BAYESIANISM—at present one of the most widespread positions, especially in the philosophy of science and epistemology, in the question of the rationality of beliefs that admit of degrees, seen as a supplement of logical deduction as a theory of beliefs that do not admit of degrees. The name of this direction of thought comes from the name T. Bayes (1701–1761) who was the first to formulate the position which is the basis for ascribing degrees to beliefs on the basis of the data of experience.

The basic principle of Bayesianism is that there are beliefs that admit of degrees, and that this gradation can be represented numerically. It is presupposed that beliefs of an ideal cognoscent subject are in agreement with the axioms of the probability calculus. Generally this means that (I) they receive values from the set [0, 1]; (II) only tautological beliefs receive the value of 1; (III) the probability of logically independent beliefs is equal to the sum of the probabilities of both beliefs.

If we treat this principles as empirical hypotheses, many experimental investigations seem to contradict them. For this reason, Bayesians prefer to regard them only as certain idealizations or heuristic conventions rather than empirical hypotheses, especially that they are to be norms of rationality rather than a description or generalization of experimental data.

Among the arguments of the Bayesians for the above-mentioned principles, the best known is the so-called Dutch wager. F. P. Ramsey was the first to present a way to translation degrees of beliefs into books of wagers, e.g. p (“Tomorrow it will rain”)= 2/3 it may—granting certain presuppositions—be interpreted as preparedness to wager in a relation of 2 (units of money) against 1 (unity of money), that the sentence “Tomorrow it will rain” will turn out to be true. If someone’s belief is in disagreement with the probability calculus, one may show that in relation to this person a system of wagers can be established such that this person, independent of the final outcome, will always win.

Bayes’ theorem (B) plays the most important role in Bayesian theory. This assertion concerns how experiential data influence the degrees of beliefs we hold:
p(H \ E)= p(H)p(E \ H)
, (B)
where H is the verified hypothesis; E is the data of experience; p(H/E) is the probability that H is true to the extent E is true; p(H) is the probability that H is true before the data of experience are collected; p(E) is the probability that E is true independent of the considered hypothesis.

Bayes’ theorem as a consequence of axioms of the probability calculus is not controversial. Its application to the evaluation of epistemic situations, on the other hand, raises doubts. There are many convincing historical proofs that Bayes introduced this assertion to counteract Hume’s inductive skepticism, especially to show that there is a possibility of a strict and formal development of inductive inferences.

The reference of Bayes’ theorem to real beliefs or beliefs known from the history of science is connected with the acceptance of the logical omniscience of the subject of beliefs, which requires a knowledge of all tautologies [p(T) = 1] and contratautologies [p(K) = 0]. This is not a realistic presupposition, but this theory is treated as normative, not descriptive, in connection with which certain idealizations are admissible. A more serious problem is connected with the fact that this assertion concerns beliefs taken synchronically in a certain temporal moment. On the other hand, inductive inferences concern beliefs taken dynamically, and so changes in beliefs in the face of new experiential data. In connection with this, the Bayesians introduce the so-called rule (C) of the conditionalization of beliefs:

Q(H) = P(H/E) (C)

where Q(H) is a new degree of belief concerning the truth of H; P(H/E) is the prior degree of belief concerning the truth of H actualized after new experiential data have been provided.

The rule for the conditionalization of beliefs, although intuitively convincing, does not possess argumental support such as, e.g., the Dutch wager, since it is shown that no general form of this wager can be provided for the dynamic relation between beliefs. It cannot be shown that a person who in the actualization of beliefs does not make use of the rule (C) will accept a system of wagers that in every outcome brings him a loss.

Bayesianism is most often identified as a certain theory of confirmation, and at the same time as the most ambitious attempt to provide a uniform and general explanation of scientific knowledge. The degree of the belief in a scientific hypothesis by experiential data is identified with the conditional probability of this hypothesis: p(H \ E), and the way it is calculated is presented in accordance with Bayes’ assertion (B). Such an approach allows one to explain many problems in the philosophy of science. If the experiential data E follow logically from the hypothesis H, then in accordance with the probability calculus p(E \ H) = 1. Therefore, if the experiential data E have actually been observed, then in accordance with Bayes’ assertion (B): p(H \ E) > p (H). This may be interpreted as follows: E, by increasing the degree of probability of H, confirms this hypothesis.

If E is a predication that logically follows from H, then in accordance with Bayes’ assertion (B): p(H / E)/p(H) = 1/p(E). The less expected is E, that is, the less the probability of E, the more it confirms H. This therefore explains the practice generally accepted in science that the less expected the consequences of a given theory are, and these are asserted right after the formulation of the theory, the most probable is the theory.

The Bayesian theory of confirmation explains in similar manner why a greater or more varied set of experiential data provides greater confirmation for a hypothesis that does a lesser or less varied set. It also explains how it is that certain hypotheses are ad hoc and in connection with this are more weakly confirmed than hypotheses that are not ad hoc. It also provides a clear solution to Hempel’s paradox of the crows and provides a representation of many of the theses of the historicized philosophy of science that was initiated by T. Kuhn.

When it holds that probability should be interpreted as someone’s degrees of beliefs, Bayesianism opens itself to the accusation of subjectivism. Since the only limitations placed on beliefs are the axioms of the probability calculus, then every scientist may have an arbitrary degree of belief concerning the truth of a given hypothesis. In response, the Bayesians refer to the so-called convergence theorem: independent of what probability scientists initially assign to a given hypothesis, they always arrive at a probability close to unity if the hypothesis is true (respectively, a probability of close to zero, if it is false), insofar only as they actualize their degrees of convictions in light of accumulating experiential data in accordance with the rule of the conditionalization of beliefs (C).

The basic problem in the application of the rule of conditionalization (C) is connected with cases where the initial probabilities of a hypothesis or experiential data receive extreme values, i.e., 1 or 0. Then regardless of what experiments are performed, the degree of the probability of the hypothesis or experiential data are never subject to change. An illustration of this difficulty is the so-called problem of discovered data. Even before Einstein’s formulation of the general theory of relativity (GTR), Mercury’s trajectory was well described as being not in conformity with the predictions of Newton’s theory of gravitation. The GTR provided a good explanation of the deviation in the trajectory, which was recognized as one of the most important facts confirming the general theory of relativity. The Bayesian theory of confirmation does not explain this. According to the Bayesian theory of confirmation (cf. (B)) the GTR’s degree of affirmation did not change, since the probability of Mercury’s trajectory, which was known before the GTR, bore a value of 1.

In view of the admissibility of more conditions apart from the axioms of probability, which would limit rationally held degrees of beliefs, there is an enormous wealth of variations of Bayesianism: from radically subjectivistic (personalism) to radically objectivistic (maxent). Subjectivists, e.g. C. Howson and P. Urbach, resolutely defend positions that axioms of probability can be the only conditions imposed on beliefs that can be non-arbitrarily justified (e.g., by the Dutch wager). Objectivists, e.g., T. E. Jaynes and R. Carnap, use the idea of the least amount of information or logical structure of propositions to provide criteria for limiting the way in which probabilities are assigned to beliefs, which (criteria) are purely objective and independent of subjective beliefs.

Bayesianism finds many practical applications, especially in the philosophy of medicine where it serves to model prognostic and diagnostic inferences, and more generaly, for inferences from uncertain data. According to Bayes’ theorem (B) and the rule of conditionalization (C), original probabilities are actualized and changed under the influence of data that come from clinical tests. The Bayesian approach turns out to be more effective, sometimes much more effective, than the opinions of experienced physicians, so long as it is limited to a narrow field of research. In application to broader and more complex fields, the quantity of data required by the Bayesian approach increases exponentially, which sets limits on the applicability of the approach.

Bayesianism is also a certain theory of learning. Toward the end of the twentieth century, there were empirical studies on whether the Bayesian representation of learning is the quickest and most effective method of learning, a method formed by evolution. The fact that Bayesianism is a theory of learning is sometimes the basis for an objection against the Bayesian philosophy of science and epistemology. It is thought that they are theories concerned with the acquisition of beliefs but have no connection with the epistemic appraisal of the status of the beliefs and provide no epistemological or philosophical explanations apart from indicating their genesis. Moderate Bayesians hold that it is not a complete philosophy of science and epistemology but only a certain idealizing model, but a model that allows a clear and precise explanation and solution for many problems that result from a simplified treatment of beliefs as exclusively true or false.

R. Carnap, Logical Foundations of Probability, Ch 1950, 19622; R. S. Ledley, L. B. Lusted, Reasoning Foundations of Medical Diagnosis, Science 130 (1959), 9–21; W. C. Salmon, Bayes’s Theorem and the History of Science, in: Historical and Philosophical Perspectives of Science, Minneapolis 1970, 68–86; C. Glymour, Theory and Evidence, Pri 1980; P. Horwich, Probability and Evidence, C 1982; C. Howson, P. Urbach, Scientific Reasoning: The Bayesian Approach, La Salle (Il.) 1989, Ch 19932; A. Franklin, Experiment, Right or Wrong, C 1990; J. Earman, Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory, C (Mass.) 1992; K. F. Schaffner, Clinical Trials and Causation: Bayesian Perspectives, Statistics in Medicine 12 (1993), 477–494.

Paweł Kawalec

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