#6 by hackenslash » Sep 22, 2021 10:27 am
Bayes Theorem is, in a loose way, an approach to the problem of induction (though, like falsification, doesn't actually solve it).
In essence, as you know, inductive reasoning is reasoning of the sort in which true premises and valid inference leads to a conclusion that, while we can't assert it as true, we can have a measure of confidence in, with subsequent validating observations increasing that confidence.
Bayes Theorem is a quantitative approach to this process, allowing us to literally put a number on our confidence based on what we've observed. It's also a strong safeguard against naïve conclusions in probability, by ensuring that we look at genuine statistical prevalence.
Can't recall if Grant talks about this example in the above video (he certainly does elsewhere), but there's a brilliant exposition by the awesome Daniel Kahneman, famous in these parts for showing the true value of eyewitness testimony (it has none) and oodles of other examples of cognitive bias (his work was the foundation of that famous experiment on cognitive priming, wherein interviewers were primed to a positive or negative reaction to somebody based only on the temperature of a cup that had been handed to them some moments prior to their meeting).
He asked a question about whether, based on certain characteristics (being studious, bookish, preferring one's own company), it was more probable that a person was a librarian or a farmer. We might suppose that these traits are more prevalent in librarians and therefore conclude naïvely that it was more likely a librarian. However, if you consider the comparative distribution of librarians and farmers in the population, wherein farmers vastly outnumber librarians in raw numbers, and take the percentage of farmers who have these traits, you discover that, even thought these traits occur in a tiny percentage of farmers and a high percentage of librarians, the distribution is such that it's significantly more probable that the subject is a farmer.
It's brilliantly elegant, and even knowing it exists is often a solid bulwark against naïve conclusions.