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Result · GENERAL-FIRST-VARIATION

A cell transfer has an explicit first variation

Proved herepublication: internal

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Meaning

The first variation connects changes in cell moments to the information criterion’s local geometry.

Statement

Moving infinitesimal mass from cell a to b gives dI=[(s-μ_a)(s-μ_a)^T-(s-μ_b)(s-μ_b)^T]dε; for differentiable F the directional derivative is the difference of the corresponding G=∇F(I) quadratic forms.

generic population stationarity engine

Relationships

Rests on

Enables

Local graph
Local map of Infinitesimal cell-transfer identity1 prerequisites to the left, 4 results it enables or questions it raises to the right, 0 boundary cases below.Quantized Fisher-information identityQuantized Fisher-info…Population stationary D geometryPopulation stationary…Ds first-order gradient is an efficient semimetricDs first-order gradie…Repeated minimum eigenvalues make one-transfer first-order E geometry non-identifyingRepeated minimum eige…Population stationary geometry for profiled DsPopulation stationary…Infinitesimal cell-transfer identityInfinitesimal cell-tr…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

Proof

G1. Infinitesimal cell-transfer identity

Move infinitesimal probability mass dεd\varepsilon at score ss from cell aa to bb. Then

dIq=[(s−μa)(s−μa)⊤−(s−μb)(s−μb)⊤]dε.\boxed{ dI_q= [(s-\mu_a)(s-\mu_a)^\top-(s-\mu_b)(s-\mu_b)^\top]d\varepsilon. }

For differentiable F(I)F(I) with symmetric G=∇F(I)G=\nabla F(I),

dFdε=(s−μa)⊤G(s−μa)−(s−μb)⊤G(s−μb).\boxed{ \frac{dF}{d\varepsilon} =(s-\mu_a)^\top G(s-\mu_a) -(s-\mu_b)^\top G(s-\mu_b). }

A shared GG makes pairwise boundaries affine because the common s⊤Gss^\top Gs term cancels.

From First-order and finite screening · chapter contents