Source
Chaloner and Verdinelli (1995), LIT-1525, §4.2 (approximations), §4.4 (local optimality), §5.1 (equivalence theorem) and §7.2 (turning point). Read in NOTE-1161. The Elfving argument below is the record’s.
What was actually shown
The approximation (§4.2). With the posterior approximated by N(θ̂, [nI(θ̂, η)]⁻¹), and the predictive distribution of θ̂ by the prior, expected Shannon information is −(k/2)log(2π) − k/2 + ½∫ log det{nI(θ, η)} p(θ) dθ. That gives the criterion φ₁(η) = ∫ log det{nI(θ, η)} p(θ) dθ. Squared-error loss for g(θ), with gradient c(θ), gives φ₂(η) = −∫ c(θ)ᵀ{nI(θ, η)}⁻¹c(θ) p(θ) dθ. These are averages of local criteria. With p a point mass at θ₀ they become Chernoff’s locally optimal criteria det I(θ₀, η) and −c(θ₀)ᵀI(θ₀, η)⁻¹c(θ₀) (§4.4). Each is concave on the space of design measures, so the equivalence theorem certifies a global optimum by directional derivatives toward one-point designs (§5.1).
The turning point (§7.2), re-derived. E[y | x] = θ₀ + θ₁x + θ₂x² on x ∈ [−1, 1], with f(x) = (1, x, x²). The turning point is g(θ) = −θ₁/(2θ₂), so ∇g = (0, −1/(2θ₂), θ₁/(2θ₂²)). At a best guess with g = ½, θ₁ = −θ₂, so ∇g = −(1/(2θ₂))(0, 1, 1). The local criterion is c-optimality for c = (0, 1, 1).
- The design. Put mass ½ at x = 1 and ½ at x = 0. Then ½·f(1) − ½·f(0) = (0, ½, ½) = c/2.
- Its optimality, by Elfving. A design is c-optimal iff c/ρ is on the boundary of the convex hull of {±f(x)} and is reached by its support points with signs. Take h = (−1, 0, 2). Then hᵀf(x) = 2x² − 1 ∈ [−1, 1] on [−1, 1], so the hyperplane hᵀz = 1 supports the hull. And hᵀ(c/2) = 1, with hᵀf(1) = 1 and hᵀ(−f(0)) = 1, so c/2 lies on the face reached by +f(1) and −f(0). The design is c-optimal, with ρ = 2 and asymptotic variance 4σ²/n for cᵀθ. The only other point where |2x² − 1| = 1 is x = −1. Writing c/2 as a combination of +f(1), +f(−1) and −f(0) forces the weight on f(−1) to zero, so the optimal design is unique.
- What it estimates. M(η) = ½f(0)f(0)ᵀ + ½f(1)f(1)ᵀ has rank 2, with range spanned by (1, 0, 0) and (1, 1, 1). Only θ₀ and θ₀ + θ₁ + θ₂ are identified, so θ₁ + θ₂ is too. That is enough for the linear function cᵀθ but not for the turning point −θ₁/(2θ₂), which needs θ₁ and θ₂ separately. As the review says, “it is clearly impossible to fit a quadratic regression to two data points and estimate the turning point.”
What could have failed: the locally optimal design for a nonlinear function might have been estimable whenever its linearization is, as in linear c-optimality, where cᵀθ “is always estimable”. The example shows that the linearization at the guess can be estimable while the function is not.
What this does not say
- Not that Bayesian designs always avoid the failure. A prior spread over θ moves the c(θ) being averaged and so tends to add support points. The review shows this in examples (Chaloner and Larntz) and says “It is unclear, however, under what circumstances this is so.”
- Not how good the approximations are. “Not much is known in general about how well these criteria … perform empirically.” Simulations are cited for special cases only.
- Not anything about sequential design beyond the review’s argument that nonlinear problems should gain from it (§5.5).
Connections
- CASE-159. The mixture response p_π = Σ πᵢpᵢ is linear in π, but its Fisher information is not constant: I_rare(π) = 0.64/(p(1 − p)) with p = 0.1 + 0.8π. So designing probes for authority shares is a nonlinear design problem in this sense. A design must be judged at a guess of π or averaged over a prior on it. In the two-specialist case the choice does not depend on π, because the routine context carries zero information at every π. With more processes it does, and the equivalence theorem certifies the optimum (NOTE-1161 computes one).
- QUESTION-031. A criterion averaged over a prior answers “which probes, given what we expect”, not “which probes separate”. The turning point example is a case where the best probes for a linearized question fail the real one.
- THEORY-956 is the linear case, where none of this arises.