The end of the family is the member the algorithm cannot reach

Every member of the Φₚ family optimised on the same 11² candidates, and two eigenvalues of each answer. The upper curve is the smallest eigenvalue of the information matrix — the quantity E-optimality maximises — which rises from 0.0993 at the D end to 0.1994 at p = 64. The lower curve is the gap between that eigenvalue and the next one up, which falls from 0.0611 to 0.0011. A smallest eigenvalue is not differentiable where it is repeated, and the family is driving the gap to zero: the one criterion here whose meaning fits in a sentence is the one whose optimum sits on a corner of its own surface. The candidate grid is on a slider and it answers a narrower question than it looks. On this square region, refining an odd grid from seven to eleven moves nothing at all — the optimum's support is the corners, the edge midpoints and the centre, and every odd grid from five up contains all of them. An even grid has no centre point and cannot reach the answer at any member of the family. On a disc, where the boundary passes through no grid point, refinement does move it.

The information a single run at time t carries about the rate of an exponential decay, (∂η/∂θ)² = t²·exp(−2θt), at three values of θ. Each curve has one maximum and it is at t = 1/θ exactly — marked, and found by a search over 8,001 settings that was never told the formula. Nothing in a linear model behaves this way: there the information matrix is X′X and the parameters are not in it, so a design can be chosen once and used whatever the answer turns out to be. Here the design is optimal at a guess, and the three curves are three different experiments for one model.

Two designs for the same two-parameter model, scored at every true value of K across a range of 16-fold. The local design is the two-point optimum for a guess of K = 1: it reaches 100% there and 66.7% at the worst point of the range. The hedged design maximises the average of log|M| over a prior spanning a factor of 4 either side, uses 3 settings rather than two, and is never below 75.4%. What it costs is 10.4 points at the one value the local design was built for — which is the whole trade, and it is only available to somebody willing to say how wrong the guess might be.