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The goal is worthy, but the product is inadequate to say the least. This thing is littered with typos, and enough of the exposition is sufficiently irrelevant o

The goal is worthy, but the product is inadequate to say the least. This thing is littered with typos, and enough of the exposition is sufficiently irrelevant or incorrect to be unintuitive. That said, I like the graphics and layout. For example, when he discusses power sets in order to introduce sigma algebras, he implies that a sigma algebra is a better-behaved alternative to a power set. However, a power set is always itself a sigma algebra (after all, even a power set of an uncountable set still is closed under complements and countable unions). Later, when discussing probability distributions, he writes: > [W]e want this allocation [of a conserved quantity] to be self-consistent – the allocation to any collection of disjoint sets, A_n ∩ A_m=0, n≠m, should be the same as the allocation to the union of those sets, > ℙπ[∪(n=1 to N) A_n]=∑(n=1 to N)ℙπ[A_n]. The condition `A_n ∩ A_m=0, n≠m` is actually incorrect, since A_n and A_m are sets and 0 is an integer. The author means the empty set, but typo'd. Sometimes he frequently uses words like "conserved" or "well-defined" without giving us a clue as to what these mean. In what context are probabilities "conserved"? What distinguishes "well-defined" from "not well-defined"? I'm a software engineer. A non-trivial amount of my time is devoted to reading code and finding bugs. Sloppy reasoning, inconsistencies and outright errors like that are big red flags to me. It doesn't help that the whole section on sigma algebras is somewhat irrelevant, since he doesn't really explore measure theory as the basis for modern probability. IMO a better resource is the series of "Probability Primer" videos from mathematicalmonk on YouTube[1]. He does an excellent job (IMO) of covering all pertinent pre-requisites and being mostly rigorous without necessarily proving every single fact or exhaustively covering all edge and corner cases. He also makes a good effort to recommend advanced (and rigorous) treatments of the subject (and ancillary ones like measure theory). A readable version of this YouTube series would be a great resource, and if Michael Betancourt is reading, I'd encourage him to pursue that in his next iteration of this product. https://www.youtube.com/playlist?list=PL17567A1A3F5DB5E4 https://www.youtube.com/playlist?list=PL17567A1A3F5DB5E4

soVeryTired 9y ago
> It doesn't help that the whole section on sigma algebras is somewhat irrelevant, since he doesn't really explore measure theory as the basis for modern probability. Christ, practically all of measure theory is irrelevant for applied work, in much the same way that an engineer shouldn't care about the definition of a real number. There's a model of real analysis due to Solovay that used the axiom of dependent choice instead of the full axiom of choice. In the Solovay model, all sets are measurable. Thus any results that require measure theory inherently depend on the axiom of choice. I'd be worried if I was relying on Choice as an applied scientist. Edit: same goes for Lebesgue vs. Riemann integration. To quote Richard Hamming: Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
- psoy 9y ago
  It matters very much in computational / quant finance
  - soVeryTired 9y ago
    It matters by convention, because the textbooks are written that way. My point is that you don't need that level of formal rigour to do applied work. You can derive the Feynman-kac formula via a scaling limit of discrete-time Markov chains. Add some levy process (a.k.a compound Poisson processes) and you're basically done. If you want to be ultra-rigourous in your definitions, then you need measure theory, yes. But even Einstein didn't need that for his description of Brownian motion. If a scaling limit is good enough for him, it's good enough for me.

soVeryTired 9y ago

> It doesn't help that the whole section on sigma algebras is somewhat irrelevant, since he doesn't really explore measure theory as the basis for modern probability. Christ, practically all of measure theory is irrelevant for applied work, in much the same way that an engineer shouldn't care about the definition of a real number. There's a model of real analysis due to Solovay that used the axiom of dependent choice instead of the full axiom of choice. In the Solovay model, all sets are measurable. Thus any results that require measure theory inherently depend on the axiom of choice. I'd be worried if I was relying on Choice as an applied scientist. Edit: same goes for Lebesgue vs. Riemann integration. To quote Richard Hamming: Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.

psoy 9y ago
It matters very much in computational / quant finance
- soVeryTired 9y ago
  It matters by convention, because the textbooks are written that way. My point is that you don't need that level of formal rigour to do applied work. You can derive the Feynman-kac formula via a scaling limit of discrete-time Markov chains. Add some levy process (a.k.a compound Poisson processes) and you're basically done. If you want to be ultra-rigourous in your definitions, then you need measure theory, yes. But even Einstein didn't need that for his description of Brownian motion. If a scaling limit is good enough for him, it's good enough for me.