diff --git a/blueprint/src/sections/f_divergence.tex b/blueprint/src/sections/f_divergence.tex index 61a04056..42198695 100644 --- a/blueprint/src/sections/f_divergence.tex +++ b/blueprint/src/sections/f_divergence.tex @@ -115,13 +115,13 @@ \section{Conditional f-divergence} By Lemma~\ref{lem:rnDeriv_compProd} and Corollary~\ref{cor:rnDeriv_value}, \begin{align*} D_f(\mu \otimes \kappa, \mu \otimes \eta) -&= \int_{p} f\left(\frac{d (\mu \otimes \kappa)}{d (\mu \otimes \eta)}(p)\right) \partial(\mu \otimes \kappa) +&= \int_{p} f\left(\frac{d (\mu \otimes \kappa)}{d (\mu \otimes \eta)}(p)\right) \partial(\mu \otimes \eta) \\ -&= \int_{p} f\left(\frac{d \kappa}{d \eta}(p)\right) \partial(\mu \otimes \kappa) +&= \int_{p} f\left(\frac{d \kappa}{d \eta}(p)\right) \partial(\mu \otimes \eta) \\ -&= \int_x \int_y f\left(\frac{d \kappa}{d \eta}(x, y)\right) \partial \kappa(x) \partial \mu +&= \int_x \int_y f\left(\frac{d \kappa}{d \eta}(x, y)\right) \partial \eta(x) \partial \mu \\ -&= \int_x \int_y f\left(\frac{d \kappa(x)}{d \eta(x)}(y)\right) \partial \kappa(x) \partial \mu +&= \int_x \int_y f\left(\frac{d \kappa(x)}{d \eta(x)}(y)\right) \partial \eta(x) \partial \mu \\ &= \mu\left[D_f(\kappa(x), \eta(x))\right] = D_f(\kappa, \eta \mid \mu) @@ -158,13 +158,13 @@ \section{Data-processing inequality} By Lemma~\ref{lem:rnDeriv_compProd}, \begin{align*} D_f(\mu \otimes \kappa, \nu \otimes \kappa) -&= \int_{p} f\left(\frac{d (\mu \otimes \kappa)}{d (\nu \otimes \kappa)}(p)\right) \partial(\mu \otimes \kappa) +&= \int_{p} f\left(\frac{d (\mu \otimes \kappa)}{d (\nu \otimes \kappa)}(p)\right) \partial(\nu \otimes \kappa) \\ -&= \int_{p} f\left(\frac{d \mu}{d \nu}(p_X)\right) \partial(\mu \otimes \kappa) +&= \int_{p} f\left(\frac{d \mu}{d \nu}(p_X)\right) \partial(\nu \otimes \kappa) \\ -&= \int_x \int_y f\left(\frac{d \mu}{d \nu}(x)\right) \partial \kappa(x) \partial \mu +&= \int_x \int_y f\left(\frac{d \mu}{d \nu}(x)\right) \partial \kappa(x) \partial \nu \\ -&= \int_x f\left(\frac{d \mu}{d \nu}(x)\right) \partial \mu +&= \int_x f\left(\frac{d \mu}{d \nu}(x)\right) \partial \nu \\ &= D_f(\mu, \nu) \: .