Normal Distribution
nothing is called 95% until it has been counted
Series

Equivalence — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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
  2. 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
  3. The certificate when 4 runs are already spent. a 2² factorial has already been run and 2 further runs are to be placed. The stationarity condition is no longer max d = p; it is max d = (p − λ·tr(M⁻¹M_fixed))/(1 − λ) with λ = 0.6667 the share of runs already spent, which is 7.0985 here. The search reaches 7.098508790 against it, and the largest value anywhere on a 41×41 grid is 7.098508790.

    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
  4. 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

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