# Subjective Probability Does Not Exist

The title is an inversion of De-Finetti’s famous statement that “Probability does not exist” with which he opens his famous treatise on Probability. My paper on “Subjective Probability Does Not Exist“ (revised and updated Dec 2019), discussed below, shows that the arguments used to establish the existence of subjective probabilities, offered as a substitute for frequentist probabilities, are flawed.

The existence of subjective probability is established via arguments based on coherent choice over lotteries. Such arguments were made by Ramsey, De-Finetti, Savage and others, and rely on variants of the Dutch-Book, which show that incoherent choices are irrational – they lead to certain loss of money. So every rational person must make coherent choices over a certain set of especially constructed lotteries. Then the subjectivist argument shows that every coherent set of choices corresponds to a subjective probability on part of the decision maker. Thus we conclude that rational decision makers must have subjective probabilities. This paper shows that coherent choice over lotteries leads to weaker conclusion than the one desired by subjectivists. If a person is forced to make coherent choices for the sake of consistency in certain specially designed environment, that does not “reveal” his beliefs. The decision may arbitrarily chose a “belief”, which he may later renounce. To put this in very simple terms, suppose you are offered a choice between exotic fruits Ackee and Rambutan, neither of which you have tasted. Then the choice you make will not “reveal” your preference. But preferences are needed to ensure stability of this choice, which allows us to carry it over into other decision making environments.

The distinction between “making coherent choices which are consistent with quantifiable subjective probabilities” and actually having beliefs in these subjective probabilities was ignored in era of dominance of logical positivism, when the subjective probability theories were formulated. Logical positivism encouraged the replacement of unobservables in scientific theories by their observational equivalents. Thus unobservable beliefs were replaced by observable actions according to these beliefs. This same positivist impulse led to revealed preference theory in economics, where unobservable preferences of the heart were replaced by observable choices over goods. It also led to the creation of behavioral psychology where unobservable mental states were replaced by observable behaviors.

Later in the twentieth century, Logical Positivism collapsed when it was discovered that this equivalence could not be sustained. Unobservable entities could not be replaced by observable equivalents. This should have led to a re-thinking and re-formulation of the foundations of subjective probability, but this has not been done. Many successful critiques have been mounted against subjective probability. One of them (Uncertainty Aversion) is based on the Ellsberg Paradox, which shows that human behavior does not conform to the coherence axioms which lead to existence of subjective probability. A second line of approach, via Kyburg and followers, derive flawed consequences from the assumption of existence of subjective probabilities. To the best of my knowledge, no one has directly provided a critique of the foundational Dutch Book arguments of Ramsey, De-Finetti, and Savage. My paper entitled “Subjective Probability Does Not Exist” provides such a critique. A one-hour talk on the subject is linked below. The argument in a nutshell is also given below.

The MAIN ARGUMENT in a NUTSHELL:

Magicians often “force a card” on a unsuspecting victim — he thinks he is making a free choice, when in fact the card chosen is one that has been planted. Similarly, subjectivists force you to create subjective probabilities for uncertain events E, even when you avow lack of knowledge of this probability. The trick is done as follows. I introduce two lotteries. L1 pays $100 if event E happens, while lottery L2 pays $100 if E does not happen. Which one will you choose? If you don’t make a choice, you are a sure loser, and this is irrational. If you choose L1, then you reveal a subjective probability P(E) greater than or equal to 50%. If you choose L2, then you reveal a subjective probability P(E) less than or equal to 50%. Either way, you are trapped. Rational choice over lotteries ensures that you have subjective probabilities. There is something very strange about this argument, since I have not even specified what the event E is. How can I have subjective probabilities about an event E, when I don’t even know what the event E is? If you can see through the trick, bravo for you! Otherwise, read the paper or watch the video.. What is amazing is how many people this simple sleight-of-hand has taken in. The number of people who have been deceived by this defective argument is legion. One very important consequence of widespread acceptance of this argument was the removal of uncertainty from the world. If rationality allows us to assign subjective probabilities to all uncertain events, than we only face situations of RISK (with quantifiable and probabilistic uncertainty) rather then genuine uncertainty where we have no idea what might happen. Black Swans were removed from the picture.

POSTSCRIPT: I have now written a substantially revised and updated version of this paper. The main difference is that this version provided the COUNTER-ARGUMENT to De-Finetti and Savage, without providing their argument, assuming that this is known. I discovered that very few people know or understand the original arguments of the subjectivists, even though they accept their conclusions. So the current version of the people spells out the original arguments in complete detail, and THEN provides the counter argument. For a sequence of nine posts which discuss this whole matter in great detail, see Foundations of Probability 1, to Foundations of Probability 9.

I like your nutshell. A slight modification would be if instead of choosing a lottery, you decide to let a coin-toss choose it for you. It is then even clearer that the ‘choice’ reveals nothing about your preference. One might extend the argument further: let L3, L4 be copies of L1, L2. Which will you choose? If your first choice revealed a preference then either your subjective probability is 50% or you should choose the copy. But I would choose the other lottery, since then I am guaranteed $100.

Where the usual argument does get you, is that if you have preferences they should be probabilistic. But why should you have definite preferences?

There is no utlity to choose among two lotteries with same payoff and same zero information(subjective probablity) but, if you guarantee me 1 dollar more(101 vs 100) underlyng probabilities (information) become less important. For equal uncertainty we prefer “the one dollar more” lottery. De Finetti rules… 🙂

Regards,

The WEA just emailed me a copy of this as a ‘new comment’. I have no idea why. I’d be interested if anyone could tell me how to form a ‘subjective probability distribution’ for the possibilities of how this happened, or why anyone might take note of any such thing. 😉