In the the above-titled article, David Spiegelhalter writes:
Attempts to put numbers on chance and uncertainty take us into the mathematical realm of probability, which today is used confidently in any number of fields. . . . And yet, any numerical probability, I [Spiegelhalter] will argue — whether in a scientific paper, as part of weather forecasts, predicting the outcome of a sports competition or quantifying a health risk — is not an objective property of the world, but a construction based on personal or collective judgements and (often doubtful) assumptions. Furthermore, in most circumstances, it is not even estimating some underlying ‘true’ quantity. Probability, indeed, can only rarely be said to ‘exist’ at all. . . .
I agree and disagree with this statement.
I agree that “probability doesn’t exist,” in the same sense that “real numbers don’t exist” and “straight lines don’t exist” and “angles don’t exist.” First off, none of these concepts exist in isolation; they’re defined as part of a larger structure. Beyond that, “probability,” “real numbers,” “straight lines,” and “angles” are fictitious; they’re part of mathematical models. Mathematical models are constructs that we use to understand the world. If you draw a “triangle” on the ground with three “straight lines,” then their “angles” will add up to “180 degrees”–except they won’t, because the Earth is not “perfectly flat.” Mathematical models are not reality; they’re tools that we use to describe reality.
We can use mathematical models to make deductive inferences, which is useful for two reasons:
1. We can use these inferences to make decisions and predictions,
2. We can compare these predictions to reality and use discrepancies to inform improvements to our models.
Where I disagree, implicitly, with Spiegelhalter is in his singling out of “probability” for special treatment. Everything he writes about “not an objective property of the world, but a construction based on personal or collective judgements and (often doubtful) assumptions” could apply to just about any area of mathematics.
Spiegelhalter also talks about subjectivity. For that, I refer you to my article with Christian Hennig published in 2017, Beyond subjective and objective in statistics.
What is subjective about basic calculations like chance of flipping three heads in a row or rolling a 7?
The notion that the basic calculation is somehow related to the physical act of flipping a coin three times and the assumption that all the outcomes are equally likely (as we also know that the outcome is a necessary consequence of the flip).
Usually there is an appeal to physics with these arguments. You can use classical mechanics to predict your favorite game of chance and reduce it to determinism.
Not sure how quantum mechanics gets treated however. From what I recall probability is baked in.
Here is a non-paywalled version:
https://blog.kowatek.com/2024/12/26/why-probability-probably-doesnt-exist-but-its-useful-to-act-like-it-does/
Say there are three apples on a table, but I hide one under a bowl then ask someone else to count how many apples are on the table. They will count 2, while I count 3. Will we consider simple counting of items subjective as well?
> Will we consider simple counting of items subjective as well?
I suppose it depends on what you mean by subjective here, but your apple example demonstrates that one source of subjectivity is the interpretation of the meaning of the question. If I answer 2, then it’s not hard to believe that I interpreted your question as “how many apples can you see on the table?”. Interpretation of the question being asked is a source of ambiguity and subjectivity can creep in.
Now, you could argue that I am wrong in that interpretation and clarify exactly what you mean, that the question is “how many apples are on the table, seen or unseen?”. Now I can recognize that my information about the state of what is on the table is incomplete. If I see two apples clear as day and a portion of the table obscured by a bowl, then I have to make a subjective assessment about the probability that additional apples are hidden by the bowl. There is no straightforwardly objective way to assess P( apple under the bowl ), one would have to assign some degree of belief. One could interpret my subjective answer to your question as “I believe that there are 2 apples under the table with probability p and 3 apples with probability 1 – p”.
Now I take issue with the author claiming that the coin flip is an example of aleatoric uncertainty, when it clearly is an example of epistemic uncertainty about the initial conditions of the physical system of the coin flip.
I’m not sure how this logic extends to projections, but let’s say I have *already* flipped a coin but closed my fist over the coin so that no-one (including myself) has seen the outcome (which is how the question is framed in the article). What is the probability the coin landed heads?
The intuitive answer is 50% since (we assume) each side has an equally likely outcome. And that’s a perfectly rational answer. But some would argue, hey, the coin has already landed – therefore the “chance” it has landed heads is already certain. The fact that no-one has seen the outcome is not important; it’s already happened, and our perception is just that – our perception.
Therefore, the “50%” answer is based purely on perception – i.e., subjectivity. The actual *objective* answer is either 0% or 100%.
This is a super tricky part about teaching statistics. When we fit a regression to existing data, we are making the counterfactual move to consider that the data might have been otherwise with some probability, even though we have observed how it actually turned out. So depending on what we’re doing, we might answer Matt Skaggs’s question below, “What is the probability that the Mariners won their baseball game last night?” differently. If we’re fitting it into a Bradley-Terry model of team abilities, it still makes sense to treat the probability as something other than zero (Seattle lost 7–5 to the Royals last night—they’re battling Cleveland, for whom Mitzi and I are rooting, so from today’s perspective, there’s no uncertainty).
Thanks all. Now we need an example of something that is *not subjective*, using the same definition.
Geiger counter readings.
That is the best example of a probability being something physical, as predicting individual nuclear decays is hopelessly difficult. But it could be like a complicated coin toss.
I guess I don’t understand “subjective” then. Looking at apples on a table can be subjective but not counting click sounds. And no two Geiger counters will have the exact same counts either.
It really seems like a meaningless term in this context, like “doesn’t exist”.
I can do things with probability that I cannot do with angles and lines, especially by playing with time:
Given all the possibilities, what are the odds that the universe would turn out exactly the way it did? [Answer: unity or an infinitesimally small number, your choice]
What are the odds that the Mariners won yesterday? [Answer: Um, did you look at the paper this morning? Be honest!]
What are the odds that the last Concord ever flown would crash? [Answer: I would say unity, but pretty much any damn number works just fine here, because the reader won’t know what it means anyway.]
> What are the odds that the Mariners won yesterday?
Bad example, we know this to be 54%.
The difference stated formally: probability lacks symmetry at t = 0, and is undefined at t > 0.
The triangle drawn on the ground is the same after I measured it as it was before, and the measurement result is the same today as it was when I made it yesterday, but the numerical probability of a Mariner’s win in yesterday’s game changed from .54 to 1.0.
This notion goes back at least a couple centuries.
I usually just quote this. Laplace was a pre-postmodernist in that regard.
If you want more elaboration, John Stuart Mill bounces back and forth across eight editions of his book on logic, but eventually agrees with Laplace.
P.S. I don’t understand why Andrew thinks Spiegelhalter is singling out probability here as I don’t see anything in Spiegelhalter’s words calling out probability as special. The real question here is whether probability is something we learn about or something we construct, and this same issue comes up with respect to the rest of mathematics and all of human language and its relation to the world.
And “something we learn about and something we construct” need not be mutually exclusive alternatives.
A special aspect of probability is that probability is essentially always not only about what actually happens or is the case, but also about what doesn’t happen or isn’t the case but could’ve.
Regarding “probability doesn’t exist” I’m always surprised that nobody seems to find the term “existence” problematic. If probability “existed”, what would that actually mean? Would we need to make reference to parallel worlds in which the things happen that could happen in ours but don’t?
The quantum mechanics many-worlds enthusiasts make a mess of this issue. They like to talk about things happening in parallel worlds. But as you say, probability is also about what does not happen, and in many-worlds theory everything happens. So they have to reject the whole notion of probability as it is understood here. You might think that they are assigning probabilities to different worlds, but no, they do not do that.
This blog singles out the social sciences for “special treatment” :)
Andrew writes, “Everything he writes about “not an objective property of the world, but a construction based on personal or collective judgements and (often doubtful) assumptions” could apply to just about any area of mathematics.”
James Franklin’s book An Aristotelian Realist Philosophy of Mathematics is really excellent. He takes the opposing, realist position. He provides a number of interesting examples, like these:
“Mathematics provides, however, many prima facie cases of necessities that are directly about reality. One is the classic case of Euler’s bridges…Euler proved that it was impossible for the citizens of Konigsberg to walk exactly once over (not an abstract model of the bridges but) the actual bridges of the city.
“To take another example: it is impossible to tile my bathroom floor with (equally sized) regular pentagonal lines.”
And what gets into is that his bathroom floor has the shape of a nearly-Euclidean plane, and its a provable proposition of geometry that one can’t tile a nearly-Euclidean plane with nearly-pentagonal tiles.
But to be clear Franklin does not say that probabilities are real (he advocates the typical realist position…that probabilities are part of epistemology).
Next up: “Why a true null effect doesn’t exist (but it is useful to act like it does)”? Ha!