Lang and I spent a long time discussing Tremaine's objection to the Bayesian program for showing that a set of angles (say) are drawn from the flat (uniform) distribution between 0 and 2 pi. If the N-dimensional probability distribution for N angles phi is just the product of N flat distributions, then every N-dimensional point is equally likely, whether it corresponds to spread-out phases or concentrated phases. Lang and I came up with some clever things to say about this, but we don't yet have an answer that will satisfy Tremaine. At some level, the problem comes down to the problem that a frequentist can ask are the data consistent with the flat hypothesis? whereas the Bayesian needs to ask is the flat hypothesis better than X? where X is a well-specified alternative.

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