From da4476318290453a247521b1244ba18feb34cf33 Mon Sep 17 00:00:00 2001 From: Yuchen Pei Date: Fri, 15 Mar 2019 20:35:55 +0100 Subject: minor changes --- posts/2019-03-14-great-but-manageable-expectations.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) (limited to 'posts/2019-03-14-great-but-manageable-expectations.md') diff --git a/posts/2019-03-14-great-but-manageable-expectations.md b/posts/2019-03-14-great-but-manageable-expectations.md index d812043..889f674 100644 --- a/posts/2019-03-14-great-but-manageable-expectations.md +++ b/posts/2019-03-14-great-but-manageable-expectations.md @@ -375,6 +375,7 @@ I will break the proof into two parts: $c_1$, $c_2$ and the function $C(c_1, c_2)$ are important to the practicality and usefulness of Conjecture 0. +Part 1 can be derived using Conjecture 1. We use the notations $p_I$ and $q_I$ to be $q$ and $p$ conditioned on the subsampling index $I$, just like in the proof of the subsampling theorems (Claim 19 and 24). Then @@ -385,16 +386,16 @@ $$D_\lambda(q_I || p_I) = D_\lambda(p_I || q_I) = D_\lambda(\mu_0 || \mu_0) = D_\lambda(\mu_1 || \mu_1) = 0 & I \in \mathcal I_\notin \end{cases}$$ -and that $p = |\mathcal I|^{-1} \sum_{I \in \mathcal I} p_I$ and +Since $p = |\mathcal I|^{-1} \sum_{I \in \mathcal I} p_I$ and $q = |\mathcal I|^{-1} \sum_{I \in \mathcal I} q_I$ and -$|\mathcal I_\in| = r |\mathcal I|$. +$|\mathcal I_\in| = r |\mathcal I|$, by Conjecture 1, we have Part 1. **Remark in the proof**. As we can see here, instead of trying to prove Conjecture 1, it suffices to prove a weaker version of it, by specialising on mixture of Gaussians, -to have a Claim 26 without any conjectural assumptions. +in order to have a Claim 26 without any conjectural assumptions. I have in fact posted the Conjecture on [Stackexchange](https://math.stackexchange.com/questions/3147963/an-inequality-related-to-the-renyi-divergence). -Now let us prove Item 2. +Now let us verify Part 2. Using Claim 27 and Example 1, we have -- cgit v1.2.3