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

Least favourable prior — where it appears

The weighting of parameter values that makes an averaged criterion as bad as possible, and under which the averaged design is the maximin one. It is the bridge between averaging a criterion and protecting its worst case, and it exists because the two problems are duals.

Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.

Four designs, scored on the one parameter that was wanted. Every design scored by its Ds-efficiency for K at 13 true values across a 16-fold range. The peaked curve is the subset design built at the guess K = 1: 100% there and 42.1% at the worst point of the range. The flat curve is the maximin-Ds design, never above 64.8% and never below 61.2%. Between them is the maximin design for the pair — a robust design, protecting something else, and worth 41.9% at worst here. The lowest curve is the D-optimal design at the guess, which is what an experimenter who wanted K and looked up a design for the model would actually run: 29.1% at the worst point, against 61.2% available.

Protecting one parameter over a range

A design for a non-linear model is optimal at a guess. A design for one of its parameters over a range of guesses is a worst case of a ratio of two determinants, and it is not a special case of either problem it is made of.

guarantee · Local design The optimum is a tie, and the tie is at both ends. The maximin design's efficiency across the range, and underneath it the prior that makes the averaged criterion as bad as possible. The efficiency curve is flat to within 2.1 points, and the minimum 78.74% is attained at K = 0.25 and 0.79 and 0.89 and 1.00 and 1.12 and 4.00 rather than at a single value: if it were attained once, the design could be moved towards that value and the worst case improved, so a tie is what having finished looks like. The bars are the least favourable prior's weights, computed by a completely different route — an averaging problem solved under the weighting that hurts most — and it puts its weight exactly where the ties are, reaching 78.63% against the direct search's 78.74%.

Where the minimum is attained

A design that protects a range is finished when its worst case is a tie. That is a checkable property rather than a description, it is why the search cannot climb a derivative, and it is the same corner the criteria field found at the end of the Φₚ family.

robust · Local design The guarantee, as the basis is allowed more functions. The lower line is the best worst case over the six named shapes for a basis of each size, found by scoring every subset of the dictionary — an exact answer, since the problem is finite. One function guarantees 2.3%, which is nearly nothing; three guarantee 59.0% and the basis that does it is the covariate, its square and its cube, with no indicator in it. The upper line is the same problem with the basis drawn rather than fixed, which is worth 2.09 times as much at two functions and 1.32 at three. The two lines converge because a basis large enough to protect everything has nothing left to randomise over.

Which shapes are worth protecting

Choosing a basis by its worst case is a finite problem with an exact answer. The answer has no tie in it, which a maximin optimum is supposed to have — and the tie comes back, along with twice the guarantee, when the basis is drawn rather than chosen.

basis · Blocking What a floor costs a drawn basis. The expected guarantee of a basis drawn from a mixture restricted to bases that each remove at least a stated share of every shape, solved at each distinct floor. 2 functions, of 276 bases: 40.8% at a floor of 0.0%; the best fixed basis floors at 0.0%. 3 functions, of 2024 bases: 61.3% at a floor of 0.0%, 50.0% at a floor of 31.8%, 50.0% at a floor of 34.7%, 49.7% at a floor of 40.5%, 42.4% at a floor of 42.4%; the best fixed basis floors at 42.4%. 4 functions, of 10626 bases: 79.2% at a floor of 0.0%, 77.1% at a floor of 31.8%, 77.1% at a floor of 34.7%, 77.1% at a floor of 35.1%, 77.1% at a floor of 35.6%, 77.1% at a floor of 37.0%, 77.1% at a floor of 38.3%, 77.1% at a floor of 40.5%, 76.5% at a floor of 41.0%, 76.5% at a floor of 42.4%, 76.5% at a floor of 42.8%, 76.5% at a floor of 42.9%, 76.5% at a floor of 43.5%, 76.5% at a floor of 44.7%, 76.5% at a floor of 45.3%, 76.5% at a floor of 46.0%, 76.5% at a floor of 46.8%, 76.5% at a floor of 47.2%, 76.5% at a floor of 50.0%, 76.5% at a floor of 56.0%, 76.5% at a floor of 63.7%; the best fixed basis floors at 63.7%.

A floor for the draw

A basis drawn at random guarantees more on average than any fixed one, at the price of leaving some shape wholly untouched in a large share of trials. Requiring every basis in the draw to keep a floor looks like a smooth trade between the two. It is a staircase, and the first step does nearly all the work. No basis has a floor between zero and 31.83%, so there is nothing to give up until a floor of that size is asked for. And at four functions, seven bases tie at the fixed basis's own 63.66%: a mixture of them keeps that floor exactly and adds thirteen points to the guarantee.

product · Optimum The textbook (Wald) interval's coverage among studies whose count of 40 lands in the tails. Coverage of the nominal 95% textbook (Wald) interval at 40 trials, conditional on the count being at most k or at least 40 − k, computed exactly over the counts for every true proportion. With k = 2 its highest value anywhere is 88.96%; With k = 5 its highest value anywhere is 94.90%; With k = 8 its highest value anywhere is 95.57%. A set whose highest value stays below 95% is a set of counts on which the label is wrong whatever the proportion is.

A count that bets against its interval

A reader who sees only the count can bet against an interval's 95%, and win whatever the proportion is, exactly when some set of counts carries less than 95% at every proportion. Out of forty trials the textbook interval has one — a count of five or fewer, or thirty-five or more, covers at most 94.90% — and the score interval has none: a prior exists under which every one of its forty-one intervals is 96.46% credible, and that prior is the proof that no bet can be found.

intervals · Coverage

Named alongside it

The objects these essays reach for when they reach for this one.

Maximin designBasis functionsBayesian optimal designCovariate balanceD-optimalityDesign measureEquivalence theoremInformation matrixLocally optimal designMichaelis–MentenThe non-linear modelAllocation rule

All concepts