Probability theory is the special case of quantum mechanics in which ones algebra of observables is commutative.

This is horribly misleading. Bayesian probability can be applied perfectly well in a universe that obeys MWI while being kept completely separate mathematically from the quantum mechanical uncertainty.

As a mathematical statement, what Baez says is certainly correct (at least for some reasonable mathematical formalisations of ‘probability theory’ and ‘quantum mechanics’). Note that Baez is specifically discussing quantum statistical mechanics (which I don't think he makes clear); non-statistical quantum mechanics is a different special case which (barring trivialities) is completely disjoint from probability theory.

Of course, the statement can still be misleading; as you note, it's perfectly possible to interpret quantum statistical physics by tacking Bayesian probability on top of a many-worlds interpretation of non-statistical quantum mechanics. That is, it's possible but (I argue) unwise; because if you do this, then your beliefs do not pay rent!

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability. (I mean probability here, not a superposition.) An alternative map is that you believe that the particle is spin-right with 50% probability and spin-left with 50% probability. (Now superposition does play a part, as spin-right and spin-left are both equally weighted superpositions of spin-up and spin-down, but with opposite relative phases.) From the Bayesian-probability-tacked-onto-MWI point of view, these are two very different maps that describe incompatible territories. Yet no possible observation can ever distinguish these! Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability. (The wavefunctions give Born probabilities for the observations, which are then weighted according to your Bayesian probabilities for the wavefunctions, giving the result of 50% every time.)

In statistical mechanics as it is practised, no distinction is made between these two maps. (And since the distinction pays no rent in terms of predictions, I argue that no distinction should be made.) They are both described by the same ‘density matrix’; this is a generalisation of the notion of quantum state as a wave vector. (Specifically, the unit vectors up to phase in the Hilbert space describe the pure states of the system, which are only a degenerate case of the mixed states described by the density matrices.) A lot of the language of statistical mechanics is frequentist-influenced talk about ‘ensembles’, but if you just reinterpret all of this consistently in a Bayesian way, then the practice of statistical mechanics gives you the Bayesian interpretation.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

What do you have knowledge of then? Or is there some concept that could be described as having knowledge of something without that thing having an actual value?

This is the weak point in the Bayesian interpretation of quantum mechanics. I find it very analogous to the problem of interpreting the Born probabilities in MWI. Eliezer cannot yet clearly answer these questions that he poses:

What are the Born probabilities, probabilities of? Here's the map - where's the territory?

And neither can I (at least, not in a way that would satisfy him). In the all-Bayesian interpretation, the Born probabilities are simply Bayesian probabilities, so there's no special problems about them; but as you point out, it's still hard to say what the territory is like.

My best answer is simply what you suggest, that our maps of the universe assign probabilities to various possible values of things that do not (necessarily) have any actual values. This may seem like a counterintuitive thing to do, but it works, and we have no other way of making a map.

By the way, I've thought of a couple more references:

• John Baez (1993), This Week's Finds #27;
• Toby Bartels (1998), Quantum measurement problem.

Baez (1993) is where I really learnt quantum statistical mechanics (despite having earlier taken a course in it), and my first (subtle) introduction to the Bayesian interpretation (not made explicit here). Note the talk about the ‘post-Everett school’, and recall that Everett is credited with founding the many-worlds interpretation (although he avoided the term ‘MWI’). The Bayesian interpretation could have been understood in the 1930s (and I have heard it argued, albeit unconvincingly, that it is what Bohr really meant all along), but it's really best understood in light of the modern understanding of decoherence that Everett started. We all-Bayesians are united with the many-worlders (and the Bohmians) in decrying the mystical separation of the universe into ‘quantum’ and ‘classical’ worlds and the reality of the ‘collapse of the wavefunction’. (That is, we do believe in the collapse of the wavefunction, but not in the territory; for us, it is simply the process of updating the map on the basis of new information, that is the application of a suitably generalised Bayes's Theorem.) We just think that the many-worlders have some unnecessary ontological baggage (like the Bohmians, but to a lesser degree).

Bartels (1998) is my first attempt to explain the Bayesian interpretation (on Usenet), albeit not a very good one. It's overly mathematical (and poorly so, since W*-algebras make a better mathematical foundation than C*-algebras). But it does include things that I haven't said here, (including mathematical details that you might happen to want). Still (even for the mathematics), if you read only one, read Baez.

Edit: I edited to use the word ‘world’ only in the technical sense of an interpretation.