add testing bounds

blueprint/src/sections/kl_divergence.tex

@@ -149,46 +149,4 @@ \section{Chain rule and tensorization}
 
 \end{proof}
 
-\section{Change of measure}
-
-\begin{lemma}
-  \label{lem:llr_change_measure}
-  %\lean{}
-  %\leanok
-  %\uses{}
-  Let $\mu, \nu$ be two measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\beta \in \mathbb{R}$. Then
-  \begin{align*}
-  \nu(E) e^{\beta} \ge \mu(E) - \mu\left\{ \log\frac{d \mu}{d \nu} > \beta \right\} \: .
-  \end{align*}
-\end{lemma}
-
-\begin{proof}
-\begin{align*}
-\nu(E)
-&\ge \mu\left[\mathbb{I}(E) e^{- \log\frac{d \mu}{d \nu} }\right]
-\\
-&\ge \mu\left[\mathbb{I}\left(E \cap \left\{\log\frac{d \mu}{d \nu} \le \beta\right\}\right) e^{- \log\frac{d \mu}{d \nu} }\right]
-\\
-&\ge e^{- \beta}\mu\left(E \cap \left\{\log\frac{d \mu}{d \nu} \le \beta\right\}\right)
-\\
-&\ge e^{- \beta}\left( \mu(E) - \mu\left\{ \log\frac{d \mu}{d \nu} > \beta \right\} \right)
-\: .
-\end{align*}
-\end{proof}
-
-\begin{corollary}
-  \label{cor:kl_change_measure}
-  %\lean{}
-  %\leanok
-  \uses{def:KL}
-  Let $\mu, \nu$ be two measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\beta \in \mathbb{R}$. Then
-  \begin{align*}
-  \nu(E) e^{\KL(\mu, \nu) + \beta} \ge \mu(E) - \mu\left\{ \log\frac{d \mu}{d \nu} - \KL(\mu, \nu) > \beta \right\} \: .
-  \end{align*}
-\end{corollary}
-
-\begin{proof}
-\uses{lem:llr_change_measure}
-Use Lemma~\ref{lem:llr_change_measure} with the choice $\KL(\mu, \nu) + \beta$ for $\beta$.
-\end{proof}
 

blueprint/src/sections/renyi_divergence.tex

@@ -49,6 +49,18 @@ \chapter{Rényi divergences}
 Unfold the definitions.
 \end{proof}
 
+\begin{lemma}
+  \label{lem:renyi_cgf_2}
+  %\lean{}
+  %\leanok
+  \uses{def:Renyi}
+  The cumulant generating function of $\log\frac{d\mu}{d\nu}$ under $\mu$ is $\alpha \mapsto \alpha R_{1+\alpha}(\mu, \nu)$.
+\end{lemma}
+
+\begin{proof}
+Unfold the definitions.
+\end{proof}
+
 \begin{definition}[Conditional Rényi divergence]
   \label{def:condRenyi}
   %\lean{}

blueprint/src/sections/testing.tex

@@ -59,7 +59,109 @@ \section{Generic lower bounds}
 The inequality is an application of Lemma~\ref{lem:testing_bound_renyi_mean} for $\alpha = 1/2$. The equality is Lemma~\ref{lem:renyi_half_eq_log_hellinger}.
 \end{proof}
 
-\subsection{Lower bounds on the maximum}
+\section{Change of measure}
+
+\begin{lemma}
+  \label{lem:llr_change_measure}
+  %\lean{}
+  %\leanok
+  %\uses{}
+  Let $\mu, \nu$ be two measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\beta \in \mathbb{R}$. Then
+  \begin{align*}
+  \nu(E) e^{\beta} \ge \mu(E) - \mu\left\{ \log\frac{d \mu}{d \nu} > \beta \right\} \: .
+  \end{align*}
+\end{lemma}
+
+\begin{proof}
+\begin{align*}
+\nu(E)
+&\ge \mu\left[\mathbb{I}(E) e^{- \log\frac{d \mu}{d \nu} }\right]
+\\
+&\ge \mu\left[\mathbb{I}\left(E \cap \left\{\log\frac{d \mu}{d \nu} \le \beta\right\}\right) e^{- \log\frac{d \mu}{d \nu} }\right]
+\\
+&\ge e^{- \beta}\mu\left(E \cap \left\{\log\frac{d \mu}{d \nu} \le \beta\right\}\right)
+\\
+&\ge e^{- \beta}\left( \mu(E) - \mu\left\{ \log\frac{d \mu}{d \nu} > \beta \right\} \right)
+\: .
+\end{align*}
+\end{proof}
+
+\begin{corollary}
+  \label{cor:kl_change_measure}
+  %\lean{}
+  %\leanok
+  \uses{def:KL}
+  Let $\mu, \nu$ be two measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\beta \in \mathbb{R}$. Then
+  \begin{align*}
+  \nu(E) e^{\KL(\mu, \nu) + \beta} \ge \mu(E) - \mu\left\{ \log\frac{d \mu}{d \nu} - \KL(\mu, \nu) > \beta \right\} \: .
+  \end{align*}
+\end{corollary}
+
+\begin{proof}
+\uses{lem:llr_change_measure}
+Use Lemma~\ref{lem:llr_change_measure} with the choice $\KL(\mu, \nu) + \beta$ for $\beta$.
+\end{proof}
+
+\begin{lemma}
+  \label{lem:renyi_change_measure}
+  %\lean{}
+  %\leanok
+  \uses{def:Renyi}
+  Let $\mu, \nu$ be two measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\alpha,\beta \ge 0$. Then
+  \begin{align*}
+  \nu(E) e^{R_{1+\alpha}(\mu, \nu) + \beta} \ge \mu(E) - e^{-\alpha \beta} \: .
+  \end{align*}
+\end{lemma}
+
+\begin{proof}
+\uses{lem:llr_change_measure}
+Use Lemma~\ref{lem:llr_change_measure} with the choice $R_{1+\alpha}(\mu, \nu) + \beta$ for $\beta$. Then by a Chernoff bound,
+\begin{align*}
+\mu\left\{ \log\frac{d \mu}{d \nu} - R_{1+\alpha}(\mu, \nu) > \beta \right\}
+&= \mu\left\{ \exp\left(\alpha\log\frac{d \mu}{d \nu}\right) > \exp\left(\alpha R_{1+\alpha}(\mu, \nu) + \alpha \beta\right) \right\}
+\\
+&\le e^{-\alpha R_{1+\alpha}(\mu, \nu) - \alpha \beta} \mu\left[\left(\frac{d \mu}{d \nu}\right)^\alpha \right]
+\end{align*}
+Then $\mu\left[\left(\frac{d \mu}{d \nu}\right)^\alpha \right] = \nu\left[\left(\frac{d \mu}{d \nu}\right)^{1+\alpha} \right] = e^{\alpha R_{1+\alpha}(\mu, \nu)}$. We obtained
+\begin{align*}
+\mu\left\{ \log\frac{d \mu}{d \nu} - R_{1+\alpha}(\mu, \nu) > \beta \right\}
+\le e^{- \alpha \beta}
+\end{align*}
+\end{proof}
+
+\begin{lemma}
+  \label{lem:llr_change_measure_add}
+  %\lean{}
+  %\leanok
+  %\uses{}
+  Let $\mu, \nu, \xi$ be three probability measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\beta_1, \beta_2 \in \mathbb{R}$. Then
+  \begin{align*}
+  \mu(E) e^{\beta_1} + \nu(E^c) e^{\beta_2} \ge 1 - \xi\left\{ \log\frac{d \xi}{d \mu} > \beta_1 \right\} - \xi\left\{ \log\frac{d \xi}{d \nu} > \beta_2 \right\} \: .
+  \end{align*}
+\end{lemma}
+
+\begin{proof}
+\uses{lem:llr_change_measure}
+Two applications of Lemma~\ref{lem:llr_change_measure}, then sum them and use $\xi(E)+\xi(E^c) = 1$.
+\end{proof}
+
+\begin{lemma}
+  \label{lem:renyi_change_measure_add}
+  %\lean{}
+  %\leanok
+  %\uses{}
+  Let $\mu, \nu, \xi$ be three probability measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\alpha, \beta \ge 0$. Then
+  \begin{align*}
+  \mu(E) e^{R_{1+\alpha}(\xi, \mu) + \beta} + \nu(E^c) e^{R_{1+\alpha}(\xi, \nu) + \beta} \ge 1 - 2 e^{-\alpha \beta} \: .
+  \end{align*}
+\end{lemma}
+
+\begin{proof}
+\uses{lem:renyi_change_measure}
+\end{proof}
+
+
+\section{Lower bounds on the maximum}
 
 \begin{lemma}
   \label{lem:testing_bound_tv_hellinger_max}
@@ -97,6 +199,23 @@ \subsection{Lower bounds on the maximum}
 Use Lemma~\ref{lem:testing_bound_renyi_mean} and remark that $\mu(E)^\alpha + \nu(E^c)^{1 - \alpha} \le 2\max\{\mu(E), \nu(E^c)\}^{\min\{\alpha, 1 - \alpha\}}$.
 \end{proof}
 
+
+\begin{lemma}
+  \label{lem:testing_bound_renyi_one_add}
+  %\lean{}
+  %\leanok
+  %\uses{}
+  Let $\mu, \nu$ be two probability measures on $\mathcal X$ and let $E$ be an event on $\mathcal X$. Let $\alpha > 0$. Then
+  \begin{align*}
+  \max\{\mu(E), \nu(E^c)\} \ge \frac{1}{4}\exp\left( - \inf_{\xi \in \mathcal P(\mathcal X)}\max\{R_{1+\alpha}(\xi, \mu), R_{1+\alpha}(\xi, \nu)\} - \frac{\log 4}{\alpha}\right) \: .
+  \end{align*}
+\end{lemma}
+
+\begin{proof}
+\uses{lem:renyi_change_measure_add}
+\end{proof}
+
+
 \section{Product spaces}
 
 \begin{corollary}