Mathematical methods inspired by an eighteenth century minister (8)
“Bayesian” is a word that has gained a lot of attention recently, though my experience tells me most people aren’t exactly sure what it means. I’m fairly confident that there are many more crossword-puzzle enthusiasts than Bayesian statisticians — but I would also note that the overlap is larger than most would imagine. In fact, anyone who has ever worked on a crossword puzzle has employed Bayesian reasoning. They just aren’t (yet)aware of it. So I’m going to explain both how intuitive Bayesian thinking is, and why it’s useful, even outside of crosswords and statistics.
But first, who was Bayes, what is his “law” about, and what does that mean?
Sound of a Conditional Reverend’s Dog (5)
“Bayes” of statistical fame, is the Reverend Thomas Bayes. He was a theologian and mathematician, and the two works he published during his lifetime dealt with the theological problem of happiness, and a defense of Newton’s calculus — neither of which concern us. His single posthumous work, however, was what made him a famous statistician. The original title, “ A Method of Calculating the Exact Probability of All Conclusions founded on Induction,” clearly indicates that it’s meant to be a very inclusive, widely applicable theorem. It was also, supposedly, a response to a theological challenge posed by Hume — claiming miracles didn’t happen.
Wonders at distance travelled without vehicle upset (8)
“Miracles”, Hume’s probabilistic argument said, are improbable, but incorrect reports are likely— so, the argument goes, it is more likely that the reports are incorrect than that the miracle occurred. This way of comparing probabilities isn’t quite right, statistically, as we will suggest later. But Bayes didn’t address this directly at all.
Taking a risk bringing showy jewelry to school (8)
“Gambling” was a hot topic in 19th century mathematics, and Bayes tried to answer an interesting question; when you see something happen several times, how do can you figure out, in general, the probability of it occurring? His example was about throwing balls onto a table — you aren’t looking, and a friend throws the first ball. After this, he throws more, each time, telling you whether the ball landed to the left or right of the first ball. After a doing this a few times, you still have’t seen the table, but want to know how likely is it that the next ball land to the left of that original ball.
To answer this, he pointed out that you get a bit more information about the answer every time a ball is thrown. After the first ball, for all you know the odds are 50/50 that the next one will be on either side. after a few balls are thrown, you get a better and better sense of what the answer is. After you hear the next five balls all land to the left, you’ve become convince that the next ball landing to the left is more likely than landing to the right. That’s because the probabilities are not independent — each answer gives you a little bit more information about the odds.
But enough math — I’m ready to look at a crossword.
Could wine be drunk by new arrival? (6)
“Newbie” is how I’d prefer to put my ability with crossword puzzles. But as soon as I started, I noticed a clear connection. The method of reasoning I practice and endorse as a decision theorist are nearly identical to the methods that are used by people in this everyday amusement. So I’ll get started on filling in (only one part of) the crossword I did yesterday, and we’ll see how my Bayesian reasoning works. I start by filling in a few easy answers, and I’m pretty confident in all of these. 6 Down — Taxing mo. for many, 31 Across — Data unit, 44 Across — “Scream” actress Campbell.
The way I’ve filled these in so far is simple — I picked answers I thought were very likely to be correct. But how can I know that they are correct? Maybe I’m fooling myself. The answer is that I’ve done a couple crosswords before, and I’ve found that I’m usually right when I’m confident, and these answers seem really obvious. But can I apply probabilistic reasoning here?
Distance into which vehicle reverses ___ that’s a wonder (7)
“Miracles,” or anything else, according to Reverend Bayes, should follow the same law as thrown balls. If someone is confident, that is evidence, of a sort. Stephen Stigler, a historian of math, argues that Bayes was implying an important caveat to Hume’s claim — the probability of hearing about a miracle increases each time you hear another report of it. That is, thee two facts are, in a technical sense, not independent — and the more independent accounts you hear, the more convinced you should be.
But that certainly doesn’t mean that every time a bunch of people claim something outlandish, it’s true. And in modern Bayesian terms, this is where your prior belief matters. If someone you don’t know well at work tells you that they golfed seven under par on Sunday, you have every reason to be skeptical. If they tell you they golfed seven over par, you’re a bit less likely to be skeptical. How skeptical, in each case?
We can roughly assess your degree of belief— if a friend of yours attested to the second story, you’d likely be convinced, but it would take several people independently verifying the story for you to have a similar level of belief in the first. That’s because you’re more skeptical in the first place. We could try to quantify this, and introduce Bayes’ law formally, but there’s no need to bring algebra into this essay. Instead, I want to think a bit more informally — because I can assess something as more or less likely without knowing the answer, without doing any math, and without assigning it a number.
When you hear something outlandish, your prior belief is that it is unlikely. Evidence, however, can shift that belief — and enough evidence, even circumstantial or tentative, might convince you that the claim is plausible, probably, or even very likely. And in a way it doesn’t matter what your prior is, if you can accumulate enough different pieces of trustworthy evidence. And that leads us to how I can use the answers I filled in as evidence to help me make further plausible guesses.
I look at some of the clues I didn’t immediately figure out. I wasn’t sure what 6 Across — Completely blows away, would be; there are lots of 4-letter words that might fit the clue. Once I get the A, however, I’m fairly confident in my guess, conditional on this (fairly certain) new information. I look at 31 Down — Military Commission (6), but I can’t think of any that start with a B. I see 54 Across — Place for a race horse, and I’m unsure — there are a few words that fit — it could be “first”, “third”, “fifth,” “sixth” or “ninth”, and I have no reason to think any more likely than another. So I look for more information, and notice 54 Down — It might grow to be a mushroom (5, offscreen). “Spore” seems likely, and I can see that this means “Sixth” works — so I fill in both.
At this point, I can start filling in a lot more of the puzzle, and the pieces are falling in to place — each word I figure out that fits is a bit more evidence that the others are correct, making me confident, but there are a few areas where I seem stuck.
Being stuck is evidence of a different sort — it probably means at least one of two things — either I have something incorrect, or I’m really bad at figuring out crosswords. Or, of course, both.
At this point I start revisiting some of my earlier answers, ones I was pretty confident about until I got stuck. I’m still pretty confident in 39 Down — Was at one time, but ___ now. “Isn’t” is too obvious of an answer to be wrong, I think. On the other hand, 38 Down — A miscellany or collection, has me stumped, but two Is in a row also seem strange. 37 Down — Small, fruity candy, is also frustrating me; I’m not such an expert in candy, but I’m also not coming up with anything plausible. So I look at 50 Across — A tiny part of this?, again, and re-affirm that “Bit” seems like it’s a good fit. I’m now looking for something that can give me more information, so I rack my brains, and 36 Across — Ho Chi Min’s capital, comes to me: Hanoi. I’m happy that 39 Down is confirmed, but getting nervous about the rest.
I decided to wait, and look elsewhere, filling in a bit more where I could. My progress elsewhere is starting to help me out.
Now, I need to re-evaluate some earlier decisions and update my beliefs again. It has become a bit more complex than evaluating single answers — I need to consider the joint probability of several different things at once. I’ll unpack how this relates to Bayesian reasoning afterwards, but first, I think I made a mistake.
I was marginally confident in 50 Across — A tiny part of this? as “bit”, but now I have new evidence. I’m pretty sure Nerb isn’t a type of candy, but “Nerd” seems to fit. I’m not sure if they are fruity, so I’m not confident, and I’m still completely at a loss on 38 Down — A miscellany or collection. That means I need to come up with an alternative for 50 Across; “Dot” seems like an unlikely option, but it fits really well. And then it occurs to me; A dot is a little bit of the question mark. That’s an annoying answer, but it seems a lot more likely than that “Nerb” is a type of candy. And I’m not sure what Olio is, but there’s really nothing else that I can imagine fitting. And there are plenty of words I don’t know. (As I found out later, this is one of them.)
At first, I had a high confidence that “Bit” was the best answer for 50 Across — I had a fairly strong prior belief, but I wasn’t certain. As evidence mounted, I started to re-assess. Weak evidence, like the strange two Is in a row, made me start to question the assumption that I was right. More weak evidence — remembering that there is a candy of some sort called Nerds, and realizing that “Dot” was a potential answer, made me revise my opinion. I wasn’t strongly convinced that I had everything right, but I revised my belief. And that’s exactly the way a Bayesian approach should work; you’re trying to figure out which possibility is worth betting on.
That’s because all of probability theory started with a simple question that a certain gambler asked Blaise Pascal; how to we split the pot when a game gets interrupted. And historians who don’t think Bayes was trying to formulate a theological rebuttal to Hume suggest that he’s really responding to a question posed by de Moivre — from whose book he may have learned probability theory, which we need to mention in order to figure out why I’d pick “Dot” over “Bit” — even though I think it’s a stupid answer. But before I get there, I’ve made a bit more progress — I’m finished, except for one little thing.
31 Down — Military Commission. That’s definitely a problem — I’m absolutely sure Brevei isn’t the right answer, and 49 Down, offscreen, is giving me trouble too. The problem is, I listed all the possible answers for 54 Across — Place for a race horse, and the only one that started with an “S” was sixth.
Conviction … or what’s almost required for a conviction (9)
“Certainty” can be dangerous, because if something is certain, almost by definition, it means nothing can convince me otherwise. It’s easy to be overconfident, but as a Bayesian, it’s dangerous to be so confident that I don’t consider other possibilities — because I can’t update my beliefs! That’s why Bayesians, in general, are skeptical of certainty. If I’m certain that my kid is smart and doing well in school, no number of bad grades or notes from the teacher can convince me to get them a tutor. In the same way, if I’m certain that I know how to get where I’m going, no amount of confused turns, circling, or patient wifely requests will convince me to ask for directions. And if I’m certain that “Place for a race horse” is limited to a numeric answer, no number of meaningless words like “Brevei” can change my mind.
High payout wagers (9)
“Perfectas” are bets placed on a horse race, predicting the winner and second place finisher, together. If you get them right, they payoff can be really significant — much more than bets on horses to win or to place. In fact, there are lots of weird betting terms in horse racing, and by excluding them from consideration, I may have been hasty in filling out “sixth.” My assumption of having compiled and exhaustive list of terms was premature. Instead, I need to reconsider once again — and that brings us to why, in a probabilistic sense, crosswords are hard.
Disreputable place for a smoke? (5)
“Joint” probabilities are those that relate to multiple variables. And when solving the crossword, I’m not just looking to answer each clue, I’m looking to fill in the puzzle — it needs to solve all of the clues together. Just like figuring out a Perfecta is harder than picking the right horse; putting multiple uncertain questions together is where joint probabilities show up. But it’s not hopeless; as you figure out more of the puzzle, you reduce the remaining uncertainty. It’s like getting to place a Perfecta bet after seeing 90% of the race; you have some pretty good ideas about what can and can’t happen.
Similarly, Bayesians, in general, collect evidence to constrain what they think is and isn’t probable. Once enough balls have been thrown to the left of that first one, you get pretty sure the odds aren’t 50–50. The prerequisite for getting the right answer, however, is being willing to reconsider your beliefs — because reality doesn’t care what you believe.
And the reality is that 31 Down is Brevet, so I need an answer to 54 Across — Place for a race horse that starts “St”. And that’s when it hit me — sometimes, I need to simply be less certain I know what’s going on. The race horse isn’t running, and there are no bets. It’s in a stall, waiting patiently for me to realize I was confused.
A Final Note
I’d note three key lessons that Bayesians can learn from crosswords, since I’ve already spent pages explaining how Crossworders already understand Bayesian thinking. And they are lessons for life, ones that I’d hope crossword enthusiasts can apply more generally as well.
- The process of explicitly thinking about what you are uncertain of, and noticing when something is off, or you are confused, is useful to apply even (especially!) when you’re not doing crossword puzzles.
- Evaluating how sure you are, and wondering if you are overconfident in your model or assumptions, would have come in handy to those predicting the 2016 election.
- Being willing to actually change your mind when presented with evidence is hard, but I hope you’d rather have a messy crossword than an incorrectly solved one.
A Postscript for Pedants
Scrupulously within the rules, but not totally restrictive
“Strict” Bayesians are probably annoyed about some of this — at no point in the process did I get any new evidence. No one told me about any new balls thrown, I only revised my belief based on thinking. A “Real Bayesian” starts with all the evidence already available, and only updates when new evidence comes in. For a non-technical response, it’s sufficient to note that computation and thought takes time, and although the brain roughly approximates Bayesian reasoning, the process of updating is iterative. And for a technical version of the same argument, I’ll let someone else explain that there are no real Bayesians. (And thanks to Noah Smith for that link!)
The crossword clues were a combination of info from http://www.wordplays.com/crossword-clues/, and my own inventions.
The crossword is an excerpt from Washington Post Express’s Daily Crossword for January 11th, 2017, available in full on Page 20, here: https://issuu.com/expressnightout/docs/express_01112017