Fix double-counted hyperspherical moment Jacobian - #5455
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FlorianPfaff
enabled auto-merge (squash)
August 28, 2026 19:12
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Bug
AbstractHypersphereSubsetDistribution.moment_numerical()explicitly multiplied its hyperspherical-coordinate integrand by the surface-element factors (for examplesin(phi2)on S2), then passed that already-weighted function tointegrate_fun_over_domain(). The latter delegates tointegrate_fun_over_domain_part(), which applies the hyperspherical surface-element Jacobian again.This computes second moments against the wrong measure for dimensions >= 2. In particular, for a uniform distribution on S2,
trace(E[x x^T])should be 1 because||x|| = 1, but the double Jacobian yieldspi/4instead.Fix
Pass the Cartesian moment integrand in hyperspherical coordinates directly to
integrate_fun_over_domain()and let the integration helper apply the surface Jacobian exactly once.Regression test
Add a uniform-S2 identity check:
E[x x^T] = I / 3trace(E[x x^T]) = 1The test is skipped on JAX, consistent with the existing numerical hyperspherical integration limitation.