Two faces of probability

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was produced by the Winton programme for the public understanding of risk based in the Statistical Laboratory in the University of Cambridge. The aim was to help improve the way that uncertainty and risk are discussed in society, and show how probability and statistics can be both useful and entertaining.

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One of the curious features of probability is that in everyday speech it is used in two rather different circumstances: first, when there is some essential randomness, and second, when our uncertainty is due to lack of knowledge.

This central ambiguity is what Hacking (1975 p.12) Hacking1975 calls the “Janus-faced” nature of probability, after the Roman god with two faces. On the one hand probability is seen as something objective and due to the essential lack of predictability in the situation, such as before throwing a die, tossing a coin and so on. On the other hand probability may express our belief in whether something is true or not, which may be dependent on our judgments and the state of our knowledge, such as when a die has been tossed but covered up.

This “Janus-faced” aspect of probability is an issue that is still far from resolved, though in most practical applications of probability the distinction does not make much difference as the mathematics is unaffected. But two rival schools of statistics, known as "frequentist" and "Bayesian", broadly follow from the two different philosophical interpretations (though the precise relation between the statistical work on Bayesianism and frequentism and their philosopical interpretations of the same name are slightly complicated and the statistical schools should not be directly equated with the philosophical interpretations.)

The two faces are a feature of almost any discussion on the philosophical foundation of probability. Unfortunately philosophers have not been consistent in naming them. One of the options is to follow Karl Popper Popper1934 and talk about “objective” versus “subjective” probabilities, though this faces the problem that the “subjective” interpretation itself has traditionally been split into an objective and a subjective side. Hacking (1975) instead talks of “aleatoric” probabilities (derived from the latin word for dice), and “epistemological” probabilities (epistemology meaning the study of how we find out about the world). Gillies (1973) Gillies1973 calls them “scientific” and “logical” probabilities, though he later (Gillies 2000) Gillies2000) calls them "objective" and "epistemological" . Probabilities based on essential randomness have also been termed "ontological" or "phenomenological".

As an illustration of the two interpretations, the early 19th century mathematician Laplace gives an interesting example of the two faces of probability which I think neatly illustrates the quite considerable effect that philosophical interpretation can have on the estimation of probability. Imagine I am given a coin that I am told is biased, though I don't know which way. What should be my assessment of the probability of throwing heads? According to a subjective interpretation, since I have no reason to favour one side or the other, I should give it a probability of ½. On the other hand I can interpret the probability as something objective and inherent in the situation. Since I know the coin is biased, but I don't know which way, I simply can't say what the probability is; I can however say that it definitely is not ½. (Example quoted in Gillies 2000)

Probabilities that we face in everyday life are often combinations of the two types of probability, depending on the circumstances and individual interpretation. Consider the following questions and try to decide whether the uncertainty in each situation is aleatoric (essential randomness) or epistemic (due to our insufficient knowledge of the situation) , or possibly a combination of the two.

  1. Which numbers should I choose for the lottery?
  2. Which lottery scratch-card should I buy?
  3. Was there a conspiracy to kill Princess Diana?
  4. What cards will I have in my bridge hand?
  5. Should I draw another card in blackjack?
  6. Has this patient got cancer?
  7. Might I have a stroke or heart attack in the next 10 years?