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> while there is no such similar statement for a method that depends on the experimenter's personal beliefs

The posterior probability P(H|D) is exactly this kind of statement. You say "Based on the data, I am 78% sure that the coin is biased". I think this is both more direct and interpretable.

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By "similar statement" I meant a statement about how often the method will lead you to the incorrect conclusion. If you can't quantify that, it seems to me you don't have any basis for claiming your method "works".

If that is what you meant, than you are getting in the realm of hypothesis testing. The best equivalent of a p-value is then an anabashedly named "Bayes factor" [1], which is a ratio of posterior probabilities for competing hypotheses.

https://en.wikipedia.org/wiki/Bayes_factor

Thanks, didn't know about this.

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