Here is something like a proof without (a lot of) words:
Remember that the Strong Law of Large Numbers (SLLN) says that it is almost certain that between the mth and nth observations in a group of length n, the relative frequency of Heads will remain near the fixed value p, whatever p may be, and be within the interval [p-e, p+e], for any e > 0, provided that m and n are sufficiently large numbers. That is,
Keep in mind that for large n almost all priors are irrelevant compared to the likelihood. If priors are irrelevant for large n, then they are still irrelevant for small n, even if they have more pull. Although, for small n, as you may have expected, most frequentist and even Bayesian analyses (almost any type of analysis honestly) are of dubious value.
I personally think a more interesting discussion in statistics is parametric vs. nonparametric.
Thanks for reading!
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