Bayesian approach in the context of falsifiability

Among the attempts to dismantle the first horn of Hume’s fork, Henderson mentions Kant’s approach discussed later and that of Bayes. Bayesian logic is not ampliative: it does not provide more information than that contained in the premises and observations. As mentioned in Henderson’s analysis, this attempted justification requires a synthetic a priori proposition which, in this case, is a way of determining the a priori probabilities required by Bayes’ rule. The attempt fails because no way, including the principle of indifference, seems generally acceptable.

Colin Howson takes the Bayesian approach. In the same way that Popper transformed Hume’s problem, which is that of the justification of knowledge, into that of finding the usefulness of deductive logic in the scientific process, Howson transforms Hume’s problem into that of finding the usefulness of Bayesian logic in this approach. Like Popper, he accepts that induction (i.e., an amplifiative logic) is impossible. He does not seek to attack Hume’s argument. He proposes that the Bayesian logic of probabilities is the (non-amplifiative) logic of an inductive reasoning in which the truth of the conclusion depends on the truth of the premises, as is also the case in a deductive logic. In the same way that Popper insisted on the distinction between the logic of the deductive phase (which is the subject of falsifiability) and methodology, Howson insists that the problems of methodology, the falsifiability of conjectures and observations, do not concern the logic of the Bayesian approach. In this approach, the various hypotheses and their a priori probabilities constitute an empirical conjecture. This approach is consistent with the principle, adopted by Popper, that science works by empirical conjectures that we tend to believe a priori, but which are falsifiable.

An important difference is that Bayesian logic uses prior guesswork and observations to calculate posterior probabilities. According to Jan-Willem Romeijn, Bayesian logic serves to ensure consistency between the a priori and a posteriori conjectures. Chalmers mentions examples presented by Howson which illustrate that Bayesian logic seems to explain the scientific process. Thus, Popper would be mistaken in suggesting that Bayesian logic can play no role.

However, Chalmers criticizes Howson’s strict way of applying Bayesian logic. According to Howson, the solution to Hume’s problem is to “eliminate the connection between induction and truth”. Like Miller, Chalmers believes science should not sever the link with truth. The starting premises must be compatible with correct a posteriori hypotheses. Nothing in Bayesian logic allows us to reject the initial premises, even if they deductively contradict the observations. Like critical rationalism, Bayesian logic does not propose to test conjectures to feed a critical discussion with the aim of possibly proposing better a priori conjectures. To explain how difficult it is to establish correct premises, Henderson compares scientific experiments with the simple experience of extracting a ball from an urn. One can lack imagination in the choice of hypotheses: even if the premise is that the urn contains white and black balls, a red ball can be drawn, assuming that it is even a ball. Henderson raises deeper difficulties. For example, the assumption that the urn does not change from one time to the next does not correspond to the fact that the instruments, their precision, etc. may change from experience to experience. What you learn about the urn is valid for future extracted balls, but what you learn about the instruments used in one experiment may not necessarily apply to other experiments that use improved technology. We learn from our mistakes. The next experiences are not necessarily identical. This is just one example that illustrates how a premise can be incorrect. Bayesianism is not in question, because it is the premises that do not hold, but saying so does not offer a methodology or a philosophy to correct a conjecture that is invalid and must be modified.

For this approach to join Popper’s “conjectures and refutations” principle, there would need to be a way to use Bayesian conclusions as mathematical information in a critical discussion whose goal is to possibly replace the a priori conjecture with another. This other conjecture would be in a sense also a posteriori, but not necessarily the one that is calculated by Bayesian logic, since other elements than the original observations and hypotheses can be taken into account. This does not contradict Bayesian logic, because this logic cannot be blamed when it is the premises that are at issue. According to Miller, Bayesian logic is silent on the need to challenge them and the situation is summed up by saying that Bayesian logic has epistemological value, but little methodological value.

(Includes texts from Wikipedia translated and adapted by Nicolae Sfetcu)