Couple of folks (Mike W, Dan Hughes) asked me to comment on the article “The concept of probability is not as simple as you think” by Nevin Climenhaga.
Three popular theories analyse probabilities as either frequencies, propensities or degrees of belief. Suppose I tell you that a coin has a 50 per cent probability of landing heads up. These theories, respectively, say that this is:
- The frequency with which that coin lands heads;
- The propensity, or tendency, that the coin’s physical characteristics give it to land heads;
- How confident I am that it lands heads.
These are three of the biggies, it’s true. Climenhaga gives examples of the quirks and difficulties of these definitions.
Climenhaga’s first example starts with “Adam flips a fair coin that self-destructs after being tossed four times.” He then shows how confusion enters, but I don’t think he saw how he introduced some confusion with that “fair”, for it follows that saying a coin is “fair” is saying it has a probability of coming up heads 50% in the propensity definition; that is, “fair” implies propensity.
Yet he also takes it that since the coin came up three out of four times propensity implies the probability is 3/4. But if that were so if it came up zero times out of four, or even out of one, the probability would be 0.
There is a mathematical definition of frequency, which involves limits of subsequences embedded in infinite sequences which we needed not detail. The math is fine, but do such sequences or even subsequences exist in actuality rather potentially? No. That makes using a probability-as-frequency impossible because no unique sequences exist for anything (especially coin flips).
The other difficulty with frequency is this. Suppose we have four beings in a room, A-D, and all are the same species. A-C are Martians. What is D? Martian, too. Logic demands this and nobody disputes that. But there is no frequency because there are no Martians. Yet we can still do logic, and we can still do probability. Change the first premise to four beings, A-D, three Martians and one Venusian. Then what is probability D is a Martian? Simple: 3/4. But there is no frequency again, because there is no sequence, and there are no aliens.
Confidence it closer to the mark, but it has whiffs of subjectivism, which would mean the amount beans you had in your stomach changes the probability.
Climenhaga and I agree with what probability is: degree-of-support. Which is to say, logic. My own views and proofs and full arguments are in this award eligible book. Here’s Climenhaga:
Here, probabilities are understood as relations of evidential support between propositions. ‘The probability of X given Y’ is the degree to which Y supports the truth of X. When we speak of ‘the probability of X’ on its own, this is shorthand for the probability of X conditional on any background information we have. When Beth says that there is a 50 per cent probability that the coin landed heads, she means that this is the probability that it lands heads conditional on the information that it was tossed and some information about its construction (for example, it being symmetrical).
Yep. This also implies, as I believe, there is no such thing as “The probability of X” on its own; i.e. there is no unconditional probability.
Because they turn probabilities into different kinds of entities, our four theories offer divergent advice on how to figure out the values of probabilities. The first three interpretations (frequency, propensity and confidence) try to make probabilities things we can observe — through counting, experimentation or introspection. By contrast, degrees of support seem to be what philosophers call ‘abstract entities’ — neither in the world nor in our minds.
We disagree about that, for I say in our minds is exactly where probability is.
Suppose we’re on a jury. How are we supposed to figure out the probability that the defendant committed the murder, so as to see whether there can be reasonable doubt about his guilt?…
…Here we are concerned with the probability of a cause (the defendant committing the murder) given an effect (his fingerprints being on the murder weapon). Bayes’s theorem lets us calculate this as a function of three further probabilities: the prior probability of the cause, the probability of the effect given this cause, and the probability of the effect without this cause.
I’d state this differently. I say that if we knew the cause, then we don’t need the probability; which is to say, the probability is easy and extreme, either 0 or 1, depending on the proposition (“He did it”, “He’s innocent”).
And, of course, we don’t need Bayes per se. It’s a handy tool, but it’s not strictly necessary. If you want to go hard into this, you can read about presuming innocence and Bayesian priors.
Let me clarify: probability is a measure of our understanding of cause. Cause has four aspects: there is the formal cause, material, efficient, and final. The more we know of these the closer the probability is to 1. The less we know, the fuzzier the probability. As homework, do the Martian/Venusian example with respect to cause (hint: authors cause).
There’s a whole collection of articles on the subject here.