Confused by the Odds: How Probability Misleads Us by David Lockwood

Author:David Lockwood
Language: eng
Format: epub
ISBN: 9798886450040
Publisher: Greenleaf Book Group Press
Published: 2023-01-15T00:00:00+00:00

A Theory That Must Be Wholly Rejected

In addition to his beliefs concerning what he called “improving the human stock,” Fisher was also a vehement opponent of the application of Bayes’ theorem to problems that involve probability.11 In his landmark 1925 book, Statistical Methods, there is no favorable mention of Bayes’ theorem. On the contrary, Fisher wrote that “the theory of inverse probability is founded upon an error and must be wholly rejected.”12 Fisher believed that Bayes “reduces all probability to a subjective judgement.”13 That “subjective judgment” is the initial prior probability.

In the example of throwing balls on square tables, Bayes made an initial guess that the first ball landed in the middle of the table. Fisher was of the view that statistics should be based on “objective” facts, not “subjective” priors, which depended on the opinion of whoever set the initial prior. In circumstances in which the prior cannot be clearly deduced, Fisher believed Bayes’ theorem not only had no value—it was misleading. He believed calculations that began with “subjective” priors, such as the first ball that lands in the middle of the table, invalidated Bayes’ theorem. Those who share this antipathy toward Bayes’ theorem are known as “frequentists,” and Fisher was the leading proponent of this view.

But I think Fisher and the frequentists missed the point. Bayes and Price were not claiming that the prior is anything more than a guess and agreed that initial prior probabilities are often subjective. In our example of tossing a coin, if we knew the coin was fair, then there would be no need to test for fairness. It is because the prior is unknown that we compute inverse probability in the first place.

Furthermore, I believe no priors are truly objective anyway. Even deductive conclusions, such as those related to a coin toss, are based on inductive reasoning (see Chapter 4). Our intuitions may be right, but then again probability can mislead us. Scientists often have differing opinions about and approaches to testing a particular hypothesis. But after repeated and varied experiments, a consensus generally develops. On the other hand, frequentists hold firm convictions about what should happen, and when the unexpected arises, they often try to explain away an experimental outcome. In my view, it is better to rely on repeated trials using Bayes’ theorem than preconceptions about the expected result.

We have discussed the example of the toss of a fair coin. Fisher claimed that over time the ratio of heads to tails will converge to 1:1. In an idealized world, that is true. But in the real world, I have argued there is no such thing as a truly random process, and an actual coin toss will tend to favor either heads or tails. But we cannot practically know down to a molecular level all the factors that determine the outcome to foresee on which side the coin is more likely to land. Hence, there is no way to “objectively” estimate the prior probable outcome of tossing a coin.

By contrast, we

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