Sunday, 16 June 2019

Laplace's rule of thumb

Laplace's rule of succession is only a rule of thumb. The idea is that since probabilities cannot be zero or one, if one has seen a hundred white swans and no black swans, one should not assign a probability of 100/100=100%. Instead, one assigns 100/101 = 99.01%. n/n+1, not n/n.

This is more sensible than assigning 100% because it is open to the possibility of error, and makes it possible to change your mind. If your prior probability is zero or one, no amount of evidence can change your mind according to Bayes' theorem.

However, in the real world, events are rarely exclusive. There are many alternatives to a hypothesis. We should not assign equal probability to the alternatives. For example, if you had only seen five white swans, there might exist red, green, blue, orange or black swans. According to a naive version of Laplace's rule, we would expect a random swan to be white with only 50% probability, assigning 10% each to red, green, blue, orange and black.

Clearly, a more sensible prediction for the colour of a random adult swan in Europe is more like 99% white, 1% black and close to zero for red, green, blue and orange, even if you have only seen fifty swans in your lifetime.

No comments:

Post a Comment