Monday, 19 April 2021

Using Bayes' theorem to figure out whether Bayes' theorem is an "obscure math theorem".

An edited abstract of our book was published on Sunday and caused statisticians everywhere to lose their minds. One particular tweet has gone viral.

The issue was due to the headline implying that Bayes' theorem was an "obscure math theorem". The reason it produced so much heat is the phrase is debatable in meaning, like "the dress".

What exactly we infer from this phrase is coincidently, also related to Bayes' theorem. Does it imply that it is obscure for a math theorem? Or that math theorems are not very well known anyway? Or perhaps "obscure" means hard to understand*?

Baye's theorem is certainly well known for a theorem, it is taught in many introductions to stats classes. But then again, what percentage of the population would have taken stats classes, not a huge amount? What percentage of the population is low enough to qualify for something to be called "obscure" anyway? 

Perhaps you think the meaning is clear so let's change it around. Let's say the headline was Bayes' theorem is a "well-known math theorem". Do you honestly think no one would reply: "Well-known! I haven't heard of it before!!!"

These phrases are lexically ambiguous and it's something I have written about before. If you hear someone say that the "priest married my sister" you may be unsure if the priest was the person who conducted the ceremony or is your sister's husband/wife. Usually, you can work this out from the context (e.g. if the priest was catholic then it's extremely unlikely to be the latter). This is how Bayesian reasoning works, we update our beliefs given new information and we do it all the time without thinking. 

However, sometimes this reasoning can go wrong and lead to a lot of issues like the above. If you know that Bayes' theorem is well known for a theorem, it may affect how you think most people would interpret the phrase "obscure maths theorem". Knowing this information biases you: it makes it much harder to think how others (who do not know this information) would think about the phrase. 

This is also related to something called the curse of knowledge (another thing I have written about before). This happens in conversation when you mistakingly believe that they know what you know. For example, some people pointed out that P(A|B) = (P(B|A)P(A))/P(B) is really simple as it's just multiplication and division. But unless you know that the "|" means conditional on and the "P" means probability, it's just completely incomprehensible. It isn't simple unless you know what the symbols mean, it is like saying this is simple, これは簡単です.**

The irony of all this is that journalists (including Tom) often get annoyed at readers who think that they write the headline of an article. Journalists know that it is usually the editor that writes the headline as they have direct experience of this. However, how common is this knowledge amongst the general population?

Saying this, I can also understand people's anxiety that the headline may be misleading. I do think it highlights a broader issue with the disconnect between headline writers and those who write the article.

Overall though, the fact that the Observer wanted to commission a piece about Bayes' theorem in the first place is fantastic. Let's build positively on this rather than arguing about it into obscurity. 

*Some people were concerned that the phrasing of the headline made it sound like it was more difficult to understand which could put people off. Another way of looking at is that for many, maths seems difficult to understand and acknowledging this may make people feel more confident. I have no idea which way is the right way to look at this.

**This is the Japanese for "this is simple".

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