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Result · DS-SCHUR

Profiled information is a Schur complement

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Meaning

The profiled determinant measures efficient information for the parameters of interest after nuisance adjustment.

Scientific context

When only some parameters are of interest and the rest are treated as nuisance, the profiled information objective is the Schur complement of the full information matrix, and its log determinant equals the log determinant of the full information minus the log determinant of the nuisance block.

This is classical design theory, due to Näther and Reinsch, who developed the profiled case within optimal experimental design; it is cited here and not claimed.

Statement

For POI ψ and nuisance λ, S_ψ(I)=I_ψψ-I_ψλ I_λλ^{-1} I_λψ and F_s=logdet S_ψ(I)=logdet I-logdet I_λλ when the relevant blocks are nonsingular.

Closest prior work

Nearest prior work

Relationships

Enables

In the library

Carried by ProfiledDOptimality. Reference.

Every object and refusal

Local graph
Local map of Ds objective is the logdet of efficient/profiled Schur information0 prerequisites to the left, 6 results it enables or questions it raises to the right, 0 boundary cases below.Full-data efficient-score dominationFull-data efficient-s…Exact finite Ds one-point objective oracleExact finite Ds one-p…Full in-bin Ds profiling is singular when K<=dFull in-bin Ds profil…Ds first-order gradient is an efficient semimetricDs first-order gradie…Ds-efficiency relative to unbinned profiled inferenceDs-efficiency relativ…Variational form of the profiled objectiveVariational form of t…Ds objective is the logdet of efficient/profiled Schur informationDs objective is the l…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

DS0. Profiled objective and Schur notation

Let θ=(ψ,λ)\theta=(\psi,\lambda) and

Fs(I)=log⁡det⁡Sψ(I)=log⁡det⁡I−log⁡det⁡IλλF_s(I) =\log\det S_\psi(I) =\log\det I-\log\det I_{\lambda\lambda}

in the nonsingular block regime.

From Profiled Ds-optimality · chapter contents