rhymer

You're right, thanks for pointing that out. I missed adding the reference: Kammler DW. A First Course in Fourier Analysis. 2nd ed. Cambridge University Press; 2008. Figure 1.19. This visualization of the relationships between time and frequency domains is a good illustration of Pontryagin duality applied to signal processing

Be careful, the weight of Algorithm A by Efraimidis and Spirakis cannot be interpreted as the inclusion probability, and thus cannot be used in survey sampling to construct the Horvitz–Thompson estimator.

See "Remarks on some misconceptions about unequal probability sampling without replacement" by Yves Tillé.

Quoted from Tillé's conclusion: "There is a multitude of correct and fast method of sampling... there is no reason to use an incorrect method like weighted random sampling where we do not control the inclusion probabilities"

It's not clear to me how easy it is to implement the "multitude of corect and fast methods" in SQL, though. Would love to see some reference implementation.

Relevance: from the cited paper, the variance of the median estimator is proportional to 1/(n * f^2), where n is the sample size and f is the density at median.

Two observations: 1. With sufficiently large n, you can control your variance to an acceptable level. 2. The factor f is outside of your control. If your problem has a small density around the median, then you'll need to throw more samples to compensate for it.

I think your concern is about #2: you can always construct a pathological distribution to make the sampling-based approach unattainable. The paper provided guidance on what to expect when you apply the sampling-based approach.

Asymptotic properties of quantile estimators are widely studied [1]. The key is to have a sufficiently large sample size.

[1] Bahadur, R. R. (1966). A note on quantiles in large samples. Annals of Mathematical Statistics, 37, 577–580.

My takeaway is to avoid mixing the frequentist and Bayesian approaches. Choose one method: either follow the frequentist approach and avoid early data analysis, or use the Bayesian approach to compute posterior probabilities once data are available. Mixing the two without expertise can lead to errors.

I'm grateful for leetcode. Despite my background in electrical engineering, where I specialize in statistical signal processing, I never had the opportunity to delve into algorithm or data structure courses during my college years.

In my field, understanding signals and systems, probability, optimization, and numerical computing is very important.

Leetcode, in comparison, offers a more confined scope, making preparation more manageable and systematic, ultimately helping me break into software engineering.

Agree, this is great! I wonder if there're some ML equivalent sites that present topics in a modular way?

Just curious what are the deeper optimization topics you were hoping to see in the monograph?

Unless you water them down like some populate science books, I don't understand how you can teach probability and statistics without calculus?