Tom Griffiths

So he was thinking about some gambling-inspired examples.

So if there is a lottery which is paying off at some rate, how do you estimate the rate at which it's paying off?

But the way that he sets that up is in terms of the beliefs that you have about that lottery.

What's a reasonable estimate that you should have for the probability that it's going to pay off at the next moment, given the examples that you've seen so far?

And then that idea is developed further by Pierre-Simon Laplace, who really came up with it independently and really worked out all of the consequences of that way of thinking about how to update our beliefs.

I think there's a discrete point that happens with Bayes and Laplace just instead of working out that theory.

But you do see hints of people talking in those terms before that.

Even Wilkins, who came up with this idea of the language that you could use where you can never say anything false, talked about,

probability as a way of talking about a degree of belief.

He didn't work out the mathematical consequences, but he used that language.

Pascal famously, after he departed the world of mathematics and instead started to think about religion, made a probabilistic argument in that setting, which is really an argument about belief as well.

You see lots of hints of this prior to Bayes and Laplace, but I think they're the ones who really developed that into a theory of what we could call thought.

I think the really cool thing about Bayesian probability is that one very natural way to see it is just an extension of logic.

In logic, we talk about possible worlds.

So if you have two propositions, P and Q, you can imagine all of the possible worlds that you could be in.

You could be in a world where P is true and Q is true, a world where P is true and Q is false, a world where P is false and Q is true, and a world where both P and Q are false.

Those are the possible worlds we could live in.

And logic is really about what conclusions you can draw with certainty.

based on the information you have about what world you might be in.

So if you have got enough information to rule out some of those possible worlds such that it has to be the case that the world you're in is one where Q is true, then it's reasonable to conclude that Q is true.