But are these numbers, then, our subjective, perhaps flawed estimates of some underlying ‘true’ probability, an objective feature of the world?
I will add the caveat here that I am not talking about the quantum world. At the sub-atomic level, the mathematics indicates that causeless events can happen with fixed probabilities (although at least one interpretation states that even those probabilities express a relationship with other objects or observers, rather than being intrinsic properties of quantum objects). But equally, it seems that this has negligible influence on everyday observable events in the macroscopic world.
I can also avoid the centuries-old arguments about whether the world, at a non-quantum level, is essentially deterministic, and whether we have free will to influence the course of events. Whatever the answers, we would still need to define what an objective probability actually is.
Many attempts have been made to do this over the years, but they all seem either flawed or limited. These include frequentist probability, an approach that defines the theoretical proportion of events that would be seen in infinitely many repetitions of essentially identical situations — for example, repeating the same clinical trial in the same population with the same conditions over and over again, like Groundhog Day. This seems rather unrealistic. The UK statistician Ronald Fisher suggested thinking of a unique data set as a sample from a hypothetical infinite population, but this seems to be more of a thought experiment than an objective reality. Or there’s the semi-mystical idea of propensity, that there is some true underlying tendency for a specific event to occur in a particular context, such as my having a heart attack in the next ten years. This seems practically unverifiable.
There is a limited range of well-controlled, repeatable situations of such immense complexity that, even if they are essentially deterministic, fit the frequentist paradigm by having a probability distribution with predictable properties in the long run. These include standard randomizing devices, such as roulette wheels, shuffled cards, spun coins, thrown dice and lottery balls, as well as pseudo-random number generators, which rely on non-linear, chaotic algorithms to give numbers that pass tests of randomness.
In the natural world, we can throw in the workings of large collections of gas molecules which, even if following Newtonian physics, obey the laws of statistical mechanics; and genetics, in which the huge complexity of chromosomal selection and recombination gives rise to stable rates of inheritance. It might be reasonable in these limited circumstances to assume a pseudo-objective probability — ‘the’ probability, rather than ‘a’ (subjective) probability.
In every other situation in which probabilities are used, however — from broad swathes of science to sports, economics, weather, climate, risk analysis, catastrophe models and so on — it does not make sense to think of our judgements as being estimates of ‘true’ probabilities. These are just situations in which we can attempt to express our personal or collective uncertainty in terms of probabilities, on the basis of our knowledge and judgement.
Matters of judgement
This all just raises more questions. How do we define subjective probability? And why are the laws of probability reasonable, if they are based on stuff we essentially make up? This has been discussed in the academic literature for almost a century, again with no universally agreed outcome.
One of the first attempts was made in 1926 by the mathematician Frank Ramsey at the University of Cambridge, UK. He ranks as the person in history I would most like to meet. He was a genius whose work in probability, mathematics and economics is still considered fundamental. He worked only in the mornings, devoting his after-hours to a wife and a lover, playing tennis, drinking and enjoying exuberant parties while laughing “like a hippopotamus” (he was a big man, weighing in at 108 kilograms). He died in 1930 aged just 26, probably, according to his biographer Cheryl Misak, from contracting leptospirosis after swimming in the River Cam.
Ramsey showed that all the laws of probability could be derived from expressed preferences for specific gambles. Outcomes have assigned utilities, and the value of gambling on something is summarized by its expected utility, which itself is governed by subjective numbers expressing partial belief — that is, our personal probabilities. This interpretation does, however, require an extra specification of these utility values. More recently, it’s been shown that the laws of probability can be derived simply by acting in such a way as to maximize your expected performance when using a proper scoring rule, such as the one shown in the quiz “How ignorant am I?”.
Attempts to define probability are often rather ambiguous. In his 1941–2 paper ‘The Applications of Probability to Cryptography’, for example, Alan Turing uses the working definition that “the probability of an event on certain evidence is the proportion of cases in which that event may be expected to happen given that evidence”. This acknowledges that practical probabilities will be based on expectations — human judgements. But by “cases”, does Turing mean instances of the same observation, or of the same judgements?
The latter has something in common with frequentist definition of objective probability, just with the class of repeated similar observations replaced by a class of repeated similar subjective judgements. In this view, if the probability of rain is judged to be 70%, this places it in the set of occasions in which the forecaster assigns a 70% probability. The event itself is expected to occur in 70% of such occasions. This is probably my favourite definition. But the ambiguity of probability is starkly demonstrated by the fact that, after nearly four centuries, there are many people who won’t agree with me on that.
Pragmatic approach
When I was a student in the 1970s, my mentor, statistician Adrian Smith, was translating the Italian actuary Bruno de Finetti’s Theory of Probability. De Finetti had developed ideas of subjective probability at around the same time as Ramsey, but entirely independently. (They were very different characters: in contrast to Ramsey’s staunch socialism, in his youth de Finetti was an enthusiastic supporter of Italian dictator Benito Mussolini’s style of fascism, although he later changed his mind.) That book begins with the provocative statement: “probability does not exist”, an idea that has had a profound influence on my thinking over the past 50 years.
In practice, however, we perhaps don’t have to decide whether objective ‘chances’ really exist in the everyday non-quantum world. We can instead take a pragmatic approach. Rather ironically, de Finetti himself provided the most persuasive argument for this approach in his 1931 work on ‘exchangeability’, which resulted in a famous theorem that bears his name. A sequence of events is judged to be exchangeable if our subjective probability for each sequence is unaffected by the order of our observations. De Finetti brilliantly proved that this assumption is mathematically equivalent to acting as if the events are independent, each with some true underlying ‘chance’ of occurring, and that our uncertainty about that unknown chance is expressed by a subjective, epistemic probability distribution. This is remarkable: it shows that, starting from a specific, but purely subjective, expression of convictions, we should act as if events were driven by objective chances.
It is extraordinary that such an important body of work, underlying all of statistical science and much other scientific and economic activity, has arisen from such an elusive idea. And so I will conclude with my own aphorism. In our everyday world, probability probably does not exist — but it is often useful to act as if it does.
This article is reproduced with permission and was first published on December 16, 2024.
Down the rabbit hole
But are these numbers, then, our subjective, perhaps flawed estimates of some underlying ‘true’ probability, an objective feature of the world?
I will add the caveat here that I am not talking about the quantum world. At the sub-atomic level, the mathematics indicates that causeless events can happen with fixed probabilities (although at least one interpretation states that even those probabilities express a relationship with other objects or observers, rather than being intrinsic properties of quantum objects). But equally, it seems that this has negligible influence on everyday observable events in the macroscopic world.
I can also avoid the centuries-old arguments about whether the world, at a non-quantum level, is essentially deterministic, and whether we have free will to influence the course of events. Whatever the answers, we would still need to define what an objective probability actually is.
Many attempts have been made to do this over the years, but they all seem either flawed or limited. These include frequentist probability, an approach that defines the theoretical proportion of events that would be seen in infinitely many repetitions of essentially identical situations — for example, repeating the same clinical trial in the same population with the same conditions over and over again, like Groundhog Day. This seems rather unrealistic. The UK statistician Ronald Fisher suggested thinking of a unique data set as a sample from a hypothetical infinite population, but this seems to be more of a thought experiment than an objective reality. Or there’s the semi-mystical idea of propensity, that there is some true underlying tendency for a specific event to occur in a particular context, such as my having a heart attack in the next ten years. This seems practically unverifiable.
There is a limited range of well-controlled, repeatable situations of such immense complexity that, even if they are essentially deterministic, fit the frequentist paradigm by having a probability distribution with predictable properties in the long run. These include standard randomizing devices, such as roulette wheels, shuffled cards, spun coins, thrown dice and lottery balls, as well as pseudo-random number generators, which rely on non-linear, chaotic algorithms to give numbers that pass tests of randomness.
In the natural world, we can throw in the workings of large collections of gas molecules which, even if following Newtonian physics, obey the laws of statistical mechanics; and genetics, in which the huge complexity of chromosomal selection and recombination gives rise to stable rates of inheritance. It might be reasonable in these limited circumstances to assume a pseudo-objective probability — ‘the’ probability, rather than ‘a’ (subjective) probability.
In every other situation in which probabilities are used, however — from broad swathes of science to sports, economics, weather, climate, risk analysis, catastrophe models and so on — it does not make sense to think of our judgements as being estimates of ‘true’ probabilities. These are just situations in which we can attempt to express our personal or collective uncertainty in terms of probabilities, on the basis of our knowledge and judgement.
Matters of judgement
This all just raises more questions. How do we define subjective probability? And why are the laws of probability reasonable, if they are based on stuff we essentially make up? This has been discussed in the academic literature for almost a century, again with no universally agreed outcome.
One of the first attempts was made in 1926 by the mathematician Frank Ramsey at the University of Cambridge, UK. He ranks as the person in history I would most like to meet. He was a genius whose work in probability, mathematics and economics is still considered fundamental. He worked only in the mornings, devoting his after-hours to a wife and a lover, playing tennis, drinking and enjoying exuberant parties while laughing “like a hippopotamus” (he was a big man, weighing in at 108 kilograms). He died in 1930 aged just 26, probably, according to his biographer Cheryl Misak, from contracting leptospirosis after swimming in the River Cam.
Ramsey showed that all the laws of probability could be derived from expressed preferences for specific gambles. Outcomes have assigned utilities, and the value of gambling on something is summarized by its expected utility, which itself is governed by subjective numbers expressing partial belief — that is, our personal probabilities. This interpretation does, however, require an extra specification of these utility values. More recently, it’s been shown that the laws of probability can be derived simply by acting in such a way as to maximize your expected performance when using a proper scoring rule, such as the one shown in the quiz “How ignorant am I?”.
Attempts to define probability are often rather ambiguous. In his 1941–2 paper ‘The Applications of Probability to Cryptography’, for example, Alan Turing uses the working definition that “the probability of an event on certain evidence is the proportion of cases in which that event may be expected to happen given that evidence”. This acknowledges that practical probabilities will be based on expectations — human judgements. But by “cases”, does Turing mean instances of the same observation, or of the same judgements?
The latter has something in common with frequentist definition of objective probability, just with the class of repeated similar observations replaced by a class of repeated similar subjective judgements. In this view, if the probability of rain is judged to be 70%, this places it in the set of occasions in which the forecaster assigns a 70% probability. The event itself is expected to occur in 70% of such occasions. This is probably my favourite definition. But the ambiguity of probability is starkly demonstrated by the fact that, after nearly four centuries, there are many people who won’t agree with me on that.
Pragmatic approach
When I was a student in the 1970s, my mentor, statistician Adrian Smith, was translating the Italian actuary Bruno de Finetti’s Theory of Probability. De Finetti had developed ideas of subjective probability at around the same time as Ramsey, but entirely independently. (They were very different characters: in contrast to Ramsey’s staunch socialism, in his youth de Finetti was an enthusiastic supporter of Italian dictator Benito Mussolini’s style of fascism, although he later changed his mind.) That book begins with the provocative statement: “probability does not exist”, an idea that has had a profound influence on my thinking over the past 50 years.
In practice, however, we perhaps don’t have to decide whether objective ‘chances’ really exist in the everyday non-quantum world. We can instead take a pragmatic approach. Rather ironically, de Finetti himself provided the most persuasive argument for this approach in his 1931 work on ‘exchangeability’, which resulted in a famous theorem that bears his name. A sequence of events is judged to be exchangeable if our subjective probability for each sequence is unaffected by the order of our observations. De Finetti brilliantly proved that this assumption is mathematically equivalent to acting as if the events are independent, each with some true underlying ‘chance’ of occurring, and that our uncertainty about that unknown chance is expressed by a subjective, epistemic probability distribution. This is remarkable: it shows that, starting from a specific, but purely subjective, expression of convictions, we should act as if events were driven by objective chances.
It is extraordinary that such an important body of work, underlying all of statistical science and much other scientific and economic activity, has arisen from such an elusive idea. And so I will conclude with my own aphorism. In our everyday world, probability probably does not exist — but it is often useful to act as if it does.
This article is reproduced with permission and was first published on December 16, 2024.