This the ninth and final post of a sequence on the Foundations of Probability. This contains the final Section of the paper on “Subjective Probability Does Not Exist.”
Not finding any black swans in their neighborhoods, Europeans came to the conclusion that black swans do not exist. “All swans are white” became a universal truth, due to lack of experience with the world. Using even worse logic, uncertainty was legislated out of existence.
Section 7: Conclusions
As Kyburg (1978) states, the theory of subjective probability “is a snare and a delusion”. De-Finetti is unique in recognizing the importance of rejecting ontic probability to the construction of a subjective theory of probability. If the real world is so wild that every moment is unique, and past patterns of events are not of any value in predicting future patterns, then ontic probability does not exist. It is only in this case that subjective probability theory can be constructed. But this theory boils down to a triviality. It says that when there are no external benchmarks for probability judgments, then we can pick any arbitrary number as the probability of an uncertain event, and use this number as the probability for our risk calculations, in the context of choices over artificially constructed lotteries. This choice of an arbitrary number for choices over lotteries does not reflect any knowledge or belief, and does not actually provide any guide to action in the real world.
To make this argument more concrete, suppose climate catastrophe make our planet inhabitable, and a spaceship is launched towards one of the hundred or so exoplanets identified as being potentially hospitable for human life. This identification was based on current human knowledge, and our inability to provide probabilities for habitability is also a result of our lack of sufficient knowledge. There are no reference events to provide a basis for rough probability guesstimates. This is the situation where De-Finetti’s “probability does not exist” is applicable. However, in this situation, construction of subjective probabilities by making arbitrary choices over lotteries will not provide us with any guidance regarding which planet should be chosen as the target. Only more powerful instruments and theories which create greater information about conditions on planets at great distances from us can provide us with knowledge about such probabilities.
The question of whether or not ontic probabilities exist, and whether they are sufficiently stable that past patterns are useful in predicting future patterns, is a question about external reality. It cannot be answered by a priori considerations, or by analysis of language, or by analysis of human cognitive capabilities. The existence of quantum probabilities shows us that probability phenomena are part of the real world. A promising analysis of probability is given by Belnap (2007), who writes that “propensities can in fact be understood as objective single case causal probabilities”. There is very strong empirical evidence that some phenomena are probabilistic, not just at the quantum level, but at the macro level. For types of events where empirical evidence can be brought to bear on probability, considerations of subjective probability are irrelevant. Only well-founded beliefs, based on empirical evidence about real world events, are relevant to analysis of ontic probabilities. Massive amounts of confusion on the topic has been created by the epistemic fallacy, which denies the existence of ontic probability based on our inability to observe and measure it.
Whether or not ontic probability exists in the quantum world has been a subject of ongoing dispute among physicists. Although current consensus is on the side of probability, Einstien famously argued against this, saying that “God does not play dice with the universe”. Suppose the binary sequence of rainfall is a pseudo-random sequence. For those who have the generating key, it is deterministic. For those who don’t know the key, it appears random. In this type of a universe, both Einstein and Bohr could be right. Even though the universe may be deterministic, because of our cognitive and computational limits, our best model for it may be probabilistic. For the purposes of our present discussion, the Einstein-Bohr dispute does not matter. Even if there are hidden variables knowledge of which would render the world deterministic, quantum probabilistic models provide an accurate match to observed phenomena, so that epistemic probabilities which reflect our state of knowledge are well defined.
Failure to understand the nature of probability has had very serious consequences for the real world. Many prominent economists have argued that collective professional failure to foresee the Global Financial Crisis, occurred because economic theories assume rational agents can form correct expectations about the future. Justin Fox provides details in his meticulously researched history of modern financial economics, The Myth of the Rational Market. Central ideas of Keynes, essential to understanding his analysis of the Great Depression, were rejected by the mainstream orthodoxy among economists because of their rejection of radical uncertainty. In their forthcoming book on “Radical Uncertainty”, King and Kay emphasize the how the forgotten distinction between risk and uncertainty creates a false sense of security.
This paper has the twin goals of explaining the strong attraction of the subjectivist position, the reasons why it continues to dominate, as well as the fatal flaws in this position. The flaws at the heart of positivism have not been clearly understood by economists, with the result that most economists continue to uphold core positivist beliefs, as the survey of Hands (2003) shows. As we have shown in detail, acceptance of positivist ideas leads to the inability to formulate the ideas necessary to clarify the errors of the argument for the existence of subjective probability. The concepts of ontic and epistemic probability, and the difference between choices and preferences are meaningless according to positivist ideas. This makes it extremely difficult to see the flaws in the arguments for the existence of subjective probability.
Anscombe, Francis J., and Robert J. Aumann. “A definition of subjective probability.” The annals of mathematical statistics 34.1 (1963): 199-205.
Ariely, Dan, George Loewenstein, and Drazen Prelec. ““Coherent arbitrariness”: Stable demand curves without stable preferences.” The Quarterly Journal of Economics 118.1 (2003): 73-106.
Ariely, Dan, and Michael I. Norton. “How actions create–not just reveal–preferences.” Trends in cognitive sciences 12.1 (2008): 13-16.
Belnap, Nuel. “Propensities and probabilities.” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38.3 (2007): 593-625.
Bhaskar, R. (2013). A realist theory of science. Routledge.
Christensen, David, ‘Clever Bookies and Coherent Beliefs.’ The Philosophical Review C, No. 2, (1991), 229-47.
Cooter, R. and Rappoport, P (1984). ‘Were the Ordinalists Wrong About Welfare Economics?’ Journal of Economic Literature, 22 (2) June, 1984, pp. 507-530.
Clark, M. P., & Westerberg, B. D. (2009). “How random is the toss of a coin?.” Canadian Medical Association journal, 181(12), E306–E308. doi:10.1503/cmaj.091733
Dawid, A. P. (2004). Probability, causality and the empirical world: a Bayes–de Finetti–Popper–Borel synthesis. Statistical Science, 19(1), 44-57.
de Elía, Ramón, and René Laprise (2005) “Diversity in interpretations of probability: implications for weather forecasting.” Monthly Weather Review 133.5: 1129-1143
De Finetti, Bruno (1937) “La prévision: ses lois logiques, ses sources subjectives.” Annales de l’institut Henri Poincaré. Vol. 7. No. 1. 1937.
De Finetti, B. (1974), Theory of Probability, Vol. 1. New York: John Wiley and Sons
Galavotti, Maria Carla (2019) “Pragmatism and the Birth of Subjective Probability.” European Journal of Pragmatism and American Philosophy 11.XI-1.
Ghirardato, P., & Marinacci, M. (2001). Risk, Ambiguity, and the Separation of Utility and Beliefs. Mathematics of Operations Research, 26(4), 864-890.
Hájek, Alan, “Interpretations of Probability”, The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), Edward N. Zalta (ed.), URL= <https://plato.stanford.edu/archives/win2012/entries/probability-interpret/>.
- W. Hands (2009). Philosophy of Economics, Uskali Mäki (ed.), Vol. 13 of D. Gabbay, P. Thagard and J. Woods (eds.), Handbook of the Philosophy of Science. Elsevier: Oxford
John Kay (2019) Embrace Radical Uncertainty Blog Post
Kay, J. A., & King, M. A. (2020) Radical Uncertainty: Decision Making Beyond the Numbers. W.W. Norton
Keynes, J. M. (1921). A Treatise on Probability. MacMillan and Co. London
Frank, Knight (1921) “Risk, uncertainty and profit.” Hart, Schaffner and Marx Prize Essays.
Kyburg, H. (1978). “Subjective probability: Criticisms, reflections, and problems.” Journal of Philosophical Logic, 7(1), 157-180.
O’Neill, B. (2011) Exchangeability, correlation and Bayes’ Effect. International Statistical Review 77(2), pp. 241-250.
Ramsey, Frank P. “Truth and probability (1926).” The foundations of mathematics and other logical essays (1931): 156-198.
Savage Leonard, J. (1954). The foundations of statistics. NY, John Wiley, 188-190.
Frederick Suppe (2000) “Understanding Scientific Theories: An Assessment of Developments, 1969-1998,” Philosophy of Science 67: S102-S115.
Nassim, Nicholas Taleb. “The black swan: the impact of the highly improbable.” NY: Random House (2007).
Wong, S. (2006). Foundations of Paul Samuelson’s Revealed Preference Theory: A study by the method of rational reconstruction. Routledge.