if you start with a uniform prior over [0,1] for the probability of the coin coming up heads
I’m not. The post specifies “a coin that is either biased to land Heads 2/3rds of the time, or Tails 2/3rds of the time”—that is (and maybe I should have been more explicit), I’m saying our prior belief about the coin’s bias is just the discrete distribution {”1/3 Heads, 2⁄3 Tails”: 0.5, “2/3 Heads, 1⁄3 Tails”: 0.5}.
I agree that a beta prior would be more “realistic” in the sense of applying to a wider range of scenarios (your uncertainty about a parameter is usually continuous, rather than “it’s either this, or it’s that, with equal probability”), but I wanted to make the math easy on myself and my readers.
Ah, I see. I missed that part of the post for some reason.
In this setup the update you’re doing is fine, but I think measuring the evidence for the hypothesis in terms of “bits” can still mislead people here. You’ve tuned your example so that the likelihood ratio is equal to two and there are only two possible outcomes, while in general there’s no reason for those two values to be equal.
Thanks for this analysis! However—
I’m not. The post specifies “a coin that is either biased to land Heads 2/3rds of the time, or Tails 2/3rds of the time”—that is (and maybe I should have been more explicit), I’m saying our prior belief about the coin’s bias is just the discrete distribution {”1/3 Heads, 2⁄3 Tails”: 0.5, “2/3 Heads, 1⁄3 Tails”: 0.5}.
I agree that a beta prior would be more “realistic” in the sense of applying to a wider range of scenarios (your uncertainty about a parameter is usually continuous, rather than “it’s either this, or it’s that, with equal probability”), but I wanted to make the math easy on myself and my readers.
Ah, I see. I missed that part of the post for some reason.
In this setup the update you’re doing is fine, but I think measuring the evidence for the hypothesis in terms of “bits” can still mislead people here. You’ve tuned your example so that the likelihood ratio is equal to two and there are only two possible outcomes, while in general there’s no reason for those two values to be equal.