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Believing the unlikely

We often want to know how likely something is. I might want to know how likely it is that it will rain tomorrow morning; according to the Met Office website, at time of writing, this is 90% likely. If I’m playing poker, I might want to know how likely it is that my opponent has been dealt a “high card” hand. With no other information available, we can calculate the probability of this to be about 50%. There seems to be close link between likelihood and belief; if something is likely, you would be justified in believing it, and if something is unlikely, you would not be justified in believing it. That might seem obvious, and the second part might seem especially obvious – surely you can’t be justified in believing something that’s unlikely to be true? I will suggest here that, sometimes, you can.

Some philosophers and psychologists have argued that it can sometimes be a good thing for people to hold irrational beliefs. That may be right, but it’s not the point I want to make. I think that believing the unlikely can not only be a good thing, but can be fully rational. The reason has to do with testimony, and when we owe it to other people to believe what they say. So much of what we believe about the world is based on the testimony of others. And, while we don’t always accept what others tell us, disbelieving a person’s testimony is not something to be taken lightly. If someone tells me, say, that there’s a coffee shop nearby and, for no reason at all, I refuse to believe it, then this is not just about me – another person is involved, and I’m doing that person a wrong. If I refuse to believe what someone tells me, then I need to have a good reason for doing this.

What does this have to do with believing the unlikely? That’s best illustrated with an example: Suppose I rush back to my car one evening, after my parking has expired, expecting a ticket. When I arrive, however, I discover that some kind stranger has paid for additional parking. When I ask in a nearby shop, the shopkeeper tells me that he saw someone feeding the meter, and the person was wearing glasses. Naturally, I believe what the shopkeeper says, and it’s entirely rational for me to believe this.

Odd One Out by rawdonfox. CC BY 2.0 via Flickr.
Odd One Out by rawdonfox. CC BY 2.0 via Flickr.

Suppose I then remember reading that contact lenses are really common in my town and only 1% of the population are glasses-wearers. This will lower the probability that the kind stranger wore glasses, but I wouldn’t stop believing this. I’d think “That’s good – the shopkeeper’s information could help me find this person” not “That’s bad – I shouldn’t believe the shopkeeper anymore.” If the shopkeeper’s testimony were false, there would have to be some explanation for this; if I discovered, for instance, that he suffers hallucinations, or has a memory problem, then that could provide an explanation and could make it legitimate to doubt the testimony. But I haven’t discovered anything like that. Furthermore, if the shopkeeper was telling me something far-fetched – that he’d seen aliens or could tell the future say – then, again, I might legitimately doubt the testimony. But there’s nothing far-fetched about what the shopkeeper is saying. Think about it this way: Suppose there are 100,000 people in the town, 1000 of whom wear glasses. One of these people is the kind stranger, and it might just as well be one of the glasses-wearers as one of the others. There’s nothing “hard to believe” about this.

Evidence about the proportion of glasses-wearers doesn’t affect the credibility of the testimony, but it will affect the probabilities – it’s what psychologists and statisticians would call “base rate” information. Let’s add some numbers: Suppose that, if the person wasn’t wearing glasses, there is a 1 in 50 chance that the shopkeeper would have mistakenly said s/he was, (because of a hallucination, false memory etc.). And suppose that, if the person was wearing glasses, then the shopkeeper would be sure to say that s/he was. We now have all we need to calculate the final probability that the person wore glasses – surprisingly, it comes to only about 33.5%. (For those who are interested, the probability can be calculated using Bayes’ Theorem and the Theorem of Total Probability.) It turns out I am believing the unlikely.

If I cared only about probabilities then I wouldn’t believe what the shopkeeper says – I might believe the exact opposite. But I don’t care only about probabilities. It’s still true that there would have to be some explanation if the shopkeeper’s testimony were false and, having no inkling of any such explanation, I owe it to him to accept it. The base rate information does not change this. Try to see it from the shopkeeper’s perspective – how would you feel if you had volunteered your testimony, only to have it dismissed because a low proportion of people wear glasses?

It’s well known that people tend to neglect base rates when estimating probabilities – something known as the base rate fallacy. In the example I’ve described, many people would downplay the base rate, and end up overestimating the probability that the person wore glasses. Maybe this example is just an illustration of the fallacy: We overestimate the likelihood that the person wore glasses and, because of the link between likelihood and belief, we (mistakenly) think that one should believe this. The base rate fallacy is a large topic and I won’t try to discuss it here but, suffice it to say, we could turn this kind of story on its head: Maybe one really should believe that the person wore glasses (as there’s credible testimony to that effect), and because we (mistakenly) think that there is a link between likelihood and belief, that partly leads us to overestimate the likelihood of this.

We often want to know how likely things are. We also want to know what we should believe. If I’m right, then answering the first question does not yet give us an answer to the second.

Featured image credit: Dice by Jacqui Brown. CC BY-SA 2.0 via Flickr.

Recent Comments

  1. Sylvia

    Dear Martin,

    I enjoyed thinking about this example, but I do not agree with the conclusion. I will share my reasoning here, so maybe you can convince me otherwise.

    Laplace wrote in his Essai on probability theory: “Plus un fait est extraordinaire, plus il a besoin d’être appuyé de fortes preuves. Car, ceux qui l’attestent, pouvant ou tromper, ou avoir été trompés, ces deux causes sont d’autant plus probables que la réalité du fait l’est moins en elle-même.”
    As I am sure you know, he wrote this as an informal explanation of what we now call Bayes’ theorem. He follows the remark by an example of an urn and a witness, who may be mistaken (analogous to your example). Laplace generalizes the conclusion of the example, stating that the probability of a mistaken witness reports becomes higher as the event becomes (a priori) less likely.

    In your example:
    – “only 1% of the population are glasses-wearers” and
    – “if the person wasn’t wearing glasses, there is a [2%] chance that the shopkeeper would have mistakenly said s/he was”
    You also state: “It’s still true that there would have to be some explanation if the shopkeeper’s testimony were false and, having no inkling of any such explanation, I owe it to him to accept it.”
    However, in the example, it is more exceptional that a person wears glasses than that a shopkeeper gives a mistaken report, so wouldn’t the alleged observation that the person was wearing glasses be more in need of an explanation? It strikes me as odd that only possible explanations are given for the less unlikely event: “because of a hallucination, false memory etc.” I guess this is the point we disagree on, since you write there is nothing hard to believe about the fact that a glass-wearing person could have been the kind stranger. I agree that in the scenario there is no reason to disbelieve a glass-wearing person could do a random act of kindness, but it is far less likely for a glass-wearing person to be around in the first place. So, there may still be a base rate fallacy in the background. And it is only the part that this person was around that requires extra evidence or explanation (assuming the base rate probability for doing random acts of kindness is constant over the population). Such an explanation could be as simple as s/he lives here, or was visiting/shopping/…

    In the example, I wouldn’t (yet) believe that the person wore glasses, although I fully agree that it would be impolite to call the shopkeeper a liar (and actually, lying isn’t among the mentioned hypotheses to explain a faulty report). It would simply have become much more likely than before talking to him. Now the option that the person was a glass-wearer has become worth considering with more attention, so the report is definitely relevant for informing us about what may have happened. But why should we not follow Laplace and look for further corroborating evidence rather than believing a less likely conclusion over a more likely one?

    The fact that we should be grateful to the shopkeeper for speaking up and the fact that it would be impolite to openly doubt his statement seem to be a different matter than what is rational to believe. Since the shopkeeper’s report makes you raises the probability you assign to the glass-wearer hypothesis from 1% to , it’s not a matter of “refus[ing] to believe what someone tells me” of dismissing it! It’s just a matter of not jumping to conclusions based on a single source.

    Maybe we should also distinguish between two cases: whether or not the shopkeeper knows that there are few glass-wearers around. He may not know the statistics, but may know implicitly if he worked in the neighbourhood for long enough. If he has just moved there, he may not have noticed yet. If he knows that glass-wearers are rare, he may have been surprised and looked up an extra time and/or be extra careful in reporting this. This may decrease the probability of misreporting this, but then the assumed base rates don’t apply to him and the relevant computation would require different numbers.

    A further complication is that the base rate for glass-wearers is prefaced by “I then remember reading that”, which introduces additional probabilities regarding the accuracy of the source and your own memory.

    Best wishes,
    Sylvia

  2. Robert

    Hi Martin

    Nice piece, thank you.

    Does the notion of “testimony” not separate a “specific observation” from the “universe of possible observations”. This “testament” however introduces another universe or variable – the reliability of the witness. For example, if we bring the two together in the same space: in playing a card game I might assign a probability to the hand my opponent has; or receive information from an external observer as to what his hand might be. If the latter is my best friend, I would tend to believe it. If the latter was a stranger I would dismiss the testimony as possibly deliberately misleading. In either case my view on the probability related to the universe of possible card combinations has not changed at all. It hinges solely on my view of the reliability of the witness (which I might want to narrow the possibility on by introducing another witness of the character of the first witness).

    This naturally raises a question (that has both advanced and hindered progress in the past). “What witnesses do I choose to believe (have been conditioned to align my beliefs with)?”

    Robert

  3. Martin Smith

    Dear Sylvia,

    Thanks very much for the comment! Let me say, first of all, that I certainly agree with Laplace’s sentiment that extraordinary facts require weightier evidence. However, I think there is a sense in which the fact that the shopkeeper is testifying to is not ‘extraordinary’ at all. In the example, glasses wearing happens to be rare or infrequent in the relevant population. As such, of all the people who could have fed the meter, only a minority are glasses wearers. But the situation in which it was a glasses wearer who fed the meter doesn’t require any *more explanation* than the situation in which it was a non-glasses wearer. In that sense, these are both equally normal or ordinary events – it’s just that we would expect one to happen less often than the other. This is the same sense in which it isn’t abnormal or extraordinary for me to cut a shuffled deck of cards and find, say, an 8 of diamonds – though this is uncommon, it doesn’t require any more explanation than any of the other cards I could have found.

    In contrast, if a person testifies to P and it turns out that P is false, then there *does* have to be some special explanation for this (illusion, hallucination, false memory, deception etc.). When the shopkeeper tells me that the person wore glasses, the situation in which the person didn’t wear glasses becomes *less normal*, in the sense of demanding more explanation. At this point, you suggest that we should seek out more corroborating evidence. I do agree that it would be a good thing to acquire more relevant evidence if we can. But I also think it can be rational to believe the most normal hypothesis that is compatible with the evidence we have. And, in this case, that’s the hypothesis that the person wore glasses.

    Best

    Martin

  4. Martin Smith

    Hi Robert,

    Thanks for this! The reliability of the shopkeeper is indeed one of the crucial variables here. In the example, I suggested that he has a 2% chance of error – that is, if the person was a non-glasses wearer there is a 2% chance that the shopkeeper would mistakenly say that s/he was a glasses wearer. That makes the shopkeeper pretty highly reliable. (One thing that’s initially surprising about these sorts of cases is that the final probability can still be so low, even though the reliability is high.) If we had reason to think that the shopkeeper was even more reliable – reason to reduce the error probability further – then the final probability that the person wore glasses would increase. But even if the shopkeeper was a trusted friend I think we should be wary of making the error probability much lower than this, given what we know about the various ways that eyewitness testimony can go wrong.

    Martin

  5. Gareth Price

    I am having some of the same difficulties with this as Sylvia.

    You say that there needs to be some special explanation for the shopkeeper’s testimony to be mistaken in this case. Isn’t the explanation that you have chosen to present one of the few scenarios out of many where the shopkeeper is likely to be mistaken – that it is a selection issue?

  6. Jesse M.

    While I’m inclined to agree with the basic point about trust sometimes taking precedence over probability, I think the example with the shopkeeper confuses the issue a little in the way it was phrased. A person has a large number of visible characteristics, some more common and some more unusual, so that in a slightly paradoxical way it may actually be fairly probably that a randomly selected person will have at least one “fairly unusual” (seen in under some suitably small fraction of the population, say) visual characteristic. So, the shopkeeper may have volunteered the information about the glasses, rather than about any of the person’s other characteristics, precisely because it was that person’s most visibly unusual feature. If this is the case, then even if I ignore social trust and use pure Bayesian reasoning about probabilities, I should not treat this as undermining the probability of the shopkeeper’s story in the same way as if I had asked the shopkeeper specifically about the question of glasses (without any reason to think the person would be a glasses-wearer in advance), and only in response to my question did the shopkeeper tell me that the person did wear glasses.

  7. Martin Smith

    Hi Gareth,

    When the shopkeeper tells me that the person wore glasses, the situation in which the person didn’t wear glasses is made less normal, in the sense of demanding explanation of some kind – the shopkeeper hallucinated or had a false memory or was lying to cover up the true identity of the person etc. It can’t *just so happen* that the testimony was false and there’s no more to the story – there has to be more to the story! When I discover (or remember) that glasses wearing is rare in the relevant population, this makes it more likely that the testimony is false – but it doesn’t remove the need for an explanation in the event that it is. The situation in which the person didn’t wear glasses *still* requires just as much explanation as before.

    Thanks for the comment!

    Martin

  8. Martin Smith

    Hi Jesse,

    You make a very good point – there are two different possibilities here which we should treat in (somewhat) different ways: Either (1) I directly ask the shopkeeper whether the person was wearing glasses, and he replies that the person was, or (2) the shopkeeper simply volunteers the information that the person was wearing glasses without any specific prompting. As you suggest, the testimony in (2) could be regarded as probabilistically stronger than the testimony in (1). Certainly the prior probability of the shopkeeper providing this testimony should be reckoned lower in case (2) and, given some further assumptions, this will increase the extent to which the testimony makes probable the proposition that the person wore glasses. This is not to say of course that the proposition will end up being *likely*. I’m inclined to think that, for most realistic assignments of values to the relevant parameters, this will still be unlikely.

    Rather than arguing this further though, suppose we just turn attention to case (1) – where the testimony has a lower probabilistic strength. If I ask the shopkeeper whether the person was wearing glasses, and he tells me that the person was, I think that I would be perfectly justified in believing what he says. After all, much of the testimony that we receive is in response to direct questions. If, say, I ask the shopkeeper whether he’s had a lot of customers today, and he tells me that he has, and I ask him whether there’s a post office nearby and he tells me that there is and so on, then I would be justified in believing each of these things. And if I ask him whether the person was wearing glasses and he tells me that s/he was, then I’m justified in believing this too (in spite of what I know about the low proportion of glasses wearers).

    Thanks for the comment!

    Martin

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