One should distinguish between data driven binning and binning based on a hypothesis. If One what to test a specific hypothesis one may make the bins according to what is optimal relative to the hypothesis. If there are several hypothesis or perhaps a whole family of hypotheses it may be better to use data driven binning. I know that this is not what you ask for, but it may be useful if you could reformulate your question taking the distinction between data driven binning and hypothesis based binning into account.

Thank you, very good point. I am talking about data driven binning. For example, the binning of a data block which represents a continuous variable containing random noise. I hope I have addressed your point.

Do you mean quantile, not quintile? Quantile bins have equal probability but entropy bins show expected information, which is different, and thus would produce different bin sizes :)

Thank you to both Andrew Paul McKenzie Pegman and Monyer Abd Al-Fatlawi for your insightful comments. Andrew correctly pointed out my misuse of the word “Quintile”, when I actually meant “Quantile”. In truth, I really was fortunate that the two words were so similar, which might indicate I just can’t spell. But in reality I was confused by more than just the spelling and the link provided by Monyer help clarify several issues.

Key to this clarification was the answer to my original question: “Why shouldn't Quintile and Entropy binning (discretizing) continuous data give the same size bins for the variable under analysis?” An answer is difficult to address because the question is convoluted. That is, the premise of my question was flawed. I presented in a factual way that there are three approaches to binning which I called: “frequency”, “quintile” and “entropy”. In fact, I now believe there are only two approaches, which are formally called (per Monyer’s link: https://gerardnico.com/wiki/data_mining/discretization): 1) Equal Width and 2) Equal Frequency (aka “Quantile”).

Furthermore, I now believe that the answer to my original question is “Yes, when filtered with the new clarification of terms, they are the same”.

It appears that Shannon’s Entropy enters the binning picture by providing guidance as to the optimal number of partitions to use with the “Equal Frequency” binning option. Clark and Baron’s paper “Entropy and MDL Discretization of Continuous Variables for Bayesian Belief Networks” (http://sci2s.ugr.es/keel/pdf/specific/articulo/IJIS00.pdf) provides one such algorithms for optimization.

Please feel free to validate or refute my comments, as this is a learning experience for me.

You are right that equal width binning and equal frequency binning are the two standard methods. In Conference Paper Testing goodness-of-t via rate distortion

I present a third approach that is based on rate distortion theory. The advantage of this new method is that if one can specify a distortion function then one can calculate the optimal binning, which may be equal to equal width binning or equal frequency binning or something else.

Nice work. Is this appropriate for small data sets. I am binning time series data and each time block may only have 60 or so values representing the captured variable.? The binned data represents evidence provided to a DBN.?

Clever binning is becomes more important if the sample size is small. For the rate distortion approach the main question is whether there is a natural notion of distortion for the specific data structure.