@Strangerep and @Jano L, the question is in the context of the many attempts to understand collapse as a form of Bayesian conditioning. It's a very old idea. bhobba often mentions it casually - it was really his common use of the analogy between collapse and statistical updating, as well as his saying that the wave function doesn't need a conscious observer ("rational" might have been a better word choice) that lay behind my question, since they weren't obviously compatible to me. The analogy is hinted at in Cohen-Tannoudji, Diu and Laloe's text, and it is mentioned in the text of Wiseman and Milburn.
Some recent attempts include:
http://arxiv.org/abs/quant-ph/0106133
Quantum probabilities as Bayesian probabilities
Carlton M. Caves, Christopher A. Fuchs, Ruediger Schack
Phys. Rev. A 65, 022305 (2002)
http://arxiv.org/abs/1107.5849
Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
M. S. Leifer, R. W. Spekkens
Phys. Rev. A 88, 052130 (2013)
Bayesian coherence is a formal theory of rationality, and one of the requirements is that the prior must assign non-zero probability to the truth, which is why I believe Aaronson used that as his definition of a rational agent - one is using a Bayesian framework.
A similar line of query is what the nature of hidden variables behind quantum mechanics can be. Depending on the relationship between the hidden variables, the wave function can be interpreted as "psi-ontic" or "psi-epistemic", where I think the latter means something like "belief". One popular set of definitions is in:
http://arxiv.org/abs/0706.2661
Einstein, incompleteness, and the epistemic view of quantum states
Nicholas Harrigan, Robert W. Spekkens
Found. Phys. 40, 125 (2010)
Although maximally psi-epistemic interpretations seem to have been ruled out
http://arxiv.org/abs/1208.5132 , the existence of psi-epistemic interpretations is said to have been proved in:
http://arxiv.org/abs/1201.6554
Distinct Quantum States Can Be Compatible with a Single State of Reality
Peter G. Lewis, David Jennings, Jonathan Barrett, Terry Rudolph
Phys. Rev. Lett. 109, 150404 (2012)
http://arxiv.org/abs/1303.2834
Psi-Epistemic Theories: The Role of Symmetry
Scott Aaronson, Adam Bouland, Lynn Chua, George Lowther
Phys. Rev. A 88, 032111 (2013)
I don't know whether a psi-epistemic interpretation necessarily requires a Bayesian interpretation, but I believe the search for the possibility of a Bayesian interpretation is a motivation for psi-epsitemic interpretations.
Interestingly, there is a form of many-worlds in which Bayesian probability plays a role:
http://arxiv.org/abs/quant-ph/0312157
Quantum Probability from Subjective Likelihood: improving on Deutsch's proof of the probability rule
David Wallace