I need help: Device of imaginary results by I J Good

by BT_Uytya 1 min read5th Apr 201318 comments

5


In the chapter 5 of the Probability Theory: Logic of Science you can read about so-called device of imaginary results which seems to go back to the book of I J Good named Probability and the Weighing of Evidence.

The idea is simple and fascinating:

1) You want to estimate your probability of something, and you know that this probability is very, very far from 0.5. For the sake of simplicity, let's assume that it's some hypothesis A and P(A|X) << 0.5

2) You imagine the situation where the A and some well-posed alternative ~A are the only possibilities.

(For example, A = "Mr Smith has extrasensory perception and can guess the number you've written down" and ~A = "Mr Smith can guess your number purely by luck". Maybe Omega told you that the room where the experiment is located makes it's impossible for Smith to secretly look at your paper, and you are totally safe from every other form of deception.)

3) You imagine the evidence which would convince you otherwise: P(E|A,X) ~ 1 and P(E|~A,X) is small (you should select E and ~A that way that it's possible to evaluate P(E|~A,X) )

4) After a while, you feel that you are truly in doubt about A: P(A|E1,E2,..., X) ~ 0.5

5) And now you can backtrack everything back to your prior P(A|X) since you know every P(E|A) and P(E|~A).

 

After this explanation with the example about Mr Smith's telepathic powers, Jaynes gives reader the following exercise:

Exercise 5.1. By applying the device of imaginary results, find your own strength of
belief in any three of the following propositions: (1) Julius Caesar is a real historical
person (i.e. not a myth invented by later writers); (2) Achilles is a real historical person;
(3) the Earth is more than a million years old; (4) dinosaurs did not die out; they are
still living in remote places; (5) owls can see in total darkness; (6) the configuration of
the planets influences our destiny; (7) automobile seat belts do more harm than good;
(8) high interest rates combat inflation; (9) high interest rates cause inflation.

I have trouble tackling the first two propositions and would be glad to hear your thoughts about another seven. Anybody care to help me?

(I decided not to share details of my attempt to solve this exercise unless asked. I don't think that my perspective is so valuable and anchoring would be bad.)

 

UPD: here is my attempt to solve the Julius Caesar problem.

5