I have been reading more about the philosophical interpretations of probability and how this topic relates to the interpretation of quantum mechanics (QM). If we assume hidden-variable theories don’t work, some notion of probability is at the core of QM.
I had previously read this SEP article on the interpretation of probability. It provides a good overview of the history of the issue among philosophers, and reveals a wide variety of ideas under past and present consideration. Among physicists, however, debate on interpreting probability seems to center broadly on two conceptions: a frequency interpretation and a Bayesian interpretation.
I thought this recent post at physics musings was a good one on this topic, and I also benefited from the links (including the one to this John Baez page). Then, a recent post at Quantum Quandaries provided a link to this paper by Marcus Appleby. In it Appleby argues persuasively for the Bayesian conception and considers the implications for interpreting QM.
The frequentist conception appeals because it is intended to be objective. If we could repeat an experiment an infinite number of times we would empirically fill out the probability distribution of outcomes. The problem is we can’t do this. The Bayesian interpretation is epistemic: it shows how, given one’s prior assumption about a probability distribution, a measurement outcome serves to improve it. Appleby argues that an epistemic conception is unavoidable. He shows the problems with frequentist conceptions which try to provide an interpretation in situations with finite ensembles; here, one attempts to focus only on a pragmatically relevant finite subset of outcomes. However, this choice of subset is influenced by the context of the situation and the biases of the chooser and thus reintroduces the subjective element.
Appleby discusses the propensity interpretation of QM, which places the probability as an objective property of the system being measured. He suspects this idea usually underlies the adoption of a frequentist perspective. Propensity can, however, be made consistent with the Bayesian conception, if one gives up the idea that propensity is a directly observable property.
Appleby next discusses attempts to formulate an objective version of the Bayesian conception. Can the prior probability distribution be objectively grounded? No, because as some point you have an initial assumption which is not empirical. You cannot derive a probability from a non-probabilistic empirical fact.
Now, getting back to what it all means for our worldview: since probabilities are irreducible to objective facts, and quantum mechanics describes reality, does this mean we have to give up the idea that there is an objective real world out there? Is it true, a la most summaries of the Copenhagen interpretation, that QM only describes the content of our knowledge?
In turning to this question in the last section of his paper, Appleby surprised me by bringing up the problem of qualia from the domain of the philosophy of mind. Qualia (arguably) are irreducibility first-person phenomena which do not fit into a mechanistic view of the world. A fully objective realist view of the world has no place for qualia. And yet, Appleby says, you would say the same thing about real probability or propensity, since these are irretrievably “contaminated” by subjectivity. For him, this points to the need to give up the fully objective realism and accept that we need to find a fuller extension or development of a Copenhagen-style interpretation.
Not mentioned in this 2-year old paper is the Relational Interpretation of Quantum Mechanics (RQM), which can be thought of as such a generalization and extension of Copenhagen. This interpretation seems to fit best with the Bayesian interpretation of probability. For some more recent discussion of RQM follow some of the links in the physics musings post above and also see the recent posts in this thread at PhysicsForums.