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The “either/or” fallacy of believing in discrete models: an example of folk statistics

Psychologists talk about “folk psychology”: ideas that make sense to us about how people think and behave, even if these ideas are not accurate descriptions of reality. And physicists talk about “folk physics” (for example, the idea that a thrown ball falls in a straight line and then suddenly drops, rather than following an approximate parabola).

There’s also “folk statistics.” Some of the ideas of folk statistics are so strong that even educated people–even well-known researchers–can make these mistakes.

One of the ideas of folk statistics that bothers me a lot is what might be called the “either/or fallacy”: the idea that if there are two possible stories, the truth has to be one or the other.

I have often encountered the either/or fallacy in Bayesian statistics, for example the vast literature on “model selection” or “variable selection” or “model averaging” in which it is assumed that one of some pre-specified discrete set of models is the truth, and that this true model can be determined from the data. Or, more generally, that the goal is to estimate the posterior probability of each of these models. As discussed in chapter 6 of BDA, in the application areas I’ve worked on, such discrete formulations don’t make sense to me. Rather than saying that model A or model B might be true, I’d rather say they can both be true. Which is not the same as assigning, say, .3 probability to model A and .7 probability to model B; rather, I’m talking about a continuous model expansion that would include A and B as special cases. That said, any model I fit will have its limitations, so I recognize that discrete model averaging might be useful in practice. But I don’t have to like it.

Since I’ve been primed to see it, I notice the either/or fallacy all over the place. For example, as I discuss here, cognitive scientist Steven Sloman writes:

A good politician will know who is motivated by greed and who is motivated by larger principles in order to discern how to solicit each one’s vote when it is needed.

I can well believe that people think in this way but I don’t buy it! Just about everyone is motivated by greed and by larger principles! This sort of discrete thinking doesn’t seem to me to be at all realistic about how people behave–although it might very well be a good model about how people characterize others!

Later in his book on causal reasoning, Sloman writes:

No matter how many times A and B occur together, mere co-occurrence cannot reveal whether A causes B, or B causes A, or something else causes both. [italics added]

Again, I am bothered by this sort of discrete thinking. I’m not trying to pick on Sloman here; I’m just demonstrating how the either/or fallacy is so entrenched in our ideas of folk statistics that it comes out in all sorts of settings.

Most recently, I noticed the fallacy in the humble precincts of our blog, when, in response to Phil’s remark that having lots of kids puts a strain on the environment, commenter A. Zarkov wrote,

Believe or not, some people really like children and want a lot of them. They think of each child as a blessing, not a strain on the bio-sphere.

That’s the either/or fallacy again! As I see it, each child is a blessing and a strain on the biosphere. There’s no reason to think it’s just one or the other.

I’ll stop now. I think you get the point.

17 Comments

  1. murali says:

    Perhaps possible reasons are a) representational complexity – these models are hard to use b) insufficient data for a continuous model expansion – these models cannot be learned efficiently ?

  2. Do you mean more than "or" as a logical connective can be true when both the truth value of both sentences are true, but people tend to see or use "xor" as the logical connective?

  3. The Science Pundit says:

    The either/or fallacy is at the heart of the centuries old nature vs. nurture debate. And sure enough, people are still having that debate and commiting the same old fallacy. And you're also right that when I open my eyes to it, I can see the fallacy everywhere.

  4. David says:

    The best folk statistic I know is that approximately 42.37% of all statistics are made up on the spot.

  5. Ben Bolker says:

    @murali: I think Prof. Gelman would agree that there might be practical reasons to adopt an "either/or" approach to the actual technical bits of modeling (above: "I recognize that discrete model averaging might be useful in practice …") The problem comes when you start taking the approach, adopted for pragmatic reasons, too seriously ("… what is the probability of identifying the true model? …") [I get nervous whenever the word "true" comes up in a discussion of statistical methods ...]

  6. Paul says:

    The either/or problem makes a lot more sense when you focus on outcome rather than internal models. Yes, babies should be viewed as some mixture of blessing and biological burden, but ultimately that view translates into a "have another baby/don't" decision, and that can be characterized as one of the views overriding the other. Similarly, I'm sure if asked Sloman would concede nobody is 100% greedy or 100% principled, but the hypothetical politician is going to have to make one pitch or another: kitchen sink arguments where you appeal to every nuance of a person aren't necessarily more effective than picking a single powerful motivator and going with it.

    I agree its sloppy to let discrete outcomes color your discussion of the motivation in a discrete way, but its a useful shorthand when you're more interested in acting on the actual outcomes rather than analyzing the psychology behind the decisions.

  7. K? O'Rourke says:

    May I discretely inquire – did you or did you not – find the Dempster-Shafer stuff (A,notA,DontKnow) helpful in this regard?

    And for "one of some pre-specified discrete set of models is the truth"
    would you find "one of some pre-specified discrete set of models is least wrong (in some aspect, ties allowed)" less objectionable.

    K?
    p.s. I do like the Darwin quote "Discreteness exists only in the minds of men" but also believe it can be advantageous to entertain discrete models (which we _know_ are always wrong)
    p.s.2. I am certainly not expecting Andrew to feel obliged to answer these questions

  8. Manoel Galdino says:

    I liked what you said. It makes sense to me. However, don't you think that the discussion about identificating causal effects is related to this fallacy?

    I mean, when I device a strategy to exclude endogeneity, I am not only trying to asses if A causes B, but also thinking discretely: or A causes B, or B causes A.

    I don't know if I made myself clear, but I was wondering if all that stuff about identification (instrumental variables, regression discontinuity design etc.) isn't realted to this fallacy you talked about.

  9. Bill Jefferys says:

    When I've used model averaging, it's always been in the context of models that I know to be approximations anyway (e.g., truncated Fourier polynomials describing a complex periodic phenomenon), so I never considered any of them to be "true," nor the individual probabilities that pop out for each model to be "posterior probabilities that the model is true." Rather, I considered the averaged model to be, as I gather Andrew does, to be a kind of interpolation between the models that gives (hopefully) a more satisfactory result than I would get by adopting any particular model in the set under consideration.

  10. Morgan says:

    Can you explain your objection to "no matter how many times A and B occur together, mere co-occurrence cannot reveal whether A causes B, or B causes A, or something else causes both"? I can't fit it into the either/or fallacy.

    It seems uncontroversial to me that "mere co-occurrence" is insufficient to establish the cause of the co-occurrence. Are we interpreting the statement differently, or am I just wrong?

    ;-)

  11. noahpoah says:

    Isn't it possible that the thoughts behind the quotes are not as discrete as the statements sound? For example, the claim about politicians seems to me to allow for a more long-winded interpretation along the lines of "A good politician will know who tends to be more motivated by greed than by larger principles and who tends to be more motivated by larger principles than by greed in order to discern how to solicit each one's vote when it is needed."

    Similar rephrasing allows for a more nuanced take on biosphere straining children, though the bit about causality seems hairier.

  12. wei says:

    So how do you vote every four years?

  13. A. Zarkov says:

    Andrew writes,

    That's the either/or fallacy again! As I see it, each child is a blessing and a strain on the biosphere. There's no reason to think it's just one or the other.

    I did not write that children don't stain the biosphere. I wrote,

    … some people really like children and want a lot of them. They think of each child as a blessing, not a strain on the bio-sphere.

    My comment refers to the attitudes and beliefs that some people hold with regard to having children. They want children regardless of any effect on the biosphere. I did not assert children have no effect

    Let's not forget the larger context in that thread. Phil wrote that he once thought that Catholics have large families to increase their numbers and thus the influence of the Church. I offered some counter examples as to why that might not be true. I don't see how the "either or fallacy" applies here. Moreover the whole concept of "strain on the biosphere" is somewhat vague. It could mean the disruption of food webs which would be a problem for humans, not the biosphere.

  14. Andrew Gelman says:

    Wei:

    See Paul's comment above. Decisions can be discrete even if inferences are continuous.

    Noah:

    Maybe. But I'm inclined to believe that sloppy writing reflects and supports sloppy thinking. For example, in that politician quote, I'd prefer, instead of trying to think about which people are more motivated by greed and which by larger principles, to think about how the continuous mix varies.

    That example also illustrates another problem with either/or thinking, which is it locks out alternatives. "Greed" and "larger principles" are far from the only motivations out there!

  15. Andrew Gelman says:

    Morgan:

    I object to this part: "A causes B, or B causes A, or something else causes both."

    If A could cause B, and B could cause A, then you could also have A causing B and B causing A. And also something else causing both.

  16. Zach says:

    I think that you have to look at Sloman's work a bit more broadly. His experiments are often designed so that these are the only options within the experimental context. It's extremely unrealistic from a real world perspective (obviously, you can never fully describe a causal model), but very useful in terms of trying to figure out how people think about causality.

  17. Andrew Gelman says:

    Zach:

    I reviewed Sloman's book here for the American Journal of Sociology. My quick answer is that I can well believe his research is excellent even if he is working within what I view as a flawed conception of statistics. I'm also fully willing to believe that the either/or attitude is fundamental in human reasoning, just not that it is such a good way to understand the world. Hence my reference to "folk statistics."