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Bayesian epistemology

Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory.[1] One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. It is based on the idea that beliefs can be interpreted as subjective probabilities. As such, they are subject to the laws of probability theory, which act as the norms of rationality. These norms can be divided into static constraints, governing the rationality of beliefs at any moment, and dynamic constraints, governing how rational agents should change their beliefs upon receiving new evidence. The most characteristic Bayesian expression of these principles is found in the form of Dutch books, which illustrate irrationality in agents through a series of bets that lead to a loss for the agent no matter which of the probabilistic events occurs. Bayesians have applied these fundamental principles to various epistemological topics but Bayesianism does not cover all topics of traditional epistemology. The problem of confirmation in the philosophy of science, for example, can be approached through the Bayesian principle of conditionalization by holding that a piece of evidence confirms a theory if it raises the likelihood that this theory is true. Various proposals have been made to define the concept of coherence in terms of probability, usually in the sense that two propositions cohere if the probability of their conjunction is higher than if they were neutrally related to each other. The Bayesian approach has also been fruitful in the field of social epistemology, for example, concerning the problem of testimony or the problem of group belief. Bayesianism still faces various theoretical objections that have not been fully solved.

Relation to traditional epistemology[edit]

Traditional epistemology and Bayesian epistemology are both forms of epistemology, but they differ in various respects, for example, concerning their methodology, their interpretation of belief, the role justification or confirmation plays in them and some of their research interests. Traditional epistemology focuses on topics such as the analysis of the nature of knowledge, usually in terms of justified true beliefs, the sources of knowledge, like perception or testimony, the structure of a body of knowledge, for example in the form of foundationalism or coherentism, and the problem of philosophical skepticism or the question of whether knowledge is possible at all.[2][3] These inquiries are usually based on epistemic intuitions and regard beliefs as either present or absent.[4] Bayesian epistemology, on the other hand, works by formalizing concepts and problems, which are often vague in the traditional approach. It thereby focuses more on mathematical intuitions and promises a higher degree of precision.[1][4] It sees belief as a continuous phenomenon that comes in various degrees, so-called credences.[5] Some Bayesians have even suggested that the regular notion of belief should be abandoned.[6] But there are also proposals to connect the two, for example, the Lockean thesis, which defines belief as credence above a certain threshold.[7][8] Justification plays a central role in traditional epistemology while Bayesians have focused on the related notions of confirmation and disconfirmation through evidence.[5] The notion of evidence is important for both approaches but only the traditional approach has been interested in studying the sources of evidence, like perception and memory. Bayesianism, on the other hand, has focused on the role of evidence for rationality: how someone's credence should be adjusted upon receiving new evidence.[5] There is an analogy between the Bayesian norms of rationality in terms of probabilistic laws and the traditional norms of rationality in terms of deductive consistency.[5][6] Certain traditional problems, like the topic of skepticism about our knowledge of the external world, are difficult to express in Bayesian terms.[5]

Applications[edit]

Confirmation theory[edit]

In the philosophy of science, confirmation refers to the relation between a piece of evidence and a hypothesis confirmed by it.[18] Confirmation theory is the study of confirmation and disconfirmation: how scientific hypotheses are supported or refuted by evidence.[19] Bayesian confirmation theory provides a model of confirmation based on the principle of conditionalization.[6][18] A piece of evidence confirms a theory if the conditional probability of that theory relative to the evidence is higher than the unconditional probability of the theory by itself.[18] Expressed formally: .[6] If the evidence lowers the probability of the hypothesis then it disconfirms it. Scientists are usually not just interested in whether a piece of evidence supports a theory but also in how much support it provides. There are different ways how this degree can be determined.[18] The simplest version just measures the difference between the conditional probability of the hypothesis relative to the evidence and the unconditional probability of the hypothesis, i.e. the degree of support is .[4] The problem with measuring this degree is that it depends on how certain the theory already is prior to receiving the evidence. So if a scientist is already very certain that a theory is true then one further piece of evidence will not affect her credence much, even if the evidence would be very strong.[6][4] There are other constraints for how an evidence measure should behave, for example, surprising evidence, i.e. evidence that had a low probability on its own, should provide more support.[4][18] Scientists are often faced with the problem of having to decide between two competing theories. In such cases, the interest is not so much in absolute confirmation, or how much a new piece of evidence would support this or that theory, but in relative confirmation, i.e. in which theory is supported more by the new evidence.[6]


A well-known problem in confirmation theory is Carl Gustav Hempel's raven paradox.[20][19][18] Hempel starts by pointing out that seeing a black raven counts as evidence for the hypothesis that all ravens are black while seeing a green apple is usually not taken to be evidence for or against this hypothesis. The paradox consists in the consideration that the hypothesis "all ravens are black" is logically equivalent to the hypothesis "if something is not black, then it is not a raven".[18] So since seeing a green apple counts as evidence for the second hypothesis, it should also count as evidence for the first one.[6] Bayesianism allows that seeing a green apple supports the raven-hypothesis while explaining our initial intuition otherwise. This result is reached if we assume that seeing a green apple provides minimal but still positive support for the raven-hypothesis while spotting a black raven provides significantly more support.[6][18][20]

Coherence[edit]

Coherence plays a central role in various epistemological theories, for example, in the coherence theory of truth or in the coherence theory of justification.[21][22] It is often assumed that sets of beliefs are more likely to be true if they are coherent than otherwise.[1] For example, we would be more likely to trust a detective who can connect all the pieces of evidence into a coherent story. But there is no general agreement as to how coherence is to be defined.[1][4] Bayesianism has been applied to this field by suggesting precise definitions of coherence in terms of probability, which can then be employed to tackle other problems surrounding coherence.[4] One such definition was proposed by Tomoji Shogenji, who suggests that the coherence between two beliefs is equal to the probability of their conjunction divided by the probabilities of each by itself, i.e. .[4][23] Intuitively, this measures how likely it is that the two beliefs are true at the same time, compared to how likely this would be if they were neutrally related to each other.[23] The coherence is high if the two beliefs are relevant to each other.[4] Coherence defined this way is relative to a credence assignment. This means that two propositions may have high coherence for one agent and a low coherence for another agent due to the difference in prior probabilities of the agents' credences.[4]

Social epistemology[edit]

Social epistemology studies the relevance of social factors for knowledge.[24] In the field of science, for example, this is relevant since individual scientists have to place their trust in some claimed discoveries of other scientists in order to progress.[1] The Bayesian approach can be applied to various topics in social epistemology. For example, probabilistic reasoning can be used in the field of testimony to evaluate how reliable a given report is.[6] In this way, it can be formally shown that witness reports that are probabilistically independent of each other provide more support than otherwise.[1] Another topic in social epistemology concerns the question of how to aggregate the beliefs of the individuals within a group to arrive at the belief of the group as a whole.[24] Bayesianism approaches this problem by aggregating the probability assignments of the different individuals.[6][1]

Objections[edit]

Problem of priors[edit]

In order to draw probabilistic inferences based on new evidence, it is necessary to already have a prior probability assigned to the proposition in question.[25] But this is not always the case: there are many propositions that the agent never considered and therefore lacks a credence for. This problem is usually solved by assigning a probability to the proposition in question in order to learn from the new evidence through conditionalization.[6][26] The problem of priors concerns the question of how this initial assignment should be done.[25] Subjective Bayesians hold that there are no or few constraints besides probabilistic coherence that determine how we assign the initial probabilities. The argument for this freedom in choosing the initial credence is that the credences will change as we acquire more evidence and will converge on the same value after enough steps no matter where we start.[6] Objective Bayesians, on the other hand, assert that there are various constraints that determine the initial assignment. One important constraint is the principle of indifference.[5][25] It states that the credences should be distributed equally among all the possible outcomes.[27][10] For example, the agent wants to predict the color of balls drawn from an urn containing only red and black balls without any information about the ratio of red to black balls.[6] Applied to this situation, the principle of indifference states that the agent should initially assume that the probability to draw a red ball is 50%. This is due to symmetric considerations: it is the only assignment in which the prior probabilities are invariant to a change in label.[6] While this approach works for some cases it produces paradoxes in others. Another objection is that one should not assign prior probabilities based on initial ignorance.[6]

Problem of logical omniscience[edit]

The norms of rationality according to the standard definitions of Bayesian epistemology assume logical omniscience: the agent has to make sure to exactly follow all the laws of probability for all her credences in order to count as rational.[28][29] Whoever fails to do so is vulnerable to Dutch books and is therefore irrational. This is an unrealistic standard for human beings, as critics have pointed out.[6]

Problem of old evidence[edit]

The problem of old evidence concerns cases in which the agent does not know at the time of acquiring a piece of evidence that it confirms a hypothesis but only learns about this supporting-relation later.[6] Normally, the agent would increase her belief in the hypothesis after discovering this relation. But this is not allowed in Bayesian confirmation theory since conditionalization can only happen upon a change of the probability of the evidential statement, which is not the case.[6][30] For example, the observation of certain anomalies in the orbit of Mercury is evidence for the theory of general relativity. But this data had been obtained before the theory was formulated, thereby counting as old evidence.[30]

Bayesian statistics

Bayesian probability

Bayesian inference

Probability interpretations

Titelbaum, Michael G. (2022). Fundamentals of Bayesian Epistemology. Oxford University Press.  978-0-19-870761-5.

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