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We want all that for probability theory. We want countably infinite so that we can discuss, say, the event that a coin never or always comes up heads. We don't want uncountably infinite because it would create a big mess in the theory.

Who should I read for an introduction to that "big mess"?

So many of the cases I'm actually interested in require real valued (continuous) inputs and outcomes. While one can quantize these to create an approximation of something countable, it seems like that a much simpler theory would be possible if it was built from the ground up to handle these common non-discrete real-world cases, rather than trying to shoe-horn them into standard probability theory. I was hoping this might be the direction that Terry was headed, with the emphasis on probabilistic methods.

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I think you're misunderstanding. Standard basic (measure-theoretic) probability theory is designed to handle common non-discrete real-world cases: continuous random variables like height, temperature, etc. They're not approximated by something countable; instead theorems proving that they have the sort of behavior you'd want are established by proving them for a countable approximation, then taking limits. It is exactly like integration: prove things for step functions, then make the steps infinitely thin. The theory is clean and straightforward.

Here's where it gets less basic: say you want to look at temperature over time, but you don't want to model temperature as a variable that's measured daily, or hourly, or even every second (secondly?), but you want to model it as a process that evolves in continuous time. That's where the theory gets messy. Not necessarily at the level of a user of this theory, but definitely at the level of proving that the math you want to use is allowed.

If you actually need an introduction to that sort of probability, Lawrence Evans (Berkeley) has some old lecture notes aimed at undergrads [1] that he turned into a book [2]. If (more likely) you want standard measure-theoretic probability theory (as opposed to what's taught to undergraduates), David Pollard's book is pretty good [3].

I'm sure that @graycat will scoff at those recommendations, but his reading list would be considered excessively hardcore and time consuming even for a graduate student in math, which I'm assuming you're not.

[1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.416...

[2]: http://www.amazon.com/An-Introduction-Stochastic-Differentia...

[3]: http://www.amazon.com/Theoretic-Probability-Statistical-Prob...

ps: after looking at it again, the intro in Evans's notes is as gentle as it's going to get, so start there. And (as you'll find out, unless you're some sort of savant) this shit's hard. If you actually want to understand this stuff, graduate coursework is probably the only practical way to do it.

Thanks for your references. I have always thought that many statistics/probability based explanations are adhoc. They are adhoc because they explain pre-selected facts; and their predictions are just a confirming instances (cf. positive vs confirming instance from Larry Laudan, a philosopher of science). Your point "definitely at the level of proving that the math you want to use is allowed" hints in that direction.

Probability's hard to teach. You can give informal statements and kind of wave your hands at the underlying theory, or you can give a rigorous well-founded treatment that's intellectually satisfying. But the rigorous foundation uses math that's a step or two beyond what undergraduate math majors learn. It's not necessarily harder than what math majors see, but it's a ton of extra material to teach, when the payoff is that you can now (after half a year) prove that the conditional probability is well-defined as

Pr(A | B) = Pr(A and B) / Pr(B)

instead of just telling it to students and drawing a few diagrams that drive the point home.

But I think it's more pragmatic than ad hoc. Any deep theory of probability that doesn't deliver

Pr(A | B) = Pr(A and B) / Pr(B)

is basically useless since that's how random phenomena seem to behave in real life. Having a deeper theory is useful because it allows you to derive other implications of that theory and makes certain calculations much easier. But if the theory disagrees with phenomena that we want to model, that can be a problem.

You're right, and I misunderstood. I'm a computer programmer trying to rapidly learn enough about probability theory to be able to communicate with some theoretical statisticians regarding causality, confounding, and longitudinal data analysis. I have a decent intuitive grasp of what's happening, but no ability to convey anything with proper terminology. I could certainly use a better grasp of the basics, and I'm trying to figure out where to start. Thanks for the links.

Okay, for that stuff probability theory is too abstract. For basic basics, Edward Tufte has a $2 ebook that's pretty good: Data Analysis For Politics And Policy[1] and for terminology in causality, Rubin has a short open access paper[2]. For a freshman-stats level treatment, OpenStax college's book looks legitimate but I haven't actually read it carefully[3].

[1]: https://www.edwardtufte.com/tufte/ebooks [2]: https://projecteuclid.org/euclid.aoas/1223908042 [3]: http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de

> I'm sure that @graycat will scoff at those recommendations, but his reading list would be considered excessively hardcore and time consuming even for a graduate student in math, which I'm assuming you're not.

Probability and stochastic processes based on measure theory are not very popular in the US, even in graduate math departments.

Uh, scoff, scoff. Okay?

The full measure theoretic details of stochastic processes in continuous time can be a bit of a challenge. That topic can be important, e.g., for Brownian motion and stochastic differential equations used in mathematical finance. Of course, there is

Karatzas and Shreve, Brownian Motion and Stochastic Calculus and Chung and Williams, Introduction to Stochastic Integration. And there's much more, especially from Russia and France.

But, otherwise, usually in practice, what people are interested in is either (1) second order stationary stochastic processes, e.g., as in electronic or acoustical signals and noise. There are commonly interested in power spectral estimation, digital filtering, maybe Wiener filtering, the fast Fourier transform, etc. or (2) what is in, say, Cinlar, Introduction to Stochastic Processes.

In Cinlar, for the continuous time case, get a good introduction to the Poisson process (the vanilla arrival process, e.g., like clicks at a Geiger counter, new sessions at a Web site, and much more). Also get what else people are mostly interested in in practice, Markov processes in discrete time with a discrete state space (that is, the values are discrete).

The case of Markov processes in continuous time and discrete state space is not so tough if the jumps are driven by just a Poisson process. But there is still more in Cinlar.

And there are other good texts on stochastic processes.

For (1), look at some of the texts used by EEs. The measure theory approach is in Doob, Stochastic Processes, Loeve, Probability Theory, and several more texts by quite good authors. E.g., without measure theory, can just dive in via Blackman and Tukey, The Measurement of Power Spectra ....

With all these sources, are able to get by without measure theory. Yes, without measure theory, at some places will have to not ask to understand too much and just skip over some details to get back to the applied stuff.

But for measure theory, the Durrett text seems to get a student to that unusually quickly.

For more, at the MIT Web site, there is an on-line course in mathematical finance that avoids measure theory. They want to use the Radon-Nikodym theorem and Ito integration but still avoid measure theory. Uh, the Radon-Nikodym theorem is a generalization of the fundamental theorem of calculus. Once see it, it's dirt simple, but a good proof takes a bit or follow von Neumann's proof that knocks it all off in one stroke (it's in Rudin, Real and Complex Analysis).

All of this is fine for real-valued inputs and outcomes. It's the number of events (coin flips, measurements, etc) that we're restricting to be countable.

No, the number of events is necessarily also finite or uncountable. Indeed, it is a nice exercise that there are no countably infinite sigma algebras (extra credit for a solution!).

It's just can't take uncountably many events, take their union, and assume that the result is also an event.

No problem. For the foundations I outlined, can work just fine with continuous functions, measurable functions, stochastic processes, random variables taking values on the real line, in the complex plane, in finite dimensional real or complex vector spaces with, say, the usual topology, Hilbert and Banach spaces, etc. Can do multi-dimensional Markov processes, and much more.

And you can have each point on the real line an event. Fine. But you just can't take the uncountable union of any set of such events and assume that the result is also an event.

As for the event a random variable takes a value >= 0? Fine.

Or, let the Borel subsets of the real line be the smallest sigma algebra that contains all the open sets, e.g., all the open intervals. Then for Borel set A and real valued random variable X, can ask for the probability X is in A.

I believe you will find that you will have a solid foundation for what you want.

To see all this stuff, need more than just the sparse definitions and, instead, need an actual text and maybe a course. Recently looked at the on-line materials from MIT and didn't see such a course. Graduate probability is not all that popular in the US; stochastic processes in continuous time is still less popular.

To study graduate probability, I'd recommend a good undergraduate major in pure math with good coverage of, say, W. Rudin, Principles of Mathematical Analysis. Then good coverage of linear algebra from more than one of the best known texts. Likely also spend as much time as you can in Halmos, Finite Dimensional Vector Spaces. E.g., at one time, Halmos, Rudin, and Spivak, Calculus on Manifolds were the three main texts for Harvard's famous Math 55.

Get good with proving the theorems.

I also recommend Fleming, Functions of Several Variables.

Then, sure, Royden, Real Analysis. Couldn't be prettier.

If not in a hurry, then the real half of Rudin's Real and Complex Analysis. Especially if you like Fourier theory!

Then of the probability books, I believe that the nicest, first book is L. Breiman, Probability. He wrote that before he went consulting and came back and did CART and random forests.

Next, K. Chung, A Course in Probability Theory. Next, J. Neveu, Mathematical Foundations of the Calculus of Probability. Then, Loeve, Probability Theory.

Loeve is huge -- mostly just use it for reference or browse. E.g., it has sufficient statistics and stationary stochastic processes (the EEs love that) IIRC not in the other books.

IIRC, both Breiman and Neveu were Loeve students at Berkeley.

If do well with Breiman, then for graduate probability, likely can stop there. Else, Chung will then be fast and easy reading and reinforce what you learned in Breiman. Neveu is elegant; my favorite, but deserve extra credit for each workable exercise you can find, not actually work, you understand, just find! Sure, some of the exercises are terrific, half a course in a few lines of an exercise. E.g., he has one of those on statistical decision theory or some such. And see the Tulcea material in the back.

Then there's more that you can do on stochastic processes, potential theory via Brownian motion, e.g., for mathematical finance, stochastic optimal control, and more.

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