Source
Chaloner and Verdinelli (1995), Statistical Science 10(3):273–304, LIT-1525, §§2.1–2.2 and §5.5. Read in NOTE-1161 from the authors’ technical report. The derivations below are the record’s, written because the report’s displays did not survive extraction.
What was actually shown
Model: y | θ ~ N(Xθ, σ²I_n), σ² known, θ ∈ ℝ^k, and prior θ ~ N(θ₀, σ²R⁻¹). A design η puts proportion η_i of the n observations at x_i, so XᵀX = nM(η) = n Σ η_i x_i x_iᵀ.
The posterior. By conjugacy, θ | y ~ N(θ*, σ²D(η)) with D(η) = (nM(η) + R)⁻¹ and θ* = D(η)(Xᵀy + Rθ₀). The posterior covariance does not involve y.
Shannon information. The prior does not depend on η, so maximizing the expected Kullback–Leibler divergence from prior to posterior is maximizing U₁(η) = E[log p(θ | y, η)], the expectation over θ and y jointly. For a k-variate normal density with covariance Σ, E[log p] = −½ k log(2πe) − ½ log det Σ. Here Σ = σ²D(η) for every y, so U₁(η) = −½ k log(2πeσ²) + ½ log det(nM(η) + R). The design maximizing U₁ maximizes det(nM(η) + R).
Quadratic loss. For loss (θ − θ̂)ᵀA(θ − θ̂) with A ⪰ 0, the Bayes estimate is the posterior mean, and the posterior expected loss is tr(A Cov(θ | y)) = σ² tr(A D(η)), again for every y. So the expected utility is U₂(η) = −σ² tr{A(nM(η) + R)⁻¹}. A = ccᵀ gives Bayes c-optimality, −σ² cᵀD(η)c.
No gain from sequencing. Let n be fixed and let a sequential rule choose each x_t from the data so far. Whatever it does, the realized design η̂ is some measure in ℋ. The posterior utility given all the data is, up to constants, ½ log det(nM(η̂) + R) or −σ² tr AD(η̂), a function of η̂ alone. Its expectation is therefore at most max_η U(η), which a fixed design attains.
The prior’s weight. D(η) = n⁻¹(M(η) + n⁻¹R)⁻¹. As n grows or R shrinks the Bayesian criteria approach the non-Bayesian det M and tr AM⁻¹. A proper prior keeps nM + R invertible when nM is singular, so a design need not identify every parameter from data alone.
What could have failed: the utility could have depended on the data through the posterior, in which case a design criterion would need an average over y and would depend on θ₀. In this model the posterior covariance is free of y, which is why D, A and c-optimality are exact here and approximations elsewhere.
What this does not say
- Not for unknown variance. With a normal–inverse-gamma prior the Shannon and predictive utilities have no closed form (§2.5). Bayes A reduces to tr(nM + R)⁻¹ times a constant independent of y, and is “insensitive to the knowledge of σ²”.
- Not for nonlinear models, where the information matrix depends on θ and the criteria are approximations (THEORY-954).
- Not for E- or G-optimality, which “appear not to correspond to any utility function”.
- Not that the criteria agree. D, A and c generally select different designs. The purpose of the experiment selects among them.
Connections
- QUESTION-031. The note of 2026-10-25 uses log det G_ξ with G_ξ = Σ_c ξ(c)G_c. In a linear Gaussian probe model that is exactly U₁ up to constants. A29’s preference for a criterion that targets one contrast is c- or Ds-optimality.
- CASE-152. Its intervention do(b = b₀) gives G_aa = b₀². In a linear model, adding that observation adds b₀² to the information about a, which raises all three criteria.
- CASE-159 is not linear in this sense. Its information depends on the share being estimated (THEORY-954).