Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta,\kappa\in\mathbb{R}$ be positive with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$. Let $d=(d_{k})_{k\in\mathbb{N}}$ be an admissible sequence with $c_{k}^{-1}+d_{k}>0$ for every $k\in\mathbb{N}$, let $\Phi_{d}$ be the diagonal quadratic profile with coefficients $d$ and $V_{d}$ its gradient field, and let $c'=(c'_{k})_{k\in\mathbb{N}}$ with $c'_{k}=(c_{k}^{-1}+d_{k})^{-1}$. $\log$ is the natural logarithm, absolute convergence of series is that of Series of Real Numbers §absolute, and $H(\cdot\,|\,\cdot)$ the relative entropy of Relative Entropy of Probability Measures §relative-entropy.

1. (The new variances)¶ $c'$ is a variance sequence; ¶there is a positive $\kappa'\in\mathbb{R}$ with $c'_{k}\le\kappa'\,a_{k}$ for every $k\in\mathbb{N}$; and ¶the series $\sum_{k=1}^{\infty}\log(c'_{k}/c_{k})$ converges absolutely.

Fix such a $\kappa'$; the sets $\mathcal{D}'$ and $\mathcal{D}'_{\Sigma}$ below do not depend on this choice, being defined through relative entropy, the relative score and Fisher information with respect to $\gamma_{c'}$ only. Read A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation with $c'$ in place of $c$, so that its reference measure is $\rho'=\gamma_{c'}$, the diagonal Gaussian measure on $X$ with variances $c'$, and its set of noise-connected measures is $\mathcal{P}^{a}_{\rho'}$; and let $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ be the Gaussian entropy pair relative to $\gamma_{c'}$ with temperature $\beta$, whose hypothesis holds with $\kappa'$.

2. (The same measures)¶ $\gamma_{c'}\in\mathcal{P}^{a}_{\rho}$, and $\mathcal{P}^{a}_{\rho'}=\mathcal{P}^{a}_{\rho}$.

3. (The same domains)¶ $\mathcal{D}'=\mathcal{D}$ and $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$.

4. (The penalty)¶ For every $\mu\in\mathcal{D}$,

$$H(\mu\,|\,\gamma_{c'})=H(\mu\,|\,\gamma_{c})+\Phi_{d}(\mu)+\frac{1}{2}\sum_{k=1}^{\infty}\log\frac{c'_{k}}{c_{k}},\qquad\text{so}\qquad\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\beta\,\Phi_{d}(\mu)+\frac{\beta}{2}\sum_{k=1}^{\infty}\log\frac{c'_{k}}{c_{k}} .$$

5. (The score)¶ Let $\nu\in\mathcal{D}_{\Sigma}$, with relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$. Its relative score with respect to $\gamma_{c'}$ is the sequence of the classes $\zeta_{k}+d_{k}x_{k}$ in $L^{2}(\nu)$, and its noise score fields $Z^{a}_{\nu}$ relative to $\gamma_{c}$ and $Z'^{a}_{\nu}$ relative to $\gamma_{c'}$ (The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field) satisfy $Z'^{a}_{\nu}=Z^{a}_{\nu}+V_{d}(\nu)$; hence $\Sigma'(\nu)=\Sigma(\nu)+\beta\,V_{d}(\nu)$.