On p.95 (Sec 3.4.3), the text states that the Hessian ∇²A(η) = Cov[T(y)] is positive definite. In general a covariance matrix is only positive semi-definite, and is singular precisely when the family is non-minimal (i.e. when there exists v ≠ 0 with vᵀT(y) constant (a.s.), giving Var[vᵀT(y)] = 0 and hence a zero eigenvalue).
Positive definiteness holds specifically for minimal exponential families, where minimality rules out exactly these zero-variance directions. I don't see a minimality assumption stated in Section 3.4, so as written the claim is too strong.
Suggested fix: either state "positive semi-definite," or note that the Hessian is positive definite when the family is minimal (and state that
assumption).
On p.95 (Sec 3.4.3), the text states that the Hessian ∇²A(η) = Cov[T(y)] is positive definite. In general a covariance matrix is only positive semi-definite, and is singular precisely when the family is non-minimal (i.e. when there exists v ≠ 0 with vᵀT(y) constant (a.s.), giving Var[vᵀT(y)] = 0 and hence a zero eigenvalue).
Positive definiteness holds specifically for minimal exponential families, where minimality rules out exactly these zero-variance directions. I don't see a minimality assumption stated in Section 3.4, so as written the claim is too strong.
Suggested fix: either state "positive semi-definite," or note that the Hessian is positive definite when the family is minimal (and state that
assumption).