Bayes Confirmation Theory and Scientific Theory Testing

Introduction to Bayes' Theorem and Scientific Theory

Theorizing in science is a complex and time-consuming undertaking. The theorist uses collected evidence from some means of scientific inquiry to project a generalized case. However, there is a difficulty with this process: there is some probability that the theory will be wrong. Even if this is not a harmful outcome, it is difficult for the theorist to overcome in their professional lives. Researchers therefore want to understand the probabilities involved in the success of their theories.

Bayes' theorem addresses the probability that an event will occur — in the context of this paper, whether a theory is correct or not. Bayes considered two different events, one of which can be used to increase the probability that the other is correct. For example, suppose that a statement has a fifty percent chance of being true; that is not enough to build a theory on. To increase the probability that the statement is true, something else is added — a known quantity that relates to the previous statement — that raises the probability that the first statement was true.

As an illustration, one person (A) tells another (B) that they have just met with a friend and had lunch together. B hypothesizes that the friend was a woman, but there is only a fifty percent chance that this is so. More information is needed to confirm this supposition. In the course of the conversation, person A mentions that the friend left to have a spa day. The probability that the friend was a woman increases because B knows that seventy-five percent of people who go to spas are women. Using Bayes' theorem, the second piece of evidence further confirms the first supposition. This does not mean that B is now completely convinced that the person was a woman — twenty-five percent of spa-goers are men — but there is now approximately a sixty-seven percent chance that the individual was female.

This logic can also be applied to the probability that a hypothesis is correct, because the same pattern appears in both cases. Every theory begins with a hypothesis that has some probability of being correct and some probability of being false. A theorist who is not satisfied with the uncertainty implied by this initial probability seeks to ensure the correctness of their findings by using another piece of evidence to confirm or disprove the assumption. That additional evidence has this power because it can increase the probability that the hypothesis will be supported, or indicate that it should be abandoned. This paper examines Bayes' theorem in detail with regard to its ability to help predict the correctness of a proposed theory.

Example of Bayes' Theorem in Practice

Examples of Bayes' theorem are easy to find in the literature because it is one of the primary methods used to determine the probability of a statement being true or false. Simple probabilities — such as whether a coin will turn up heads or tails — are not affected by Bayes' theorem because no data can be added to better predict the next outcome (assuming the coin is fair). Since there are only two equally likely possibilities, it is impossible to assign one a higher probability than the other. Both have a value of one-half in the Bayesian context. Most real-world events, however, are not so simple.

A useful example is determining the probability that a test subject belongs to a particular category within a set — such as whether a participant is male or female. This is a good illustration because, as Meehl notes, "physical theories typically predict numerical values."¹ Assuming that half of the population is male and the other half female, the prior probability that a randomly chosen subject is male is 0.50. However, newly obtained data reveals that the participant is also a cigar smoker. Given that 9% of male participants admittedly smoke cigars while only 2% of female participants do, it becomes possible to determine that the probability the chosen participant is male rises to approximately 82%.² This shows how Bayes' theorem works in practice, but it does not yet answer whether the theorem can adequately confirm a scientific theory.

Challenges to Bayes' Confirmation Theory

Several challenges to the fundamental utility of Bayes' theorem have been identified in the philosophical literature. These include the values of the priors, the problem of the zero prior required for the equation, the problem of old evidence, and the observation that scientists do not actually attempt to confirm theories in a strictly probabilistic way. Bayes' theorem is considered too simplistic in certain respects because it requires specific foundational numbers to function, and those numbers are not always available when a theorist is developing a hypothesis.³

The requirement for prior evidence both helps and hurts the predictive utility of the theorem. The problem of the priors is particularly significant because it can undermine the theory entirely: there is often no logical explanation for where the prior probability value originated. As for zero priors, most probabilities have an initial value associated with them, but there is a possibility that the value of H or H/E in the theorem could be zero. If this is the case, no updated probability can be obtained. It is also difficult to know to which quantities a zero should be assigned.

The issue of old evidence appears to be the most puzzling challenge for scientists. As Rosenberg explains, "scientists who construct hypotheses by intentional 'curve fitting' are rightly criticized and their hypotheses are often denied explanatory power on the grounds that they are ad hoc."⁴ This means that the effectiveness of a theory is called into question when old — or ad hoc — evidence is used as its proof. However, Rosenberg also points out that scientists encounter a further problem when attempting to "distinguish cases like the confirmation of Newton's and Einstein's theories by old evidence from cases in which old evidence does not confirm a hypothesis because it was accommodated to the old evidence."⁵ It is therefore difficult for critics to distinguish between a theory that is genuinely confirmed by existing evidence and one that was constructed around that evidence — a distinction that is crucial before dismissing a theory outright.

The final challenge is that probabilistic theory confirmation sits uneasily with the method generally used to prove or disprove a hypothesis. In empirical theory-building, the researcher tries to gather as much evidence as possible regarding the question at hand, rather than seeking confirmatory evidence that strengthens the theory from the outside. Confirmatory evidence, in the empirical tradition, arises from the veracity of the initial hypothesis as tested against a large body of observational data.

Conclusion

Bayes' theorem is an adequate method for increasing confidence in the correctness of a theory. By its very definition, a theory can never be expected to be fully confirmed; if it were, it would be equivalent to a mathematical law. The examination of Bayes' theorem shows that evidence can progressively increase the probability that a hypothesis is true, making it a valid means of gauging, with a greater degree of certainty, whether a theory can be confirmed. It also follows that additional research — where it can be assigned a numerical quantity — can be used to further confirm the original hypothesis. This cycle of evidence can continue until it becomes highly probable that the original theory was correct. This iterative process of increasing likelihood, facilitated by Bayes' theorem, is the reason it remains a useful tool for determining the true probability of a theory's success as a predictor.

Bibliography

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Garber, D. 'Old evidence and logical omniscience in Bayesian Confirmation Theory', in J. Earman (ed.), Testing Scientific Theories, University of Minnesota Press, Minneapolis, 1983, pp. 99–132.

Garlikov, R. The Nature of the Logic of Confirmation in Science, 2000. Retrieved 24 March 2012 from http://www.garlikov.com/Science.html

Meehl, P. E. 'Theory-testing in psychology and physics: A methodological paradox', Philosophy of Science, vol. 34, 1967, pp. 103–115.

Rosenberg, A. Philosophy of Science: A Contemporary Introduction, 2nd edn, Routledge, New York, 2005.

Steel, D. Bayesian Confirmation Theory and the Likelihood Principle, 2007. Retrieved 25 March 2012 from https://www.msu.edu/user/steel/Bayes_and_LP.pdf

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