Add comparisons

[jacksonloper]

• “moment-aware" improvements incorporate higher moments or mean constraints via convexity arguments to tighten Hoeffding-type exponents within the same Chernoff framework. ➡️ i think this means like talagrand
• From, Steven G., and Andrew W. Swift. ”A refinement of Hoeffding’s inequality.” Journal of Statistical Computation and Simulation 83.5 (2013): 977-983. ▶️ This paper assumes you know $E[X_i]$ for each $i$, and all intervals are equal [0,1] intervals. We may be able to avoid comparing with this numerically as that is a lot more than we assume. Merits more thought.
• Hertz, David. ”Improved Hoeffding’s Lemma and Hoeffding’s Tail Bounds.” arXiv preprint arXiv:2012.03535 (2020). ▶️ This paper also assumes you know per var means (to know whether each variable mean is less than or more than the middle of its interval. It's not clear that numerical comparison makes much sense here.

Keeping some sources here

[jacksonloper]

For the all-known-mean-cases: cses where $\mu_i$ is known for each $i$. Things I'd want to explore:

• In this case Hoeffding tells us what the best Chernoff bound is. There will be at least one case where the best Chernoff bound is equal to ours (with the $\tau_i$ chosen according to Theorem 3.2). But there will presumably be other cases where it is even better (more uniform? less uniform?). So we can plot that, and of course explain that this only matters if you know $\mu_i$.
• Compare the Chernoff bounds from the above with From 2013 and Hertz 2020. Do they ever do better than the naive Chernoff bound you get from knowing the means?
• Compare the Chernoff bounds with what you get by getting variance bounds from Bhatia–Davis and plugging in Bennett

For the general case.

• What does Popoviciu + Bennett give you anyway?
• We need to figure out: does Talagrand apply here (e.g. if we have many different interval widths, we could always just note that they fit inside the largest interval, and then could we apply it)?
• If possible, it would be nice to get Pinelis or or Bentkus.