Abstract
The paper reviews three modes of rational inference: deductive, inductive and probabilistic. Many examples of each can be found in scientific endeavour, professional practice and public discourse. However, while the strengths and weaknesses of deductive and inductive inference are well established, the implications of the emerging probabilistic orientation are still being worked through. The paper discusses some of the recent findings in psychology and philosophy, and speculates about the implications for scientific and professional practice in general and OR in particular. It is suggested that the probabilistic orientation and Bayesian approach can provide an epistemological lens through which to view the claims of different approaches to inference. Some suggestions for further research are made.
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Appendix
Appendix
Bayes's theorem
In its simplest form Bayes's theory is formulated as follows:
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P(h|e) is known as the posterior probability of h given e.
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P(e|h) is known as the likelihood.
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P(h) is known as the prior probability of h.
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P(e) is the prior probability of e.
The change from P(h) to P(h|e) is known as Bayesian conditionalization. The likelihood divided by the prior probability of e, which conditions the prior probability of h, is referred to as the Bayesian multiplier. If we include a further variable b, we can account for any background assumptions, perhaps a background theory or some old evidence:
If two hypotheses are being compared on the basis of the same evidence e, we have:
and
We can divide these two equations to give:
That is, the ratio of the degree of confirmation of the rival hypotheses by the same evidence e is given by the ratio of the prior probabilities of the rivals multiplied by the likelihood of the rivals, or in other words the ratio of the degree to which each rival predicts the evidence. We can expand (A.5) to include background assumptions:
This formulation of Bayes's theorem yields an understanding of crucial experiments. However, more generally, each hypothesis will have its own background assumptions. We, therefore, have:
For n rival hypotheses, we have:
Thomas Bayes's (1763) theory was published posthumously. It seems likely that it was developed in response to David Hume's identification of ‘the problem of induction’. As a statement of logic, the theorem is not generally considered controversial. However, it is the use of the theory with subjective probabilities and as a metaphor for addressing epistemological questions, which is of primary importance for this paper. These usages are more controversial, but have recently received widespread attention in a variety of fields. A fuller account of Bayes's theorem can be found in Howson and Urbach (1989).
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Ormerod, R. Rational inference: deductive, inductive and probabilistic thinking. J Oper Res Soc 61, 1207–1223 (2010). https://doi.org/10.1057/jors.2009.96
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DOI: https://doi.org/10.1057/jors.2009.96