Letter-mention worthiness

A friend is working on a talk about the relationships Charles Williams had with C.S. Lewis and T.S. Eliot. He noticed that Eliot gets mentioned more in Williams’s letters than Lewis does, which might not be expected since Lewis and Williams were good friends. He counted 46 letters in Letters to Michal from Serge mentioning Eliot and 33 for Lewis. Is that a big difference, I wondered, or could it be due to chance? That’s the sort of question we use statistics for, if we can think of a way that we’re talking about a random process. The mechanical computation part of statistics is amazingly easy these days, but before I can let the computer tools loose, I have to figure out what question I’m really asking. Fortunately, that’s the fun part.

Let’s begin with a model of mentioning writers in letters. Suppose there’s some underlying thing about writers that’s generally “how important they are to Charles Williams”, which causes him to mention them in a letter or not. For statistics, we don’t need to know the precise definition of what the importance factor is in literary terms (or general-human terms). All we need is that the higher that factor is, the more letters they’ll be mentioned in.

Second assumption: Whether CW wrote a letter to his wife is determined by other parts of life than the literary-importance parameter. (Money, for instance, is a frequent topic.) If we believe that CW wasn’t motivated by his admiration for another writer to dash off a letter, then those circumstances are effectively random for our purposes.¹ Now, let’s imagine an ensemble of parallel universes in which different letter-worthy events happen and spare time comes on different days. In each of those worlds, CW writes different letters from the ones he wrote in our world, but the number of times alt-CW mentions another writer is based on that same importance factor. ² Then the mentions in each universe will fall on a curve whose shape we can calculate. From that posterior distribution, we can estimate how likely one writer is to be mentioned more than another.

Under this model, the number of letters mentioning a writer will have a binomial distribution. For a fixed set of 320 letters (like LtoMfromS), a binomial distribution has one unknown parameter in it; counting mentions in the book tells us information about that parameter.

The blurb on the flyleaf of Letters to Michal from Serge mentions six contemporary writers: Eliot, Lewis, Dorothy L. Sayers, W.H. Auden, Christopher Fry, and Edith Sitwell. Let’s run them all through the model. (It’s only one line of code apiece.) The peak of each author’s probability density is the most likely number of letters, which matches the figure in the book. What we get from these distributions is the spread — could things have been otherwise? How different are parallel universes likely to be?

The first thing we see is that the more likely a writer is to be mentioned, the more spread around their observed value there is. For example, Dame Edith gets mentioned 3 times; maybe that could have been 4 but it wasn’t going to be 10. Eliot, on the other extreme, might have been mentioned anything from 30 to 60 times under this model.

Binomial distributions for the six authors.

Eliot’s curve is definitely to the right of Lewis’s, but there’s some overlap. How likely is it that Lewis could have been mentioned more? These curves have analytic forms so we could compute it exactly, but these days it’s easier to run a simulation. I drew 10,000 numbers from each of their distributions, and Lewis’s number was higher than Eliot’s in just under 5% of them. That’s pretty solid evidence. The less-mentioned writers overlap more. Lewis was mentioned more than Sayers 99.98% of the time. Sayers is mentioned more times than Auden in 92% of our parallel universes. Auden is mentioned more often than Fry in 68% of them, which is in the range where the difference could have been just by chance.

The basic observation that started me down this path is confirmed: To Charles Williams, Eliot was almost certainly more letter-worthy than Lewis. Might be something to do with that Swedish thingy (as Paul Krugman calls his).

1. Not strictly true — there’s a letter that mentions Lewis several times because Lewis let Williams stay in his rooms at the College. But it’s close to true.
2. as Richard McElreath says, the minimum number of data points you need for a Bayesian analysis is one. That’s how many we have, so we’re in good shape.