
462 Current Trends in Bayesian Methodology with Applications
where ∇
θ
denotes the gradient. In par ticula r, for first degree polynomials
P (θ) = a
T
θ, the auxiliary function reduces to
˜g(θ) = g(θ) + a
T
z(θ). (22.10)
The linear polynomial coefficients a which minimize Va r
π
[˜g] are given by
a = −Var
−1
π
[z(θ)]Cov
π
[g(θ), z(θ)], (22.11)
where Var
π
[z(θ)] denotes the covariance matrix (of dimension n
θ
×n
θ
) o f the
control varia tes and Cov
π
[g(θ), z(θ)] = E
π
[g(θ)·z(θ)] is a vector of dimension
n
θ
× 1.
The coefficients a, as provided by (22.11), are not usually available (since
the variance and covariance are with respect to the target π) and one needs
to replace Var