Generating Simpson's Paradox with Z3
kevinlynagh.com
kevinlynagh.com
Along the same lines of "visualize your data to see what's really going on" is Anscombe's Quartet: https://en.wikipedia.org/w/index.php?title=Anscombe%27s_quar...
And then there's the Datasaurus [Dozen], which has some fun with the idea behind Anscombe's Quartet: https://en.wikipedia.org/wiki/Datasaurus_dozen (you can see it animated here: https://blog.revolutionanalytics.com/2017/05/the-datasaurus-... )
There is no contradiction in subsets having different correlations that the parent set. The apparent "paradox" arises from reading the data causally. The purpose of this lesson is to expose these assumptions in interpretation of data. Few seem to get the message though.
Consider another medicine which kills everything with kidneys, unless they have a liver (eg., which filters it). Now there is no correlation at all with an effect on the kidneys, nor will there ever be (since all animals with one have the other) unless someone deliberately impairs a liver.
It also necessarily increases the incidence of other causes of death, as those who won't die of Alzheimer's will die of something else instead.
This is a bit extreme. The tongue-in-cheek variant I like (which I first read about in the book referenced by TFA) is "no correlation without causation". In order for two things to truly co-vary (and not just by accident, or as a consequence of poor data collection/manipulation), there needs to be some causal connection between the two – although it can be quite distant.
Umm.. No, there doesn't... This idea features in the earlist 20th C. writings on statistics, but it's pseudoscience.
If one carves up the whole history of the entire universe into all possible events, then there's likely to be a (near) infinite number of pairs of events which "perfectly" co-vary without any causal connection whatsoever. Indeed, one could find two galaxies that are necessarily causally isolated and find correlated events.
This is, in part, because the properties of two casually independent systems can have indistinguishable distributions -- just by the nature of what a distribution is.
It's this sort of thinking that I was aiming to rule out: really they have nothing to do with each other. It's early 20th C. frequentist pseudoscience that has given birth to this supposed connection, and it should be thrown out all together.
Causation is a property of natural systems. Correlation is a property of two distributions. These have nothing to do with each other. If you want to "test" for causation, you need to have a causal theory and a causal analysis in which "correlation" shouldn't feature. If you induce correlation by causal intervention, i'd prefer we gave that a different name ("induced correlation") which is relevant to causation -- and it's mostly this confusion which those early eugenticists that created statistics were talking about.
But if they are not linked by a stable causal connection, wouldn't they eventually diverge, if we observe long enough?
I think it is better to think about this through mutual information rather than "correlation"[0], adding DAGs (directed acrylic graphs) also helps but are hard to draw here.
If causation exists between A and B, the two must also have mutual information. This is more akin to the vernacular form of "correlation" which is how I believe you are using it. But statisticians are annoying and restrict "correlation" to be linear. In that case, no, causation does not necessitate nor imply linear correlation (/association).
For mjburgess's universe example, I think it may depend on a matter of interpretation as to what is being considered causal here. A trivial rejection is that causation is through physics (they both follow the same physics) so that's probably not what was meant. I also don't really like the example because there's a lot of potential complexity that can lead to confusion[1], but let's think about the DAG. Certainly tracing causality back both galaxies converge to a single node (at worst, the Big Bang), right? They all follow physics. So both have mutual information to that node. *BUT* this does not mean that there is an arrow pointing from one branch to the other branch. Meaning that they do not influence one another and are thus not causally related (despite having shared causal "history", if you will).
Maybe let's think of a different bland example. Suppose we have a function f(x) which outputs a truly random discrete outputs that are either 0 or 1 (no bias). Now we consider all possible inputs. Does there exist an f(a) = f(b) where a ≠ b? I think with this example we can see believe this is true but you can prove it if you wish. We can even believe that there is a stronger condition of a having no mutual information between a and b. In the same way here, if we tracked the "origin" of f(a) and f(b) we would have to come through f (f "causes" f(a) and f(b)), but a and b do not need to be constructed in any way that relates to one another. We can even complexity this example further by considering a different arbitrary function g which has a discrete output of [-1,0,1], or some other arbitrary (even same) output, and follow the same process. When doing that, we see no "choke point" and we could even pull a and b from two unrelated sets. So everything is entirely disjoint. Try other variations to add more clarity.
[0] I also corrected mjburgess through this too because a subtle misunderstanding led to a stronger statement which was erroneous https://news.ycombinator.com/item?id=41228512
[1] Not only the physics part but we now have to also consider light cones and what physicists mean by causation
I'm not sure why you would think so. All that's required is that the process they are following to generate observables is deterministic or law-like random .
Consider a possible universe where everything is deterministic, and at t=0 N=infinity objects created each with some very large number of measurable properties. Some never change, so property p=1,1,1,1,1,1,1, etc. forever. Some change periodicially, p=1,0,1,0,1... etc.
Now I dont really see why there wouldn't be an infinite number of correlated such properties of objects with no casual relationship whatsoever.
Maybe you want to claim that the actual universe is chaotic over long time horizons, with finite objects, finite properties, etc. and as t->inf the probability of finding properties which "repeat together" goes to zero. ... like, Maybe, but that's a radical claim.
I'd say its much more likely that, eg., some electron orbiting some atom somewhere vs. some molecule spinning, etc. will always be correlated. Just because there's so many ways of measuring stuff, and so much stuff, that some measures will by chance always correlate. Maybe, maybe not.
The point is that the world does not conspire to correlate our measures when causation is taking place. We can observe any sort of correlation (including 0) over any sort of time horzion and still there be no causation.
In practice, this is very common. It's quite common to find some measurable aspects of some systems, over horizons we measure them, to "come together in a pattern" and yet have nothing to do with each other. I regard this as the default, rather than vice versa. At least every scientist should regard it as the default.. and yet, much pseudoscience is based on a null hypothesis of no pattern at all.
> correlation and causation are unrelated
You have extended the meaning of the phrase "correlation does not imply causation" to a stronger case[0]. The correct way to say this is that "correlation are not necessarily related."
The other way you might determine this was wrong is that ,,association''[2] always occurs when there is causation. So we have the classic A ⇒ B ⇏ B ⇒ A (A implies B does not imply B implies A), where ordering matters.
Last, we should reference Judea Pearl's Ladder of Causality[1].
[0] Another similar example was given to us by Rumsfield with respect to the Iraq WMD search. Where the error was changing "the absence of proof is not proof of absence" to the much stronger "the absence of evidence is not evidence of absence". It also illustrates why we might want to "nitpick" here https://archive.is/20140823194745/http://logbase2.blogspot.c...
[1] https://web.cs.ucla.edu/~kaoru/3-layer-causal-hierarchy.pdf
[2] Edit for clarity: The reason I (and Pearl) use the word "association" rather than "correlation" is because in statistics "correlation" often refers to linear relationship. So association clarifies that there is mutual information. There might be masked relationships, so non-linear. But if we are to use the standard vernacular of "correlation" (what most people think) then we could correctly say "causation implies correlation" (or more accurately, "causation implies correlation, but not necessarily linear correlation."). And of course, causation implies high mutual information, but high mutual information does not imply causation :) https://stats.stackexchange.com/questions/26300/does-causati...
This is incorrect. See Perl's work itself. Association does not occur when there is a collider. https://en.wikipedia.org/wiki/Collider_(statistics)
Since almost all variables we are measuring are on uncontrolled environments, in almost all cases, there is an opportunity to observe no association with causation.
I give an example of this above:
> Consider another medicine which kills everything with kidneys, unless they have a liver (eg., which filters it). Now there is no correlation at all with an effect on the kidneys, nor will there ever be (since all animals with one have the other) unless someone deliberately impairs a liver.
Mutual information between random variables is zero iff the two random variables are independent.
In your example, you illustrate that the MI is non-zero. Sure, it is clear that it may appear zero during sampling, but that's a different story. I fully agree that there is an opportunity to observe no association. That is unambiguously accurate. But in this scenario you presumably haven't sampled animals with damaged livers. But you can also have bad luck or improper sampling even when the likelihood of sampling is much higher! That doesn't mean that there is no association, that means there's no measured (or observed) association. The difference matters, black swans or not. Especially being experimentalists/analysts, it is critical we remember how our data and experimentation is a proxy, and of what. That they too are models. These things are fucking hard, but it's also okay if we make mistakes and I'd say the experiments are still useful even if they never capture that relationship.
If we strengthen your example to the medicine always being (perfectly) filtered out by a liver (even an impaired one) and all animals must have livers, then it does not make your case either. We will be able to prune that from the DAG. The reason being that it does not describe a random variable... (lack of distribution). I think you're right to say that there is still a causal effect, but what's really needed is to extend the distribution we are sampling from to non-animals or at least complete ones. But the point here would be that our models (experiments) are not always sufficient to capture association, not that the association does not exist.
Maybe you are talking from a more philosophical perspective? (I suspect) If we're going down that route, I think it is worth actually opening the can of worms: that there are many causal diagrams that can adequately and/or equally explain data. I don't think we should shy away from this fact (nor the model, which is a subset of this), especially if we're aiming for accuracy. I rather think what we need to do is embrace the chaos and fuzziness of it all. To remember that it is not about obtaining answers, but finding out how to be less wrong. You can defuzz, but you can't remove all fuzz. We need to remember the unfortunate truth of science, that there is an imbalance in the effort of proofs. That proving something is true is extremely difficult if not impossible, but that it is far easier to prove something is not true (a single counter example!). But this does not mean we can't build evidence that is sufficient to fill the gaps (why I referenced [0]) and operate as if it is truth.
I gripe because the details matter. Not to discourage or say it is worthless, but so we remember what rocks are left unturned. Eventually we will have to come back, so its far better to keep that record. I'm a firm believer in allowing for heavy criticism without rejection/dismissal, as it is required to be consistent with the aforementioned. If perfection cannot exist, it is also wrong to reject for lack of perfection.
If you mean to say that there are, say, an infinite number of DAGs that adequately explain reality -- and in the simplest, for this liver-kideny case, we don't see association ---- but in the "True DAG" we do.. then maybe.
But my point is, at least, that we dont have access to this True model. In the context of data analysis, of computing association of any kind, the value we get -- for any reasonable choice of formulae -- is consistent with cause or no cause.
Performing analysis as-if you have the true model, and as-if the null rival is just randomness, is pseudoscience in my view. Though, more often, it's called frequentism.
> what you mean by association here then.
> but in the "True DAG"
> But my point is, at least, that we dont have access to this True model.
Why I was pushing against your example is because I think it is important to distinguish lack of association because the data to form the association is missing or unavailable to us (which may be impossibly unavailable; and if we go deep enough, we will always hit this point) vs a lack of association because the two things are actually independent[0]. One can be found via better sampling where the other will never be found (unfortunately indistinguishable from impossibly unavailable information).
> as-if you have the true model
[0] I'll concede the to an argument of "at some point" everything is associated tracing back in time. Though I'm not entirely convinced of this argument because meta information.
If you take the example of Treatment A vs Treatment B for tumors, you can get infinite layers of seemingly contradicting statemens: - Overall, Treatment A has better average results - But if you add tumor size, Treatment B is always better - But if you add gender to size, Treatment B is always better - But if you add age category to gender and size, Treatment A is always better - etc...
It totally contradicts our instincts, and shows statistics can be profoundly misleading (intentionally or not).
I actually coded a Z3 program to prove it! The 3-variables version takes too long to resolve, but I got results for the 2-variables version (tumor size + gender):
Results can be found in this GSheet: https://docs.google.com/spreadsheets/d/1tsBhElTgXjVTeas8quar...
Code is here: https://gist.github.com/TheMrZZ/c33927ca2cc917997a67d7f84b82...
I'm currently running the 3-variables version, hopefully I'll get results this afternoon.
We can clearly see the same problems that arise in the 1-variable Simpson's paradox (widely different population sizes).
Besides Ellenberg’s mind-altering discussion of that link[1], see hints on the second page of:
https://www.qeios.com/read/XB1N2A/pdf
[1] "[the point of Simpson’s paradox] isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once."
Ellenberg, from Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else (2021)
Results can be found in this GSheet: https://docs.google.com/spreadsheets/d/1tsBhElTgXjVTeas8quar...
Code is here: https://gist.github.com/TheMrZZ/c33927ca2cc917997a67d7f84b82...
I'm currently running the 3-variables version, hopefully I'll get results this afternoon.
We can clearly see the same problems that arise in the 1-variable Simpson's paradox (widely different population sizes).
[1]: https://robertheaton.com/2019/02/24/making-peace-with-simpso...
"Mathematician Jordan Ellenberg argues that Simpson's paradox is misnamed as 'there's no contradiction involved, just two different ways to think about the same data' and suggests that its lesson 'isn't really to tell us which viewpoint to take but to insist that we keep both the parts and the whole in mind at once.'"
My own take is that any statistic has a value and a strength (in the case of averages, strength can be the number of instances averaged, for instance). You can have to keep in mind both.
When the analysis is intended to inform a decision it’s important to take the appropriate viewpoint.
https://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf
https://www.unige.ch/cisa/files/9616/7501/5107/CISA_BM_stats...
Proper multifactor analysis that accounts for all variations simultaneously is required to learn about complex phenomena.
If you get paranoid about its presence it can lead you to second guess pretty much every statistic. "I know that 4 out of 5 dentists recommend chewing X Brand gum but what if I slice the dentists by number of eyes? Maybe both one-eyed dentists and two-eyed dentists aren't so enthusiastic."
Simpson's paradox is part of a broader problem: correlation does not imply causation. In practice, it's one of many problems with making decisions based on correlations.
In a randomized control trial, with a large random sample, the odds of Simpson's Paradox coming up are astronomically low.
Good statisticians WILL second-guess ANY conclusions based on post-hoc data analysis. To frame this in scientific jargon, exploratory data analysis and correlations are great at generating hypotheses, but those need to be confirmed with methods appropriate for confirmatory analysis.
Tools like this make me feel a lot better about all the time I wasted playing with predicate logic.
Would like to know how you us Z3 to evaluate emotional happy Q. Could you apply that to evaluate a piece of content (like thread, feed, comment) and then evaluate the energy of the thing for its happiness quotient value whatever? then you can just Z3 the thread and determine the psychological predicted impact...
Maybe you could then design casula games that provide the positive happy Q vibes.
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I was in a program from when I was a baby with UCSD that was a life tracking project and they would check in with you every so often to see where you were on that trajectory - and where you were happy in life etc.
Problem is that it was also tied to Morehouse University, MK, Vic Baranco, and a bunch of other Stanford thingy's from the 70s that we all know abou these days.
I'm working on something that is trying to utilize Z3 for some financial market stuff that I hack on in my free time.
In any case, I would explain the paradox differently than the author. The author says: "The key to understanding the paradox is that the players did not bat against the same set of pitchers. A batted against 5 lefties and 12 righties; B against 2 and 11."
I would say instead that the key to understanding the paradox is to observe that both players are much better when batting against lefties and that player A batted against lefties much more often, both in absolute and relative terms. In other words, A is not as good against lefties as B but he faced a lot more of these comparatively easy pitchers.
This seems to be a pretty fundamental question but I have never seen it addressed.
For the college admissions example, the more important factor was what department you applied to. That had far more of a meaningful contribution to whether you were admitted than what your sex was. So, if folks wanted to increase admissions, you wouldn't focus on sex, you would focus on expanding departments.
That is the general trend with all of the examples. The paradox is you think focusing on X would be the important thing to focus on, but the data had hidden that there was a Z that you should instead focus on.
But let's say the effect is real. Then you have to start considering what is causing why, which is highly context dependant.
To take the example of gender bias in the grad school admissions, men were more likely to be admitted but the effect reversed when breaking down by department.
Hypotheses come to mind:
A department being easy to get into causes a higher male ratio. (Subhypotheses: Maybe men want to take it easy? Maybe women seek out prestige?)
High male ratio of applicants to a department causes it to become easier to get into. (Maybe funding flows to male departments?)
There is some kind of unknown factor that both causes a high male ratio and a high rate of admissions (Maybe a booming sector attracts men and leads to easy admissions?)
With all these possibilities I think it should become clear that there can't be a general solution to your question, you have to consider the context and dig deeper.
Being male (X) causes you psychologically to prefer STEM departments, which are more economically useful, which means they are better funded, which means they are larger, which means they can admit more people, which means you are more likely to be admitted (Y). So X is positively correlated with Y.
However, being male (X) also causes you to be less likely to be admitted in each individual department (parts of Y), e.g. because people in admissions offices have some degree of anti-male bias, known in psychology as women-are-wonderful effect [1]. So X is negatively correlated with the parts of Y (being admitted to individual departments).
So in this case, X does cause both Y, via the male psychological preference for STEM, and not-y_n (for {y_1, y_2, ..., y_n} = Y, for an n-partition of Y), via anti-male admissions bias like women-are-wonderful. So the previously most plausible initial explanation for why X causes Y (university being biased again women) is indeed false.
I guess the lesson is that detecting a case of Simpson's paradox (when partitions of a variable like Y have different direction of correlation with some other variable X than the entire variable Y) can point to causal explanations being different than what they would naively seem to be.
[1] https://en.wikipedia.org/wiki/Women-are-wonderful_effect
This is misleading. I am baseball ignorant but I feel this is a contrived and bad example for Simpson's paradox.
(UC Berkeley gender bias example is much better)
The key to understanding the paradox is that women apply mostly to popular departments therefore women are declined more often than man who apply more to unpopular departments.
Don't know anything about baseball, I would have assumed teammates battle the same pitchers.
A falsidical paradox is what most people think of as a formal paradox: from the assertions you derive a conclusion which is false (either deductively or inductively).
A veridical paradox is where you derive a true result which appears absurd. it is an informal classification based on intuition. https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox is a classic veridical paradox.