Probability In An Infinite Universe

[PatrickPowers]

Having a masters degree in probability in infinite sets I dare say I know the basics.

For probability in an infinite space it seems to me that the practical way is to declare that every event has zero probability. Sets of such events may have measure greater than zero. Such sets may be defined however you please. In real life the probability of an event appearing in that set is usually measured.

Finite sets always have measure zero, and infinite sets may also have measure zero.

What I see instead is logic based on the assumption that if an event is observed then it must have a probability greater than zero. This leads to the conclusion that there are an infinity of Earths out there identical to ours and so forth. I on the other hand say that observing one event tells one nothing about the measure of the set in which one choses to include it. The rule about membership in such a set is quite arbitrary as well.

[steelpillow]

Set theory often intersects with philosophy, and especially axiomatic systems of formal logic. In this case I would ask you some basic questions:
Please clarify your primitives and axioms. Are you talking about sets of real events, with a probability property assigned to each event?
If an event has zero probability, is it still a member of the set of real events?
On what basis do you declare the Universe to be a mathematically infinite set, when it has grown from an initial condition, at a finite rate, for a finite period, and the shortest meaningful separators are the finite Planck time and Planck length? Do you declare the Big Bang to have been the instantaneous creation of an infinite number of events at that point in time, or what?

[Keiji]

The probability of any event in an infinite set is indeed zero. This is, quite obviously, not useful.

Personally, whenever I'm dealing with probabilities on a continuous (real) number line, bounded or not, I immediately throw the idea of instantaneous probability out the window, and work purely with cumulative probabilities.

If you consider a range from a to b (with a < b) and you know the cumulative probabilities P(x < a) and P(x < b) then the probability that x is between a and b is simply P(a < x < b) = P(x < b) - P(a < x).

And since this works just fine with discrete ranges as well, you can just always work with cumulative probability regardless of whether you actually need it!

If you graph a cumulative probability distribution, you get a monotonically increasing curve between the start and end of the range, and you can consider the gradient (the derivative) to be the "instantaneous" probability. This reflects is really an approximation that x will be somewhere inside a small range centered on the point where you took the gradient: steep gradient = high probability of being near that point, shallow gradient = low probability of being near that point. So, by looking at cumulative distribution graphs, you can still reason about instantaneous probabilities, despite that they all technically work out to zero!

It's so simple - and so useful - that I'm honestly surprised that people who teach probability don't talk about cumulative probability anywhere near as much as instantaneous probability. The only reason I can think of for this is that the calculations required to actually compute the value of a distribution at some point do seem to be a heck of a lot more complicated for cumulative distributions than for non-cumulative ones. (I once needed to compute the cumulative values of the gamma distribution, and had to shell out to Python from my non-Python program to achieve this since scipy was the only available implementation I could find and I could simply not wrap my head around what would be needed to implement it from first principles. It was one of the most satisfying moments of my life when I saw this working, though!) But you don't need to be able to compute values of the distributions yourself to be able to understand the general, qualitative ideas behind them.

Of course it is possible to talk about probabilities in continuous spaces of two or more dimensions, too. In this case, I imagine these would be handled with partial derivatives, just like how the one-dimensional cumulative and non-cumulative distributions are related by taking an "ordinary" (impartial? ) derivative.

[PatrickPowers]

The probability of any event in an infinite set is indeed zero.

Consider the normal aka Gaussian distribution. The density is non-zero for all real numbers but the measure is one.

You could also if you like include some arbitrary event with probability 1/2 then for all other events rescale the Gaussian by dividing its density by two. You can do whatever you want as long as everything is non-negative and integrates/sums to one.

It appears to me that what these physicists are trying to have is a infinite uniform distribution in which all observed events have non-zero probability. I think they should start with all events having zero probability.

[steelpillow]

Set theory often intersects with philosophy, and especially axiomatic systems of formal logic. In this case I would ask you some basic questions:
Please clarify your primitives and axioms. Are you talking about sets of real events, with a probability property assigned to each event?
If an event has zero probability, is it still a member of the set of real events?

OK, I found out those definitions for myself: these are probabilities based on the definition of the sum of all event probabilities being equal to 1.
This is just set theory, there is no formal mapping to real physical events beyond the notional idea of observation or measurement.

"based on the assumption that if an event is observed then it must have a probability greater than zero," my first conclusion is that such an event is not a member of the set, as all events which are members have zero probability. Thus, no event which is a member of the set can actually be observed.

The only way forward is then to modify that assumption to, say, "if a set of events is observed then it must have a probability greater than zero." From this it is evident that any observation comprises an infinite set of events.

What infinite set would that be? I cannot see the slightest connection to "other Earths", or any other explanation, without introducing additional assumptions.

It appears to me that what these physicists are trying to have is a infinite uniform distribution in which all observed events have non-zero probability. I think they should start with all events having zero probability.

Not sure who these "other physicists" are, but I think there must be some kind of concept transfer going on here, between probabilities within a probability space and probabilities within a physical spaces; the probabilities under consideration are different in kind, and there is in fact no mapping between the two situations.

Failing to notice this might help explain the anomalies I just noted.