Prediction vs Explanation: A Puzzle
We do ten experiments. A scientist observes the results, constructs a theory consistent with them, and uses it to predict the results of the next ten. We do them and the results fit his predictions. A second scientist now constructs a theory consistent with the results of all twenty experiments.
The two theories give different predictions for the next experiment. Which do we believe? Why?
In case the puzzle isn't obvious, let me offer the straightforward argument for what I believe is the wrong answer:
Imagine a large room filled with barrels, each of which contains a lot of boxes, each of which contains several pieces of paper; each piece of paper has a scientific theory written on it. The first ten experiments let us eliminate all but one barrel, the one containing the theories consistent with the first ten experiments. The first scientist has reached into that barrel, pulled out a box, opened it, pulled out a piece of paper, and offered its theory as his.
The second ten experiments narrow the possibilities down to one box in the barrel--the one containing theories consistent with the results of the second set of experiments. The second scientist, having the results of both sets of experiments, knows which box to go to; he opens it and pulls out a piece of paper. Both pieces of paper, both theories, were selected from the same box, the one that we know contains the correct theory, since the correct theory must be consistent with all the experiments. Having been pulled from the same box, both theories should have the same probability of being the right one.
What is wrong with this argument?
(Hint: I find the concept of "false contagion" in statistics useful in making sense of the puzzle)