When seeing X suggests ‘generally ¬X’

Suppose nobody has ever told you that they like you. Suppose you are relatively uncertain about how often people like other people, and also about how often they will disclose it when they do. Suppose you are confident that these facts about your ignorance and social inexperience do not bear on whether other people like you. So as it stands you are fairly uncertain about your popularity. Suppose also that you have a deep and insatiable need for people to like you, and your pleasure is roughly linear in the number of people who like you.

Suppose one day a person tells you that they like you. If you are given to expressing emotions or making inferences, one thing you might wonder is whether this should be cause for happiness.

This is not as obvious as it first seems. A person telling you that they like you is more probable if:

  1. This specific person likes you.
  2. People like you in general
  3. People are given to expressing their liking for other people

The first two are promising. The third makes the fact that nobody else has ever said they like you a bit more damning. Just how much more damning depends on your probability distribution over different possible states of affairs. For an extreme example, suppose you had even odds on two extreme cases – people always saying they like people who they like, and people never doing so – and that many people have had a chance by now to tell you if they like you. Then you should be extremely sad if anyone tells you that they like you. The apparent update in favor of people liking you in general will be completely overwhelmed by the reverse update from flatly ruling out the possibility that all those people you have already met like you.

In general, seeing an instance of X can make X less likely, by indicating that X tends to be visible:

  • Hearing your neighbors have loud sex might lower your estimate of how often they have sex.
  • Finding a maggot in your dinner might reassure you that maggots in dinners are relatively visible (this is just a hypothetical example – in fact they are not, especially if your dinner is rice)

Conversely, failing to see X can make X more likely, by increasing the probability that it is invisible:

  • If you have never observed a person lying, it might be more likely that they are an excellent and prolific liar than it would be if you had seen them lie awkwardly once. Though not once all the excellent liars realize this and stumble sheepishly over a white lie once in a while.
  • Failing to observe phone calls  from friends for too long will often cause you to suspect they have in fact been calling you, and there is rather something wrong with your phone.
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  • http://www.facebook.com/tedsanders Ted Sanders

    Good post.

  • Paul Crowley

    Seeing an event in the news tells you that it’s rare that this happens.

    • someone

      A very important point!

      Now here’s a case study relating to the converse, where I’m genuinely unsure: 

      I’ve been told that one should avoid taking showers during a thunderstorm. But since I’ve never heard stories on the news of anyone being injured as a result of doing so, up to now I’ve been assuming the behavior isn’t as risky as the advice seems to imply. However. this post raises the possibility that this could simply be due to such injuries being too common to be newsworthy (like other types of household accidents). Now, I don’t personally know anyone who has been injured or killed as a result of showering during a thunderstorm; but neither do I know anyone who has been injured or killed by using a hair dryer in the bathtub (and I haven’t heard that on the news either) — and yet that behavior *is* obviously very dangerous. So what should I conclude?

      • lemmycaution
      • Elithrion

        More specifically, ”Ron Holle, a former meteorologist with the National Oceanic and Atmospheric Administration who tracks lightning injuries, estimates that 10 to 20 people in the United States are shocked annually while bathing, using faucets or handling appliances during storms.”

        So, seems like that’s bathing combined with everything else, and the result is usually non-lethal. The conspicuous omission of showers/specification of “bathing” suggests that that’s also safe (or at least much safer).

      • Nancy Lebovitz

        The other possibility is that people generally don’t take showers during thunderstorms, and it’s a precaution that works.

      • lemmycaution

        showers are probably safer, but it can happen

         http://www.snopes.com/science/lightningbath.asp

  • notlars

    Made me think of this old article: http://lesswrong.com/lw/ih/absence_of_evidence_is_evidence_of_absence/

    I struggled initially to reconcile the two, as they appear to be in contradiction. They aren’t, of course: the presence of the evidence still increases the estimated frequency of the underlying events, but the baseline changes.

    To borrow the above example, if one somehow found out that people tell other people that they like them exogenously, then someone telling you they like you would increase the probability that you’re well-liked (relative to a world where no one told you they liked you). However, if you find out that people tell other people that they like them only when you’re told that someone likes you, these two steps are conflated. You have to lower your estimate that you’re well-liked (because no one told you they like you in a world where this is what people do), and then raise it again to account for the fact that someone told you they liked you.

  • IMASBA

    “In general, seeing an instance of X can make X less likely, by indicating that X tends to be visible:”You can decide that this observation of X means X is often visible, meaning the lack of past observations indicates X is rare, as opposed. Or you can decide that the lack of past observations means X is rarely visible and this current observation is pure chance, so X is probably common. Which way you’re gonna go depends on your prior (how common did you believe X to be previously), so using this to update your belief about the occurence of X is circular reasoning. All you learn is that X is at least sometimes visible. The “maggot in your food” example is an exception since you know maggots are all about the same size and look alike, but in the “loud neighbor sex” example you don’t learn much since sex can be quiet or loud at different times and you don’t know the relative occurences: your prior will determine your conclusion.

    • Chrysophylax

      You could decide that “this observation of X means X is often visible, meaning the lack of past
      observations indicates X is rare, as opposed. Or you can decide that
      the lack of past observations means X is rarely visible and this current
      observation is pure chance, so X is probably common”. You could do it, but you’d be wrong. An observation can only ever be evidence in one direction (because probability mass is conserved). Your prior won’t distinguish between the two cases – if you observe X more often than your prior predicts, you have to increase both P(X) and P(X observed given X). Your posterior has to assign more probability to your observations than your prior did – you can’t adjust one component up and one down, leaving P(X observed) the same as before, because you have to update on the new evidence.

      • IMASBA

        “An observation can only ever be evidence in one direction”

        This is true, but which way the evidence points does depend on what your prior was. Your prior will determine if you observe more or less instances of X than you expect (your prior predicts).

        In the example of “loud neighbor sex” and you initially believing they have sex often, observing loud neghbor sex occasionally (not often) leads you to conclude they don’t have sex often, IF your prior belief was that sex is usually loud. If your prior belief was that sex is usually quiet you won’t be surprised if you only hear it occasionally, so your prior about the average loudness of sex does determine whether or not you end up updating your belief about the frequency of their sex life. Here observing X means X is rare only if you have good reasons to have a prior that says X is usually observable. Logical considerations (like Occam’s razor assigning a low likelyhood to invisible dragons in garages) might often give you such a good reason, but that’s certainly not always the case.

  • Matt

    In academic publishing, this is one reason why null results do not get published – because doing so is a strong signal that the nullified theory is generally believed to be true, which is what makes the null result is interesting.  Unless the paper is particularly convincing, there is a danger that readers will walk away believiing more strongly in the theory just because of the fact that the null result was published.

    • churchofrationality

      “In academic publishing, this is one reason why null results do not get published”

      I’d like to see supporting evidence for that statement.

  • anonymous

    Thanks a lot, now I’ll be scared to eat rice for weeks.

  • Dave

    Don’t worry, we do like you, Katja.

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  • http://beeminder.com Daniel Reeves

    Related: “the exception that proves the rule” http://en.wikipedia.org/wiki/Exception_that_proves_the_rule

    Which I guess is more like “observing not-X suggests ‘generally X’”.

  • stevesailer

    It’s “the exception that proves [i.e., supports] the rule [i.e., tendency]” — the existence of something that is famous for being exceptional suggests that exceptions are rare. For example, Beethoven is hugely famous for being a deaf composer, which suggests most composers aren’t deaf.

    Nobody, however, is famous for being a blind painter, so that leaves us with two alternatives: blind painters are common are blind painters are implausible.

  • srdiamond

    This could be viewed as an induction paradox: a challenge to a simplistic theory of induction. Induction depends on categorization. The grue paradox drives a similar moral.

    Other than that, I’m not sure why this is interesting, in the absence of examples where anyone has assumed otherwise.

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