No Universal Probability Space

Use common sense. This is no different from other matters of imprecise communication. Calling up on the meaninglessness of assertions about coin flips really does sound silly.

probability measure P on the measurable space ({heads, tails}, {{}, {heads}, {tails}, {heads, tails}}), P({heads}) = P({tails}) = 1/2 and P({}) = P({heads, tails}) = 0

I think you mean P({heads, tails}) = 1.

We typically say that there is a 1/2 chance of heads, but what we are implicitly saying is that given a probability measure P on the measurable space ({heads, tails}, {{}, {heads}, {tails}, {heads, tails}})

Giving a probability of one event only implies we think that particular event is possible. It doesn't say anything about what other events we are considering, so there is no necessity to describe the entire space of possibilities.

Many people act as if there is some number they can assign to any event which tells them how likely it is to occur and questions of "probability spaces" never enter their minds. What does it mean that something happens 1% of the time? I don't know; maybe that it doesn't happen 99% of the time? How is 1% of the time measured? I don't know; maybe one out of every 100 seconds?

Under a Bayesian interpretation of probability, which is generally used here, probability does not express how frequent something will occur. Instead, it represents your belief an event will occur or a proposition is true. Then p=0.01 means "possible enough to consider, but very doubtful". I think most people naturally adopt a Bayesian perspective, so I'm not sure what the problem is.

I'm surprised more people don't take this point seriously here. Really, if someone says there's a 1% chance of the defenses failing, what does that actually mean? Sure, in this context it might not matter because the point stands, but probabilities are misused this way all the time.

This sounds just like something I've always wondered about: the percentages they give in weather reports, for likelihood of rain. Does '30% chance of rain' mean that they estimate a 30% chance of getting any rain, or that they think it'll rain for 30% of the day, or what?

Please link to the news story you're referring to.

I think most LW people take a Bayesian view of probability, where probability is a consistent, numerical measure of how confident we are that a proposition is true, updated according to Bayes rule, etc. You're advocating the mathematical view of probability, where probability really means "probability measure", ie the measure on a measure space of measure 1.

It's not that one of these views is right and the other is wrong. They're actually describing two slightly differentthings. Measure-theoretic probability is something we can have a completely rigorous mathematical theory of, because it's assumptions are technical and precise. But having a mathematical theory doesn't necessarily mean you can apply it to the real world. You need knowledge and judgment to determine what real-world phenomena are modeled by your theory and how well. Measure theoretic probability models card games very well (measure space = the uniform distribution over the 52! ways that the deck could be shuffled) if the dealer is honest, but it doesn't help you decide weather to trust the dealer.

Bayesian probability is a less rigorous, but more directly-applicable-to-real-life theory. It shares a lot of terminology and theorems with measure theoretic probability, but it isn't quite the same thing. In particular, in Bayesian probability you don't have a probability space, so when you see people here talking probability without specifying the space, it's not an error, they're just being Bayesian.

Jayne's "Probability Theory" is a great book on Bayesian probability theory, but if you've got a Mathematical education, which I suspect you might, it's going to piss you off. Just ignore the parts where Jaynes bloviates about things he doesn't understand, and learn from the parts where he teaches the things he does. Most of the book is the latter.

All recursive probability spaces converge to the same probabilities, as the information increases.

Not that those people making up probabilities knows anything about that.

If you want an universal probability space, just take some universal computer, run all programs on it, and keep those that output event A. Then you can see how many of those that output event B, and thus you can get p(B|A) whatever A and B are.

This is algorithmic information theory, and should be known by any black belt bayesian.

Kim Øyhus