Say that I have a hat that contains 5 dice. One die has 4 sides numbered
1-4, one has 6 sides numbered 1-6, one has 8 sides numbered 1-8, one has 12
sides numbered 1-12 and one has 20 sides numbered 1-20. I pick a die out of
the hat without showing it to you and role it 4 times. The results of the 4
roles are 1, 6, 4, and 7. Based upon that information, you need to guess
which of the 5 dice I picked out of the hat. Let’s say 2 students, Albert
and Brian, answer that question. Albert guesses that it was the 8-sided die
that I was rolling. Brian guesses that it was the 12-sided die.
We ask Albert why he guessed 8-sides. He explains that since there was a
7 rolled, it couldn’t have been the 4-sided or the 6-sided die that I
chose—neither of those has a 7. So it had to be the 8, 12, or 20. Of those
three, the 8-sided die is the most likely one to show a 1, 6, 4, and 7. If
it was the 12-sided die, there probably would have been a role greater than
8. More so with the 20-sided die. Using this reasoning, the 8-sided die is
most likely die.
We then ask Brian why he chose 12. He said that he chose it because 12 is
his lucky number.
Now, which student had the better answer to the question, Albert or
Brian? My contention is that Albert had the better answer. Albert’s answer
is based upon a valid statistical argument. Brian’s is based upon a
haphazard guess. Answers based upon valid statistical arguments are always
better than answers based upon haphazard guesses—valid statistical reasoning
is a superior problem solving method than the method of haphazard guessing.
Let’s say that after both people make their guesses I show the die. It
turns out that Brian was right—the die has 12 sides. Even though Brian
turned out to be right, Albert’s method was still superior. *It is better to
arrive at the wrong answer through a better method than arrive at the right
answer through an inferior method. *
This is the point of the text book Loss Models when it said:
An important point is that quality is a property of the estimator and not
of the estimate. We are interested in the quality of the method, not the
quality of any particular outcome from using the method. Use of a
high-quality estimator does not ensure that realized outcomes will be
consistent with estimated outcomes. (*Loss Models: From Data to Decisions* by Klugman, Panjer, & Willmot. page 26)
What is the point of this you ask? The point is this: the church tells
you that you should "firmly believe" something, with little regard to how
you arrive at that belief. In contrast, I contend that it isn’t the ultimate
belief that matters, but rather the method used to derive it. Even if it
turns out that the die has 12 sides, an outcome that is certainly possible,
I’d still rather be Albert, knowing that I made the best decision with the
information I had available at the time. |