We are lead to post on this now because of a recent BBC Radio 4 program, More or Less, which is always about the meaning of statistics, but the Dec 30 program happened to mention the Italian statistician, Bruno de Finetti and his book, Theory of Probability, which begins thus:
PROBABILITY DOES NOT EXIST
The abandonment of superstitious beliefs about the existence of the Phlogiston, the Cosmic Ether, Absolute Space and Time, . . . or Fairies and Witches was an essential step along the road to scientific thinking. Probability, too, if regarded as something endowed with some kind of objective existence, is no less a mis-leading misconception, an illusory attempt to exteriorize or materialize our true probabilistic beliefs.So, God is dead, but what does this mean, exactly? A 2002 paper by Robert Nau on de Finetti's thesis explains that de Finetti meant that probability is nothing but a subjective analysis of the likelihood that something will happen, that probability does not exist outside the mind. That is, it's the rate at which a person is willing to bet on something happening. This is as opposed to the classicist or the frequentist's view of the likelihood of aparticular outcome of an event. That view depends on the assumption that the same event could be identically repeated many times over, and then 'probability' of a particular outcome has to do with the fraction of the time that outcome results from the repeated trials. This example from Nau's paper clarifies the differences in approach.
For example, in the case of a normal-looking die that is about to be tossed for the first time, a classicist would note that there are six possible outcomes which by symmetry must have equal chances of occurring, while a frequentist would point to empirical evidence showing that similar dice thrown in the past have landed on each side about equally often. A subjectivist would find such arguments to be suggestive but needlessly encumbered by references to superfluous events. What matters are her beliefs about what will happen on the single toss in question, or more concretely how she should bet, given her present information. If she feels that the symmetry argument applies to her beliefs, then that is sufficient reason to bet on each side at a rate of one-sixth. But a subjectivist can find other reasons for assigning betting rates in situations where symmetry arguments do not apply and repeated trials are not possible.In other words, the idea is that probability is not part of the real world, only of one's belief in the nature of that world.
What does this have to do with the risk of having a heart attack? Well, how much are you willing to bet on your chances? That is, if your chances are 20%, and you feel that's high, you might be willing to do whatever you can to reduce your cholesterol, you might take up going to the gym more regularly, quit smoking, or become a vegan. Someone else, though, might feel that 20% is not so high, and do nothing at all to alter their (what are currently considered to be) risk factors. But the physical basis of that belief is far less clear than the idea of the belief.
And, physicians advising us differ in how much they are willing to bet on the probability we'll get sick. Some are very diligent about cholesterol, some less so, some advise all men of a certain age to have a PSA test for prostate cancer, others none. They are reading the same probabilities, but what they make of them differs. Indeed, the question of interpretation is secondary to the notion of probability itself.
Yet, how do we account for the role that probabilities do play in the real world, such as in the example from which much of probability was developed, when events can be repeated: gambling. And 'bet' is the appropriate operational concept. The formal theory was largely developed in the literal context of gambling, but the same idea applies to health.
In rolling dice, we have 6 outcomes and no reason to prefer any of them (see below!). If we know about rolling, we might, in advance, decide that each possible outcome would be as likely to result. We don't know which, so we might say that in a large number of rolls, each face would come up the same number of times.
In fact, however gambling notions of probability first arose (scholars have some ideas, but we don't), by the time formal theories were being developed, there was extensive and systematic experience with past sets of rolls that actually did occur (apparently not obviously so in Roman times, where gambling with bones was thought to be related to things like how the gods viewed the gambler, etc.). We don't personally know how extensive such data, experimental or otherwise, were but the notions of equal occurrences not only seemed intuitive at the time but backed up by experience. '6' came up about 1/6th of the time in dice games, leading naturally to a theory that all sides had equal chances to arise -- fractions of the times it will arise.
Heart attack risks based on, say, cholesterol levels, are based on past experience, too. But unlike dice, people are more than simple structures. We have more than cholesterol levels. So the fraction of people with cholesterol over some level, who had heart attacks in our studies, is used to estimate the fraction of people with such level will have a coronary in the future. Yet we know very well that each person's 'ancillary' risk factors in the data on which the probability was based were different, and worse, that we simply cannot know about those risk factors in the future. So what does the genetic-risk-perveryer's probability actually mean?
We also talk in probabilistic terms when we say such things as that God probably exists (or doesn't). This clarifies the unclarity of such wording. God either does or doesn't exist, clearly without any actual 'probability', so this is really just a statement, using a serious-sounding word, of strength of belief. If you examine it closely, the same really applies to similar statements about whether we'll have a heart attack or not, or whether 5 & 6 will come up on the next roll of a pair of dice. Even those sound more rigorous, or suggest experiments or relevance to actual data, even they are based on the assumption that multiple replicates of unique events can take place, and they are based on some idea, or 'model', of the process involved such as how we measure cholesterol, how we sample people and measure their cholesterol and diets and obesity and so on), and even how dice are rolled (see next installment!).
Some statisticians and books and lectures simply assume that we know what probability is and don't attempt to define it, or to do so in practical or frequentist terms. Others try to wriggle out of this situation by saying that the frequentist terms can be disregarded and that there are other systematic ways to extract from the available data alone, some idea of what our best idea of the situation is. These are called 'likelihood' and 'Bayesian' approaches (there may be others we're not aware of), but if they stay somewhat closer to actual data, they essentially are ways to strengthen belief, and belief is in ourselves rather than a physical property of the object of belief. That is the subjectivist assertion.
Next time, we'll show how the meaning of 'probability' or the interpretation of repeated events in probability terms--even in seemingly simple cases like coin-flipping or dice-rolling--are far from clear.