Short Abstract

Abstract:

Bayesians perform inference by applying Bayes' theorem

$$P(T\vert E)=\frac{P(E\vert T)P(T)}{P(E)}\textrm{,}$$ (1)

where the prior probability $P(T)$ represents the degree of belief in theory $T$, from a set $S$ of available theories, before the occurrence of evidence $E$, the posterior probability $P(T\vert E)$ represents the degree of belief in $T$ after the occurrence of $E$, the likelihood $P(E\vert T)$ is the probability that $T$ assigns to the occurrence of $E$, and $P(E) = \sum_{T\in{}S}P(E\vert T)P(T)$.

The Bayesian world-view is found to differ from positivism in four ways. Firstly, the positivist Occam's razor insists that $P(T) = P_0\epsilon{}^n$, where $n$ is the number of postulates in $T$, and $P_0$ is a normalizing constant to ensure $\sum_{T\in{}S}P(T) = 1$, whereas in the Bayesian view, $P(T)$ is a subjective and arbitrary choice. Secondly, positivism prohibits the entry of prescriptive ideologies into knowledge, whereas Bayes' theorem allows inferences to be drawn about prescriptive ideologies and descriptive theories alike. Thirdly, a prohibition on theories $T$, for which one cannot conceive of any evidence $E$, for which $P(E\vert T) = 0$, is sometimes associated with positivism, whereas Bayesian analysis can assess many such theories. Fourthly, any positivist prohibition on the consideration of evidence obtained with an ideology in mind is another example of insistence on the assignment of particular values to prior probabilities. Therefore, Bayesians, like standpoint epistemologists, will think such a prohibition inappropriate.

Bayesians view subjectivity as entering, inevitably, into academic study by four routes. Firstly, Bayesian inference cannot be performed without the use of a prior probability distribution, mirroring standpoint epistemologists' view that previous experience always affects the interpretation of evidence. For a Bayesian, the choice of prior probability distribution is arbitrary and subjective, and any attempt to achieve objectivity by claiming that a particular prior probability distribution is correct conceals this subjective choice, and renders it immutable. Bayesians conceptualize objectivity as the achievement of similar posterior probability distributions, from a range of prior probability distributions, mirroring standpoint epistemologists' view that theoretical beliefs need to be grounded by combining the conclusions of researchers with diverse previous experiences. If evidence has been noted, for which some theories have not produced likelihood values, a prior probability distribution can be devised that approximates the posterior probability distribution that the previous evidence would have produced, mirroring standpoint epistemologists' view that thinking on the basis of evidence, which has been obtained in the course of a marginalized lives, can improve the ability of researchers to interpret other evidence, whether or not they themselves experienced those marginalized lives.

Secondly, the Bayesian statistical decision theory uses the expected reduction, which an experiment will bring about, in an uncertainty function of the prior or posterior probability distribution, to judge the worth of an experiment, and therefore to choose experiments. It can be shown that every uncertainty function is associated with an ideology, for which it measures how much an experiment's results can improve the making of policy choices to implement that ideology. Bayesians may, therefore, suspect that claims that a particular piece of research is free of ideology conceal the inevitable influence of ideology in the selection of its experiments. This suspicion is shared with standpoint epistemology, and proponents of both traditions prefer to be open about the ideological motivation of their research.

Thirdly, ideology may also enter into academic study through attempts to produce a posterior probability distribution, in which the policy that will minimize the expectation of one's own loss function is the same policy that will minimize the expectation of the loss function of someone else with decision-making power. This possibility, against which Bayesian thinking can provide some protection, has been noted by standpoint epistemologists, as an abuse of science.

Fourthly, the prior probability distribution can affect the posterior probability distribution through experiment selection. In a toy problem, it is found that whichever of two theories has the higher prior probability is able to make an experiment that confirms it appear to be the most interesting experiment to undertake, and thereby to enhance its probability. This may be related to certain recent epistemological observations, outside of the standpoint epistemology tradition.

Physicists have a tendency to study phenomena at very large and very small length scales. This can be modelled as the result of experiment selection, under statistical decision theory, using a greedy top-down approximation to the Shannon entropy as the uncertainty function.