiency 57.18%, 0 degrees of freedom for lack of fit; 7 settings: D-efficiency 94.54%, G-efficiency 61.22%, 1 degrees of freedom for lack of fit; 8 settings: D-efficiency 95.99%, G-efficiency 64.60%, 2 degrees of freedom for lack of fit; 9 settings: D-efficiency 97.40%, G-efficiency 82.76%, 3 degrees of freedom for lack of fit. The saturated design has none, and buying the first one costs about five points of efficiency to get back.

How many places a design goes

Carathéodory's bound puts an optimal design's support between six and twenty-one settings, and every design in this field that can fit the model visits exactly nine. The count is not a choice anybody makes, it decides how many degrees of freedom are left to check the model with, and the first spare setting costs six points of efficiency to get back.

part 2 · optimality

Augmenting a design that has already run

The equivalence theorem still certifies when some runs are already spent, and one number in it changes: the bound is no longer p but (p − λ·tr(M⁻¹M_fixed))/(1 − λ). It equals p again exactly when the runs already made can still be absorbed into the design that would have been chosen — so the certificate says whether the experiment is still recoverable.

part 3 · optimality

Where each criterion's optimum puts the information. the D-optimal design's smallest eigenvalue is 0.09927, attained once; the A-optimal design's smallest eigenvalue is 0.16516, attained once; the I-optimal design's smallest eigenvalue is 0.17541, attained once; the E-optimal design's smallest eigenvalue is 0.19999, attained 2 times. A criterion that reads the smallest eigenvalue has no derivative where that eigenvalue is repeated, and the E-optimal design is exactly there.

The criterion with no derivative

E-optimality maximises the smallest eigenvalue of the information matrix, and at its own optimum that eigenvalue is attained twice — which is exactly where the function has a corner. The multiplicative search this field's other three criteria use assumes a derivative that is not there, and stops at 37.2% of the optimum.

part 4 · optimality

The E-optimality certificate along the axis x₂ = 0. The certificate — a fifth of the squared distance of the point's squared radius from one, plus a fifth of the squared difference of its two squared coordinates — along the axis x₂ = 0, with its two pieces drawn alone at full weight: the radial piece, a third of the squared distance from the unit circle in squared radius, which reaches 0.333 here, and the cross piece, half the squared difference of the squared coordinates, which reaches 0.500. The certificate's largest value on the cut is 0.2000, reached at x = -1.0, 0.0, 1.0; the smallest eigenvalue of the nine-point design is 0.2.

A certificate for the worst direction

E-optimality was the one design criterion here whose answer was claimed rather than proved. It has a certificate after all, and on the square it fits on one line — (1⁄5)(1 − x₁² − x₂²)² + (1⁄5)(x₁² − x₂²)², which never exceeds 1⁄5 and touches it at exactly the nine points of the 3² grid. That proves the design with 2⁄5 of its weight at the centre, 1⁄10 at each edge midpoint and 1⁄20 at each corner is E-optimal on the whole square, and it is built from two directions mixed three to two, either of which alone fails.

part 5 · optimality