It seems such a straightforward idea: evidence confirms a theory precisely if it makes the theory more likely to be true. But once we translate this into probability theory terms, we tend to run into weird puzzles and unexpected complications.
Some of those puzzles are spurious, because they are due to the artificial constraint on probabilities that they have to sum to 1. Those puzzles disappear if we think instead in terms of odds, as the punters do at the racetrack.
My first example: could a conjunction be confirmed, even though neither conjunct is confirmed?
You might think that if the answer is YES, there must be something wrong. Let’s run the numbers.
After coming home from a long trip I was told a little about the coming weather. Tomorrow we may have rain or sleet or both, or the day may be dry. I haven’t been here recently and haven’t check the weather history — so I take each of the four possibilities to be equally likely.
Fine. Now a little later I am told that you can’t have one without the other: if there is rain there is sleet, and if there is sleet there is rain. I update, I conditionalize on this information. Now, were any of the following hypotheses confirmed by this new evidence?
A: Tomorrow it will rain
B: Tomorrow it will sleet
A & B: Tomorrow it will both rain and sleet.
Answer: surprisingly, yes, the conjunction (A & B) is confirmed, its probability doubles, while neither conjunct is confirmed.
Second example: could a conjunction be confirmed although both conjuncts are disconfirmed?
In the following diagram A and B and their conjunction receive different initial probabilities. Then the same thing happens: the new evidence is that you can’t have one without the other.
The top diagram is a ‘Muddy Venn diagram’ to represent the initial probabilities. Think of the probability mass as very fine mud, heaped proportionately on the different areas representing propositions. So a proportion 4 out of 20 of the mud (one-fifth) is heaped on the (A & B) area.
Now conditionalization is effected simply by wiping all the mud off the areas representing excluded possibilities. Replace the relevant numbers by 0. Of course the total of the probability mud has changed, we need to renormalize.
In this example, the probabilities for A and for B decreased, while the probability for (A & B) increased. The conjunction was confirmed, while each conjunct was disconfirmed.
All of this is by the ‘official’ definition of ‘confirm’ in probability terms. But if we think about these examples in terms of odds, we can see that nothing strange and significant was going on at all.
The remaining possibilities, in each example, were just: both rain and sleet, neither rain nor sleet. In the first example their odds were 1:1, and remained 1:1. In the second example their odds were 4:6, and remained 4:6
(which is of course the same as 2:3, for odds, unlike probabilities, are not constrained to sum to a specific total).
It is easy to list odds for a problem situation, focusing on the ‘ultimate’ partition of possibilities (ultimate for the particular questions at issue, in context). So these two examples would be described as follows:
This different way of looking at the situation can remove spurious puzzles.
All that happened here was that some possibilities were eliminated, and this did not change any significant relationship among the possibilities that were not eliminated.
Of course probabilities have not disappeared. The question “what is the probability of A?” is equivalent to the question “what are the odds of A to ~A?”
The shift to thinking in terms of odds will remove spurious difficulties; it will not affect significant results.
For example, if you look back to the diagram in the earlier post “Subjective probability and a puzzle about theory confirmation” you can determine the numbers (percentages) to be placed in the four main areas. Starting top left and going clockwise, those are 35, 35, 15, 15. When evidence is taken, conditionalizing on B, that changes to 0, 35, 15, 0. The odds of A to ~A change from 30:70 to 15:35, but that is the same thing.
Bas van Fraassen's philosophy blog 