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> People have been building on top of the axiomatic foundation that you outlined for decades. This is old news. Yup, essentially since Kolmogorov's 1933 paper.

by graycat 12y ago

> People have been building on top of the axiomatic foundation that you outlined for decades. This is old news. Yup, essentially since Kolmogorov's 1933 paper. The axiomatic approach I gave is just what is used in Breiman's Probability, as published by SIAM. It's a super nicely written book. I have never seen his book on statistics.

kgwgk 12y ago
I can't resist quoting Jaynes (from the closing remarks in Chapter 2 of "Probability Theory: The Logic of Science"): In 1933, A. N. Kolmogorov presented an approach to probability theory phrased in the language of set theory and measure theory. This language was just then becoming so fashionable that today many mathematical results are named, not for the discoverer, but for the one who first restated them in that language. For example, in the theory of continuous groups the term “Hurwitz invariant integral” disappeared, to be replaced by “Haar measure.” Because of this custom, some modern works—particularly by mathematicians—can give one the impression that probability theory started with Kolmogorov. [...] However, our system of probability differs conceptually from that of Kolmogorov in that we do not interpret propositions in terms of sets, but we do interpret probability distributions as carriers of incomplete information. Partly as a result, our system has analytical resources not present at all in the Kolmogorov system. This enables us to formulate and solve many problems—particularly the so-called “ill posed” problems and “generalized inverse” problems—that would be considered outside the scope of probability theory according to the Kolmogorov system. These problems are just the ones of greatest interest in current applications.
- graycat 12y ago
  I found a PDF of Jaynes and read the part at the end of chapter 2. There it said to read Appendix A, and I did that, too. Sorry, but in Appendix A, quickly Jaynes doesn't understand the Kolmogorov foundations. E.g., he confuses a trial and an event. Not good. For what he thought about "ill posed", etc., problems, I couldn't find. To me, Jaynes does not look good.
  - kgwgk 12y ago
    > For what he thought about "ill posed", etc., problems, I couldn't find. Jaynes' book has a full chapter on "Paradoxes of Probability Theory", including Borel's paradox (or Borel-Kolomogorov paradox). See also http://www.cims.nyu.edu/~csplash/data/notes/2010-14.pdf http://www.cims.nyu.edu/~csplash/data/notes/2010-14.pdf
    - grayclhn 12y ago
      Oh, that looks interesting. Thanks for posting it.

by graycat 12y ago

kgwgk 12y ago

I can't resist quoting Jaynes (from the closing remarks in Chapter 2 of "Probability Theory: The Logic of Science"): In 1933, A. N. Kolmogorov presented an approach to probability theory phrased in the language of set theory and measure theory. This language was just then becoming so fashionable that today many mathematical results are named, not for the discoverer, but for the one who first restated them in that language. For example, in the theory of continuous groups the term “Hurwitz invariant integral” disappeared, to be replaced by “Haar measure.” Because of this custom, some modern works—particularly by mathematicians—can give one the impression that probability theory started with Kolmogorov. [...] However, our system of probability differs conceptually from that of Kolmogorov in that we do not interpret propositions in terms of sets, but we do interpret probability distributions as carriers of incomplete information. Partly as a result, our system has analytical resources not present at all in the Kolmogorov system. This enables us to formulate and solve many problems—particularly the so-called “ill posed” problems and “generalized inverse” problems—that would be considered outside the scope of probability theory according to the Kolmogorov system. These problems are just the ones of greatest interest in current applications.

graycat 12y ago
I found a PDF of Jaynes and read the part at the end of chapter 2. There it said to read Appendix A, and I did that, too. Sorry, but in Appendix A, quickly Jaynes doesn't understand the Kolmogorov foundations. E.g., he confuses a trial and an event. Not good. For what he thought about "ill posed", etc., problems, I couldn't find. To me, Jaynes does not look good.
- kgwgk 12y ago
  > For what he thought about "ill posed", etc., problems, I couldn't find. Jaynes' book has a full chapter on "Paradoxes of Probability Theory", including Borel's paradox (or Borel-Kolomogorov paradox). See also http://www.cims.nyu.edu/~csplash/data/notes/2010-14.pdf http://www.cims.nyu.edu/~csplash/data/notes/2010-14.pdf
  - grayclhn 12y ago
    Oh, that looks interesting. Thanks for posting it.