Clark Glymour presents his argument against Bayesian inference as a foundation for scientific inference in the third chapter of his book *Theory and Evidence* titled “Why I Am Not a Bayesian”. By Bayesian inference, the probability for a hypothesis or theory is revised according to Bayes’ theorem as new evidence is presented. Glymour begins by defining the goals of a confirmation theory and describes how understanding probability as a “degree of belief” has become an increasingly influential philosophical viewpoint. Then he presents a three-part argument against the subjectivist Bayesian view of probability. First, Glymour eschews the rationality of the Bayesian argument that there are a priori reasons for placing restrictions on belief and inference. Secondly, he argues there are many attributes that a successful confirmation theory must possess that are not explained by Bayesian theory; these attributes include support by a variety of evidence, the superiority of a simple of theory, what determines when a piece of evidence is relevant to a hypothesis, and what makes a particular confirmation of one theory stronger than another. Finally, Glymour challenges fundamental relationships between theory and evidence that Bayesian confirmation theory fails to address without significant revision. Glymour concludes that the Bayesian scheme requires considerable revision to address the issues he identifies.

In the first part of his argument, Glymour maintains that the Bayesian argument fails to explicate the rationality of placing restrictions on belief and inference. The Dutch Book argument using betting odds is used by Glymour to illustrate that a person’s degree of belief need not satisfy the axioms of probability (for rationality). If a person’s grades of belief (which can take on any value) are mapped to degrees of belief (which are values between 0 and 1) are not probabilities, then there exists a circumstance for which the person would lose a bet in all outcomes. Thus, the Dutch Book argument does not demonstrate that degrees of belief must be probabilities; it only shows that some absurdity is avoided if a person’s degrees of belief are probabilities. Another issue with Bayesian theory is that it does not provide a connection between what is inferred and what is true. For example, given a scenario in which there existed a binomial random variable, the Bayesian approach would involve second-order probabilities, which are also degrees of belief according to Bayesians. The theoretical outcome does not tell us that a probability of 1 will be assigned to the true hypothesis (and probability 0 to the rest) in the limit as n goes to infinity, only that rational Bayesians are certain the correct value will be assigned. This is essentially meaningless. Additionally, Glymour argues that the assumptions required to make stable estimation theorems can only be removed under certain conditions, but there is no theoretical reason why those conditions should even exist. The Dutch Book argument, stable estimation theorems, and other a priori arguments are determined by Glymour to be weak and an unstable foundation for the belief in the correctness of Bayesian theory.

Glymour identifies the apparent appeal to common sense as much more persuasive to Bayesians. For one thing, it is rather reasonable to say that hypotheses are confirmed by positive instances of events that support them and if presumed to be unlikely, the occurrence of the hypothesis renders the hypothesis less likely. In addition, scientific reasoning and inference can be easily explained by Bayesian philosophers of science simply by taking assumptions into account due to the sensitivity of probabilistic measurements and the lax restrictions placed on rational degrees of belief. However, Glymour argues, that the ease with which assumptions can be brought into the Bayesian scheme allows it to give us a “theory of personal learning” rather than an explanation of scientific argument. By the Bayesian doctrine, the general foundation of an argument is used as the basis for restricting prior probabilities, creating challenges for describing the relationship between evidence and theory for the Bayesians. The Bayesian framework also does not describe why a variety of evidence is superior or the scientific preference for simplistic theories. For deciding between possible alternate hypotheses and for determining the falsity of a theory, Glymour says Bayesian theory is sufficient to explain the necessity of a variety of evidence. He does not elaborate on this and instead cites work by Abner Shimony. However, for complex theories that contain multiple, independent hypotheses, the process is related to accounting for relevance between evidence and theory; Glymour is skeptical of Bayesian framework’s ability to do so. According to Bayesian theory, simplified “deoccamized” theories--- ones in which a parameter is replaced by a function of several other parameters--- pieces of evidence will not count as explanations of these theories. By Bayesian principles, “if two theories both entail e, then (provided the prior probability of each hypothesis is neither 1 nor 0) if e confirms one of them it confirms the other” (77). Because Bayesian theory does not take into account the relation between theory and body of evidence, the likelihood of a given theory and its deoccamized counterparts will be equal. In the end, Glymour criticizes the rationality of the Bayesian approach to curve fitting, specifically Harold Jeffreys’, because it was dependent on the ordering of prior probabilities. He stated that it “requires that we proceed in ignorance of our scientific experience” (79).

By now, Glymour has twice alluded to the unsatisfactory Bayesian approach to the relation between evidence and theory. Using a Bayesian viewpoint, a person has a defined set of degrees of belief and as they gather new information revise their degrees of belief. A piece of evidence confirms a hypothesis when the probability of the hypothesis is greater after the discovery. Glymour takes exception to this approach. He argues logically that our belief in something should increase as the quality of explanation increases. However, according to Bayesian measurements, this is impossible, because “making degrees of belief probability measures in the Bayesian way already guarantees that a theory can be no more credible than any collection of its consequences” (84). Furthermore, there is no explanation in Bayesian theory for how to judge the relevance/irrelevance of hypotheses within theories. Kepler’s Laws are used as an example; Observations of a single planet are sufficient to constitute evidence for Kepler’s first and second laws, but insufficient for the third. Bayesian theory is incapable of explaining how to determine the relevance of pieces of evidence beyond merely asserting degrees of belief that create the correct results. Bayesian kinematics also fail to describe the case of old evidence confirming new theory. Glymour uses a simple example of Bayes’ theorem to illustrate his point. Let e be evidence that is known before theory T is introduced at time t. Since e is known at t, p(e)=1. It follows that p(e|T) = 1. Thus p(T|e) = p(T) * p(e,T)p(e)=p(T). The conditional probability of T given is the same as the prior probability of T. Thus, old evidence cannot confirm new theory, which is absurd. Finally, the fallacies of human logic are taken into account. “...there are always consequences of our hypotheses that we do not know to be consequences” (92). An ideal Bayesian is a perfect logician, while real humans are not. Perhaps Glymour’s biggest critique of Bayesian theory is that sufficiently explaining the relationship between evidence and theory seems to be beyond Bayesian capacities. To me, Glymour’s final argument presents the greatest challenges to the Bayesian due to the fact that they are highly irreconcilable by Bayesian confirmation theory. The fact that old evidence cannot confirm new theory is impossible according to the simplest and most fundamental Bayesian formula is troubling to me. Determining prior probabilities seems highly subjective in general, especially when the flaws in human logic were brought into account. The subjectivity became even more apparent in Glymour’s example with Kepler’s Laws. Glymour’s arguments were “sufficient to warrant considering entirely different approaches to the analysis of scientific reasoning” (68).

Ultimately, Glymour concludes that while the Bayesian scheme is correct in important aspects, it does not explicate certain key attributes of confirmation theory. These attributes include a priori reasoning made by Bayesians for restricting inference, the preference for simple theories, and the relationship between evidence and theory. Glymour does not refute Bayesian inference completely or deny its legitimacy or usefulness, however. He simply points out logical fallacies in the theory which make him skeptical.