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:

• Alex best 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.
• Bart also 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.
• Cedric had 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 prior gave him the best prediction for this sequence of data, followed by Bart, while Cedric was 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: