Do you want to become stronger in the way of Bayes? This post is intended for people whose understanding of Bayesian probability theory is currently somewhat tentative (between levels 0 and 1 to use a previous post's terms), and who are interested in developing deeper knowledge through deliberate practice.
Our intention is to form an online self-study group composed of peers, working with the assistance of a facilitator - but not necessarily of a teacher or of an expert in the topic. Some students may be somewhat more advanced along the path, and able to offer assistance to others.
Our first text will be E.T. Jaynes' Probability Theory: The Logic of Science, which can be found in PDF form (in a slightly less polished version than the book edition) here or here.
We will work through the text in sections, at a pace allowing thorough understanding: expect one new section every week, maybe every other week. A brief summary of the currently discussed section will be published as an update to this post, and simultaneously a comment will open the discussion with a few questions, or the statement of an exercise. Please use ROT13 whenever appropriate in your replies.
A first comment below collects intentions to participate. Please reply to this comment only if you are genuinely interested in gaining a better understanding of Bayesian probability and willing to commit to spend a few hours per week reading through the section assigned or doing the exercises.
As a warm-up, participants are encouraged to start in on the book:
Preface
Most of the Preface can be safely skipped. It names the giants on whose shoulders Jaynes stood ("History", "Foundations"), deals briefly with the frequentist vs Bayesian controversy ("Comparisons"), discusses his "Style of Presentation" (and incidentally his distrust of infinite sets), and contains the usual acknowledgements.
One section, "What is 'safe'?", stands out as making several strong points about the use of probability theory. Sample: "new data that we insist on analyzing in terms of old ideas (that is, models which are not questioned) cannot lead us out of the old ideas". (The emphasis is Jaynes'. This has an almost Kuhnian flavor.)
Discussion on the Preface starts with this comment.
I think this is not so important, but it helpful to think about nonetheless. I guess the first step is to define what is meant by 'Bayesian'. In my original comment, I took one necessary condition to be that a Bayesian gadget is one which follows from the Cox-Polya desiderata. It might be better to define it to be one which uses Bayes' Theorem. I think in either case, Maxent fails to meet the criteria.
Maxent produces the distribution on the sample space which maximizes entropy subject to any known constraints which presumably come from data. If there are no constraints, then one gets the principle of indifference which can also be gotten straight out of the Cox-Polya desiderata as you say. But I think these are two different approaches to the same target. Maxent needs something new -- namely Shannon's information entropy (by 'new' I mean new w.r.t. Cox-Polya). Furthermore, the derivation of Maxent is really different from the derivation of the principle of indifference from Cox-Polya.
I could be completely off here, but I believe the principle of indifference argument is generalized by the transformation group stuff. I think this because I can see the action of the symmetric group (this is the group (group in the abstract algebra sense) of permutations) on the hypothesis space in the principle of indifference stuff. Anyway, hopefully we'll get up to that chapter!
Upon further study, I disagree with myself here. It does seem like entropy as a measurement of uncertainty in probability distributions does more or less fall out of the Cox Polya desiderata. I guess that 'common sense' one is pretty useful!