Nursery Rhymes for Curious Minds

Lessons in Bayesian Statistics: The Queen of Hearts messes up her Tarts

The Queen of Hearts,

she made some tarts,

one chilly winter’s night;

she took a bite, frowned and said

they didn’t taste quite right.

The Queen of Hearts,

suspected that bad butter was the cause,

since if the butter had been bad,

her tarts would also be, of course!

She paused and wondered if those chances truly were the same,

since bad butter could not really be the only thing to blame.

She knew the root could also be the flour, milk, or salt,

so the chances were not very high that the butter was at fault.

But if she’d known the butter tasted bad right from the start,

It’s almost sure she would have made an awful-tasting tart.

When musing as to what could have been the cause of her terrible tarts, the Queen of Hearts initially makes a rookie error. She treats

• The chance that her tarts’ bad taste means the butter she used was off, and
• the chance that if she had been given off butter, the resulting tart would taste bad;

as though they were the same, and as she later correctly points out, they’re not! The Queen starts with her (prior) belief that her bad tarts are likely caused by bad butter. However, I’ve got a bit more information to add to the story:

• The Knave of Hearts, a perpetual agent of chaos, was noticed skulking around the grounds. It should be noted that the tarts’ bad taste could have been caused by all manner of different things, from milk of dubious origins to a suspicious swapping of sugar and salt.
• The Queen of Hearts had her stocks of butter freshly delivered that day.

So, yes, even though it's very likely that rancid butter will result in a terrible tart, the extra information about the Queen’s newly replenished dairy stocks and speculation of a t-artful sabotage moves her to change her thoughts around the chance that bad butter caused her woes.

The Queen takes up a new (posterior) position where it is unlikely that her poor-tasting tarts mean the butter she used had gone bad.

In short:

• In general, P(bad tart given bad butter) is not equal to P(bad butter given bad tart)
• (As far as a Bayesian approach is concerned) New information means we need to update our beliefs.