# Understanding Simpson's Paradox

An article by [Judea Pearl](http://bayes.cs.ucla.edu/jp_home.html), available [here](http://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf). It's quick at 8 pages, and worth reading if you enjoy statistics (though I think people who already are familiar with the math of causality^1^ will get more out of it than others^2^). I'll talk here about the part that I think is generally interesting:

> Any claim to a resolution of a paradox, especially one that has resisted a century of attempted resolution must meet certain criteria. First and foremost, the solution must explain why people consider the phenomenon surprising or unbelievable. Second, the solution must identify the class of scenarios in which the paradox may surface, and distinguish it from scenarios where it will surely not surface. Finally, in those scenarios where the paradox leads to indecision, we must identify the correct answer, explain the features of the scenario that lead to that choice, and prove mathematically that the answer chosen is indeed correct. The next three subsections will describe how these three requirements are met in the case of Simpson's paradox and, naturally, will proceed to convince readers that the paradox deserves the title "resolved."

I've never really liked the name "paradox," because what it seems to mean is "unintuitive phenomenon." ([Wikipedia](http://en.wikipedia.org/wiki/Paradox) puts it as "something which seems false and yet might be true.") The trouble is that "unintuitive" is a [two-place word](/lw/ro/2place_and_1place_words/), and it makes sense to [think like reality](/lw/hs/think_like_reality/), so that true things seem true to you, instead of still seeming false. (For example, when I first learned about [Zeno's Paradox](http://en.wikipedia.org/wiki/Zeno%27s_paradox), I already knew calculus, and so _Zeno's_ position was the one that seemed confusing and false.)

What I like most about Pearl's article is that it explicitly recognizes the importance of fully [dissolving](/lw/of/dissolving_the_question/) the paradox,^3^ and seems to do so. Simpson's Paradox isn't an unsolvable problem in statistics, it's a straightforward reversal effect--_only_ if you use the language of causality.

* * *

1\. [My review of Causality](/lw/emc/causality_a_chapter_by_chapter_review/) gives a taste of what it would look like to be familiar with the math, but you'd need to actually read the book to pick it up. The [Highly Advanced Epistemology 101 for Beginners](http://wiki.lesswrong.com/wiki/Highly_Advanced_Epistemology_101_for_Beginners) sequence is relevant, and contains Eliezer's attempt to explain the basics of causality in [Causal Diagrams and Causal Models](/lw/ev3/causal_diagrams_and_causal_models/).

2\. Pearl discusses how you would go about using simulations to show that do calculus gives you the right result, but leaves it as an exercise for the reader.

3\. [How An Algorithm Feels From Inside](/lw/no/how_an_algorithm_feels_from_inside/) is probably a better place to start than Dissolving the Question, and I can't help but echo a question from it: "So what kind of math design corresponds to \[Simpson's Paradox\]?"

See also: bentarm's explanation of [Simpson's Paradox](/lw/3q3/simpsons_paradox/).