Ethan Epperly

@ethanepperly.bsky.social

PhD candidate in applied math at Caltech interested in computational linear algebra he/him

Gaussian integration by parts. Let z be a standard Gaussian random variable. Then E[zf(z)] = E[f'(z)].

$We start by computing $x^\top (A\circ M)x$: $$x^\top (A\circ M)x = \sum_{i,j=1}^n x_i (A\circ M)_{ij} x_j = \sum_{i,j=1}^n x_i A_{ij} M_{ij} x_j.$$Now, we may rearrange the sum, use symmetry of $M$, and repackage it as a trace $$x^\top (A\circ M)x = \sum_{i,j=1}^n x_i A_{ij} x_j M_{ji} = \tr(\operatorname{diag}(x) A \operatorname{diag}(x) M).$$This the trace formula for quadratic forms in the Schur product.$

$Proof of the Schur product theorem: The Kronecker product $A\otimes M$ of two psd matrices is psd. The entrywise product $A\circ M$ is a principal submatrix of $A\otimes M$: $$A\circ M = ((A\otimes M)_{(i+n(i-1))(i+n(i-1))} : i = 1,\ldots,n).$$All principal submatrices of a psd matrix are psd, so $A\circ M$ is psd.$