Three friends –Alex, Bart, and Cedric– each assign their own prior distribution to a binomial chance parameter θ. Let’s say that θ is the chance that Harriet bakes a *vegan* pancake rather than a bacon pancake. Alex assigns θ a beta(300,3) prior distribution, indicating a strong belief in high values of θ (i.e., Alex predicts that almost all of Harriet’s pancakes will be vegan). Bart assigns θ a beta(100,3) prior distribution, also indicating a strong belief in high values of θ (but less strong than that of Alex). Finally, Cedric assigns θ a beta(3,2) prior distribution, reflecting a very weak belief in high values of θ (i.e., Cedric has little information and refuses to issue precise predictions). For your convenience, the prior distributions are visualized below.

After specifying the prior distributions, Harriet starts baking pancakes. *All* of her pancakes turn out to be vegan. Harriet is quite industrious and who does not like a tasty pancake? Consequently, the unbroken sequence of vegan pancakes soon runs into thousands, hundreds of thousands, and then *millions*. The question is this: which of the three friends best predicted the observed unbroken sequence of vegan pancakes? The answer will be presented next week.

## The answer from ChatGPT-4-turbo

When presented with the above scenario, ChatGPT-4-turbo drew the following conclusion:

Alexbest predicted the unbroken sequence of vegan pancakes. His prior Beta(300,3) was heavily concentrated near θ = 1, indicating he was already expecting almost all pancakes to be vegan.Bartalso had a strong prior belief in θ ≈ 1, but it was not as concentrated as Alex’s. So, Bart predicted the sequence well, but not as well as Alex.Cedrichad a weak prior belief in high values of θ, meaning his predictions were far less aligned with the observed data. It took Cedric much more data to adjust his belief toward θ = 1.In summary,

Alex’s priorgave him the best prediction for this sequence of data, followed byBart, whileCedricwas the least aligned with the observed sequence of vegan pancakes.

ChatGPT-4-turbo sounds eminently sensible. On the other hand, if this was the correct intuition, I would hardly have bothered to write this post. Right?

## The prior distributions from Alex, Bart, and Cedric

A stacked density plot may help obtain the proper intuition for the three prior beta distributions of interest: