TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The weighted information matrix is positive definite exactly at full sensitivity rank

Statement

For the finite nonlinear observation design of Finite nonlinear observation designs, local identifiability, and weighted information, fix $θ \in Θ$ and an approximate design $ξ$ . The matrix $M_{θ} (ξ)$ is positive definite, meaning symmetric with $v^{T} M_{θ} (ξ) v > 0$ for every nonzero $v \in R^{p}$ , if and only if the stacked support sensitivity Jacobian $J_{ξ} (θ)$ has trivial kernel, equivalently full column rank.

Facts & Assumptions

Given: The objects and hypotheses of Finite nonlinear observation designs, local identifiability, and weighted information, at a fixed $θ$ and design $ξ$ .

[F1]

Finite nonlinear observation designs, local identifiability, and weighted information: $M_{θ} (ξ) = \sum_{x} ξ_{x} J_{x} (θ)^{T} W_{x} J_{x} (θ)$ , each support weight is positive, and each $W_{x}$ is symmetric positive definite.

[F2]

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes: for compatible real matrices and vectors, products associate and $(J_{x} v)^{T} W_{x} (J_{x} v) = v^{T} J_{x}^{T} W_{x} J_{x} v$ by the coordinate product rule.

[F3]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $∥ z ∥_{2}^{2} = z^{T} z$ . The strict positivity of $z^{T} W_{x} z$ for nonzero $z$ follows from the positive-definiteness hypothesis on $W_{x}$ in [F1].

Proof

technique · direct

1.1F1F2F3givenalgebra

For any $v \in R^{p}$ , [F1] and [F2] give $v^{T} M_{θ} (ξ) v = \sum_{x \in supp (ξ)} ξ_{x} (J_{x} (θ) v)^{T} W_{x} (J_{x} (θ) v)$ . Every term is nonnegative by [F3], and a term is strictly positive exactly when $J_{x} (θ) v \neq 0$ , since $ξ_{x} > 0$ on the support and $W_{x}$ is positive definite.

2.1step 1.1algebra∎

If $J_{ξ} (θ) v = 0$ , every support component $J_{x} (θ) v$ is zero, so step 1.1 gives $v^{T} M_{θ} (ξ) v = 0$ . Conversely, if $v^{T} M_{θ} (ξ) v = 0$ , the finite sum of nonnegative terms in step 1.1 can vanish only if every support component $J_{x} (θ) v = 0$ , hence $J_{ξ} (θ) v = 0$ . Thus the quadratic form is positive on every nonzero $v$ exactly when $\ker J_{ξ} (θ) = {0}$ , as claimed.

Depends on

Used by

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Sources

Leah R. McLean et al., Estimability Analysis and Optimal Design in Dynamic Multi-scale Models of Cardiac Electrophysiology, §2.1 (standard reference, not scraped)