A parable on induction: The Village Census

Bertrand Russell's Census Officer

A reader mentioned a parable by Bertrand Russell about "induction by simple enumeration". Through my infinite resources, it just so happens that I have access to the exact quote:

Induction by simple enumeration may be illustrated by a parable. There was once upon a time a census officer who had to record the names of all householders in a certain Welsh village. The first that he questioned was called William Williams; so were the second, third, fourth, . . . At last he said to himself: "This is tedious; evidently they are all called William Williams. I shall put them down so and take a holiday." But he was wrong; there was just one whose name was John Jones. This shows that we may go astray if we trust too implicitly in induction by simple enumeration.

- Bertrand Russell, The History of Western Philosophy, page 543

In my unrelenting quest to make everything more tediously mathematical, I wish to analyze this parable using Bayes' Theorem. But first, let us simplify it just a bit. Let us assume that there are only five villagers (all men), and the census officer has questioned four of them.

An aside on notation

I was originally planning to explain my point in plain English, but I found it highly cumbersome. Therefore, I will be using a little notation from probability theory. Firstly, it is useful to denote several claims using letters:

• W: all five villagers are named William Williams.
• J: only four of the villagers are named William Williams, and the last is not.
  
• E: The four villagers that the census officer questioned are named William Williams.

We will use the letter P to denote the probability of a claim. For instance, P(W) is the probability that all five villagers are named William Williams. P(W) is just a number between 0 and 1 (where 1 means that W absolutely must be true). P(W) is called the "prior" probability of W. It's called the "prior" probability because it is prior to any evidence. P(W) is the probability that W is true before we learn about the census officer's new observations.

When the census officer questions the villagers, we find that all four people he questioned were named William Williams. In other words, we find that claim E is true. If we want to know the probability of W before the census officer questions the villagers, then we need P(W). If we want to know the probability of W after the census officer questions the villagers, then we would denote this probability as P(W|E). P(W|E) is called "the probability of W given E".

Back to the village

What we see in this parable is that a census officer has collected a piece of evidence: claim E is true. What should he conclude from this piece of evidence? In the parable, he concludes that all villagers are named William Williams, but is this conclusion reasonable? In other words, is P(W|E) high? We might compare the officer's conclusion with other conclusions he might have had. For instance, would it have been better to conclude that only four of the villagers were named William Williams? In other words, is P(J|E) high? Which is higher, P(W|E) or P(J|E)?

To compare these two probabilities, it might be best to take the ratio between the two. Let's find P(W|E)/P(J|E). If the value of P(W|E)/P(J|E) is greater than 1, that means that, given the census officer's observations, he was justified in concluding that all villagers are named William Williams. If the value is less than 1, then he was not justified in his conclusion. According to Bayes' Theorem, P(W|E)/P(J|E) = P(W)/P(J) * P(E|W)/P(E|J).

Basically, this means that the officer's conclusion depends on two factors. The first factor, P(W)/P(J), is the ratio of prior probabilities of W and J. This means that if we know that W is much more likely than J before learning about the results of the census, then will still be much more likely after learning about the census. After all, the census is just one piece of evidence, which may not be enough to overcome our previous prejudices. What could be the source of these prejudices? Perhaps we know from the last census that J was true last year. Or perhaps they all adhere to some religion that requires them to all share a surname. But in absence of anything like that, we may assume that P(W)/P(J) is not too big or too small. It's about equal to 1.

The second factor, P(E|W)/P(E|J), is easy to calculate. P(E|W) is nothing other than the probability of E given W. If W is true, then E is necessarily true, so P(E|W) is simply 1. P(E|J) is the probability of E given J. If J is true, then it is only by lucky coincidence that the census officer only found villagers named William Williams. Assuming that the census officer questioned four random villagers, then P(E|J) is equal to 1/5. Therefore, the second factor, P(E|W)/P(E|J), is equal to 5.

Going back to Bayes' theorem, we know that P(W|E)/P(J|E) = P(W)/P(J) * P(E|W)/P(E|J). Based on my analysis, this equal to 5. Since it's greater than 1, that means that the officer is justified in concluding that all the villagers are named William Williams.

But hold on! Recall that I said that P(W)/P(J) is "about" 1. I'm a physicist, so when I say something is "about" 1, I really mean that it could be anywhere between 0.01 and 100, maybe even more or less. There is a lot of uncertainty which I completely ignored. With this in mind, the value of P(W|E)/P(J|E) can be anywhere between 0.05 and 500. The census officer's conclusion doesn't look so solid now!

Another comparison

Previously, we compared the two claims W and J. But what happens if we introduce a third claim C?

• C: The four villagers which the census officer questioned are named William Williams. The last one has a different name.

Claim C is distinct from J because it's more specific. Claim J only states that one of the villagers is not named William Williams. Claim C states that a specific villager, the one which the officer skipped, is not named William Williams.

Let us compare claim C to claim W. As before, we will calculate P(W|E)/P(C|E),* which is equal to P(W)/P(C) * P(E|W)/P(E|C).

*The advanced reader would note that this is also exactly equal to P(W|E)/P(J|E), our previous result.

Our first factor in the equation is P(W)/P(C). We're comparing the prior probabilities of two claims. W claims that the last villager is named William Williams. C claims that the last villager is not named William Williams. All things considered, C seems more likely than W, since there are a lot of names out there, and only one of them fits W. Nevertheless, in absence of anything which would cause major prejudice, I will say that P(W)/P(C) is "about" 1, same as I did before.

The second factor, P(E|W)/P(E|C), is easy to calculate. Both W and C necessarily imply E, therefore P(E|W) = P(E|C) = 1. The ratio is just 1.

Therefore, when we compare the claims C and W, we find that P(W|E)/P(C|E) is "about" 1, same as P(W)/P(C). The ratio of probabilities is the same before and after the results of the census. The evidence tells us nothing about the comparison of claim C to W. The census officer's conclusion is looking even weaker than before!

Of course, the officer's argument was weak from the start. But before, the weakness was hidden in the word "about".

There moral of the story is that you have to be very careful with Bayesian analysis. Sometimes, the conclusion seems to depend on what angle you look at it. There is always a degree of uncertainty in the prior probabilities that you can never eliminate. The best way to do it is with more evidence, evidence, evidence. For instance, if the officer had simply questioned the last villager, then his evidence would likely overwhelm any previous uncertainty.

Searching in closets

I've joked with some of my friends that I get withdrawal symptoms from physics. After I’ve finished all my physics homework, that’s when the doubts begin. What am I doing in my life? Is there something I’m missing out on? Why can’t more of life’s problems be solved by elementary considerations of differential equations?

And so my thoughts turned to something that I usually completely ignore: romance. I’m afraid that I’m going to sound like some utterly confused teenager when I talk about this. I just don’t get any of it really. But then, what is the point of accumulating readers if I can't occasionally inflict my life upon them?

To be direct, I was entertaining the possibility that I’m gay. The answer, I believe, is no, I’m not. How do you even know if you’re gay? People just seem to know. They feel it inside. They feel an attraction to people of the same sex. I don’t really feel any of that. So I guess it was silly of me to ever worry about it in the first place.

But that was not immediately obvious to me. See, I deeply fear the possibility that I might be anything other than straight. I am very supportive of the LGBT movement, and I would never hate anyone for being queer. But if it were me, I might hate myself for it. It's not what I would want to be. I'm special and unique in quite a number of ways already, and I don't need more than that. I have a lot of respect for homosexuals who come to be comfortable with themselves, because I'm not sure I could accomplish that myself.

I strongly believe in critical thinking as a guiding principle in all of life's decisions. Therefore, when fear comes into play, rather than ignoring it, I keep a very close eye on it. You never know when a fear might lead to denial. If I fear a conclusion, I might dismiss it far too quickly, and then conjure up a rationalization after the fact. So when fear comes into play, I become more cautious, and do not immediately trust my own judgments. On the other hand, fear can also have the opposite effect. I could end up focusing disproportionately on a remote possibility, just because I fear it. I suppose that's exactly what happened this particular time.

But that’s not the end of it. There was a reason why I considered that I might be gay in the first place. I may not ever have felt attracted to males, but I’ve hardly felt any attraction to females either. So by the same logic which says I’m not gay, I could also argue that I’m not straight.

Story time! Remember back in middle school when everyone started getting crushes? I remember it happening to the other kids. I had a large group of good friends, about half girls, half boys. At first I was oblivious, but soon I realized there was all sorts of activity going on between various of my friends. Mostly, I just heard snippets of gossip while listening in on friends’ conversations. It was all about who made a cute couple, who got unexpectedly dumped, and so forth. I couldn’t really keep track of the drama myself, because I was rarely part of that particular conversation. I mostly concerned myself with other things.

Incidentally, I did have my own girlfriend one time. We didn't do a whole lot. I was too young. I couldn't figure out how I was supposed to react to anything. Nevertheless, I make a point of never regretting actions that are more than a few years old. Lots of young teens are plenty awkward, and nothing needs to be made of it.

But nothing has happened since, not that I’m aware of. I’ve always believed that romance is something that happens to you, and then you start worrying about it. Nothing has ever happened, so I’ve never worried about it. I've never had a crush, never thought anyone was "hot", or felt the urge to get to know any particular person intimately. It was not a matter of being too nervous to act upon my desires, it was a matter of not having them to begin with. In this respect, I have not changed much since middle school.

I suppose there are many factors I could blame. I am an introvert after all. I probably spend more time doing physics homework than partying (because I enjoy the former more than the latter). But being introverted does not mean being socially unsuccessful. Introverts are simply less concerned about seeking social activities. I may be introverted, but I am not shy, and I have plenty of friends. So I don’t think it’s for want of meeting new people.

With all that in mind, I’ve been considering a new possibility: am I asexual? The answer, I believe, is no, I’m not. But again, fear is in play, and I do not immediately trust my own judgment.

Asexuality is defined on AVEN, the Asexual Visibility and Education Network. The definition is someone who does not experience sexual attraction. However, many asexuals experience romantic attraction of a nonsexual nature, or experience sexual arousal which is not connected to other people. Asexuality first showed up in Kinsey’s famous studies of sexuality. In Kinsey’s study, he placed everyone on a scale from 0 to 6, where 0 is heterosexual, and 6 is homosexual. But he also placed about 1% of the population into a category “X” because they had "no socio-sexual contacts or reactions" (from Wikipedia). Since then, they’ve been ignored by all but a few studies, and by the public. I know I had never heard of it.

AVEN contends that asexuality is an intrinsic quality of a person. Just like homosexuality, it is not a choice, and it is unlikely (though possible) for it to change. But without much scientific study done on the subject, I am skeptical of these conclusions. I suppose it is plausible, and I believe it. On the other hand, in the case of a specific individual, how does one distinguish between asexuality and clueless romantic incompetence?

I honestly do not believe that I am asexual. The trouble is, I can't seem to express exactly why I feel this way. It sure makes me suspicious when I can't make my reasons explicit, not even to myself.

And if that weren't enough, there's also Grey-A, which is the blurry boundary between sexuality and asexuality. Ugh, now I'm thinking I'm demisexual, meaning I never get attracted to people based on looks or other instantly available information. Life kinda sucks for demisexuals because they usually only become attracted to people who are already friends. No matter how friendly I am, that's a relatively small set of people. And then it's like you're crossing some sort of awkward friends boundary.

Sigh...

At least writing all that made me feel a little better (even if my parents may be reading this :-/ ).