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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 ra

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=10.1.1.416.1184&rep=rep1&type=pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.416... [2]: http://www.amazon.com/An-Introduction-Stochastic-Differential-Equations/dp/1470410540 http://www.amazon.com/An-Introduction-Stochastic-Differentia... [3]: http://www.amazon.com/Theoretic-Probability-Statistical-Probabilistic-Mathematics/dp/0521002893 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.

raincom 11y ago
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.
- grayclhn 11y ago
  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.
nkurz 11y ago
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.
- grayclhn 11y ago
  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 https://www.edwardtufte.com/tufte/ebooks [2]: https://projecteuclid.org/euclid.aoas/1223908042 https://projecteuclid.org/euclid.aoas/1223908042 [3]: http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de
graycat 11y ago
> 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).

raincom 11y ago

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.

grayclhn 11y ago
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.

nkurz 11y ago

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.

grayclhn 11y ago
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 https://www.edwardtufte.com/tufte/ebooks [2]: https://projecteuclid.org/euclid.aoas/1223908042 https://projecteuclid.org/euclid.aoas/1223908042 [3]: http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de http://cnx.org/contents/30189442-6998-4686-ac05-ed152b91b9de

graycat 11y ago

> 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).