Sekhar Ramakrishnan writes:
I wanted to relate an episode of informal probabilistic reasoning that occurred this morning, which I thought you might find entertaining.
Jan 6th is the Christian feast day of the Epiphany, which is known as Dreikönigstag (Three Kings’ Day), here in Zürich, Switzerland, where I live (I work at ETH). There is a tradition to have a dish called three kings’ cake in which a plastic king is hidden in one of the pieces of the cake. Whoever finds the king gets some privileges that day (like deciding what’s for dinner).
Two years ago, the bakery we get our three kings’ cake from decided to put a king in *every* piece of cake. They received many complaints about this, and last year they returned to the normal tradition of one king per cake. Today, we were speculating on whether they were going to try their every-piece-a-king experiment again this year.
My 12-year-old son picked the first piece of cake: he had a king! He said, “It looks like they probably did put a king in every piece again this year.” We had a cake with 5 pieces, so, assuming that one king per cake and five kings per cake are equally likely, I get a posterior probability of 5/6 that there was a king in every piece. I thought it was interesting that my son intuitively concluded that a king in every piece was more likely as well, even though he hasn’t had any formal exposure to statistics or statistical reasoning.
As it turns out, though, there was only one king in the cake — my son just got lucky!
Indeed, there are some settings where probabilistic reasoning is intuitive.
This a fun idea.
OK. But I really dislike that 50/50 prior. While a twelve year old wouldn’t be expected to understand it, his parent probably should have reasoned that small businesses like bakeries are unlikely to repeat a counter-traditional practice which brought opprobrium on them from their customers previously. I would have put the prior as something more like 95/5 in favor of just one king.
This, and it nicely illustrates the meaning of “prior information”.
This reminds me of Richard Royall’s book on likelihood inference which was very influential on me. He gives the example of taking a deck of cards, pulling out a card, getting, say, the 7 of hearts, and then pointing out that on a pure likelihood basis, the odds of a normal deck versus a deck consisting only of the 7 of hearts is 1:52. The likelihood of a deck consisting entirely of 7s of hearts has jumped 52-fold. He then discusses the importance of the prior, which ought to be different if you bought the deck in a store versus from a magician, for example.
Nothing has jumped. Only if he had mentioned a deck consisting only of the 7 of hearts before pulling out this specific card…
He covers that objection. Yes, you need two hypotheses to create a likelihood ratio, but there is no reason that you have to pick those hypotheses before you see the data. Now, whether someone is going to be *convinced* by forking paths hypotheses you only created after you saw the data is obviously very problematic — no one is. But Royall argues (to me) persuasively that likelihood ratios are just calculations.
Jonathan:
Yup, the likelihood ratio is what it is (conditional on the model being correct). What to do about it is another story.
Seven months for the blog lag. That’s my guess.
My money’s on -22 days.
Is the kid’s intuition actually any more interesting than if it had been wrong? The kid had a 50-50 chance of his intuition being consistent with formal probability. At least, it *seems* that way until you realize he was born on a Swiss holiday when hospitals deliver to every family a intuitive baby!