Bayes’ Theorem: Some Intuitive Principles, Part II

In my last post, I began to explain the practical implications of Bayes’ theorem in terms of simple principles. My first principle was this:

Principle I:
All things being equal, the theory which predicts the a higher likelihood for the observed data is the theory most likely to be true.

Here is the second principle:

Principle II:
All things are not equal because evidence accumulates.

I explained the first principle with the aid of a dice game. At random, I picked either a 4-sided die or a 20-sided die, rolled the die, and reported that I rolled a 3. Then I asked which die I probably rolled, and showed that a rational observer concludes that there was an 5 in 6 chance that I rolled the 4-sided die.

To illustrate my second principle, let’s change the game a little. Suppose that I roll the chosen die 10 times instead of just once. I randomly choose a die, then roll it ten times, reporting each roll to you.

Now, suppose that I report the following results: 4, 1, 1, 2, 4, 3, 2, 3, 1, 4. After getting these results, now how confident are you that I selected the 4-sided die?

As before, we are aware that it is possible to have produced results like this on the 20-sided die, assuming that conditions (or luck) had led me to roll all my numbers 4 or less. However, the odds of rolling 4 or less in 10 rolls of a 20-sided die are approximately 1 in 10 million (1 in 9765625, to be precise).

So, the answer is that we are now extremely confident that our 4-sided die theory is the correct one.

Do you see what happened? We went from 83.33% confidence after 1 roll of a number four or less to about 99.99999% confidence after rolling 10 numbers four or less on that chosen die.

This is how inductive inference works. We accumulate evidence, and while alternative theories could still be true, we accumulate confidence in our best theories to the exclusion of other, less-specific theories.

A good example of this sort of inference is provided by a typical murder trial. The court presents evidence that the suspect was at the scene at or near the time of the murder. This by itself will shift our confidence in the suspect’s guilt, but not enough to convict the suspect. If we later hear that the suspect owned the murder weapon, had reason to kill the victim, and left fingerprints and DNA evidence at the scene, then the evidence accumulates to the point that we have high confidence in the guilt of the suspect.

Before I get to the third principle, here’s a question to think about. Return to our dice game. Suppose that I roll my chosen die an 11th time, and I report that I rolled a 14. Now, what are the odds that I had chosen the 4-sided die?