The Appearance of Subjectivity in Academic Study

Having established that Bayesian inference differs from positivist inference in important respects, the theme is emphasized that, for a Bayesian, it is inevitable that there are subjective influences in academic study. The first route for subjectivity arriving in academic study is through the initial prior probability distribution, which represents the beliefs that are held, about which theory is true, before any evidence is observed, and without which no Bayesian inference can be performed. In the Bayesian perspective, the initial prior probability distribution that is used is an arbitrary and subjective choice, and the claim, in the tradition of logical probability, that a particular prior probability distribution is correct and other choices are wrong, is seen as concealing and rendering immutable this arbitrary choice.

Instead of claiming objective supremacy for a particular prior probability distribution, Bayesians seek to construct objective conclusions by gathering enough evidence that similar posterior probability distributions can be reached, starting from a wide range of different prior probability distributions. This corresponds to the construction of objectivity in standpoint epistemology, where it is argued that theory can only be sufficiently grounded by combining the conclusions of researchers whose prior beliefs are diverse.

Standpoint epistemologists further argue that the prior beliefs of researchers are formed by their previous social experience, and that those who have been marginalized in society have experiences, and therefore prior beliefs, which can lead to greater insights than are available to those who have been privileged. Bayesians can conceptualize this as an example of a common approximation in Bayesian inference, which is used when evidence $E$ has been noted, for which some theories $T$ have not developed in such a way as to provide likelihood values $P(E\vert T)$, and which cannot, therefore, be used to infer posterior probabilities $P(T\vert E)$, from prior probabilities $P_0(T)$, through Bayes' theorem; instead, a prior probability $P(T)$ is used for future inference that attempts to estimate $P(T\vert E)$. It is noteworthy that the experiences of the marginalized can be taken into account, as standpoint epistemologists argue, by marginalized and privileged alike, obviating any need for ``extra votes'' for the marginalized.

Subjectivity also enters into academic study through the role of ideology in the selection of experiments. Statistical decision theory, which provides for Bayesian experiment selection, assumes that, for a set of available theories $S$, there is an uncertainty function $U(S)$ of the prior probabilities $P(T)$, $T \in{} S$, which represents how much is to be learned about which theory is true. The same uncertainty function, denoted $U(S\vert E)$, of the posterior probabilities $P(T\vert E)$, then represents how much remains to be learned about which theory is true, after an experiment $X$, with a set of possible results $R_X$, produces result $E \in{} R_X$. Before the experiment $X$ is performed, one can know the expectation of the uncertainty after it is performed,

$$U(S\vert X) = \sum_{E\in{}R_X}P(E)U(S\vert E)\textrm{,}$$ (4)

and can therefore measure the worth of the experiment by its information, defined as the expected reduction in uncertainty

$$I(X;S) = U(S)-U(S\vert X)\textrm{,}$$ (5)

and select experiments by offsetting their information against the direct moral and financial costs of performing the experiment. The link with ideology arises through the means by which the uncertainty function is chosen. The ideology that is to be implemented is represented by a loss function $L(y;T)$, which measures how bad the state of the universe will be, from the point of view of that ideology, if a policy $y$, from a set $Y$ of possible policies, is adopted, and theory $T \in{} S$ is true. The uncertainty, the quantity that one seeks to reduce by performing experiments, is then the expectation of the loss function, when the policy that minimizes that expectation is adopted

$$U(S) = \min_{y\in{}Y}\sum_{T\in{}S}P(T)L(y;T)\textrm{.}$$ (6)

Moreover, any continuous uncertainty function, which obeys the intuitive principle that no experiment can have a negative information, can be associated with an ideology through the above equation. Bayesians may, therefore, suspect that claims that a 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.

In addition to the mechanisms discussed above, ideology may 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, i.e. of research as a coalition-building tool.

When those associated with standpoint epistemology assert that science in Nazi Germany was used to ``depoliticize'' repressive policies, they appear to be referring to to an attempt to interpret scientific evidence in a manner that induced individuals, who were not necessarily Nazis themselves, to tolerate Nazi policies. In the standpoint epistemologists' analysis, this was achieved by making one theory, in which the consequences of not following Nazi policies appeared almost as bad as those of following Nazi policies, appear to be certainly true. The probabilistic nature of truth, as espoused by Bayesians, provides some protection against such a deception. However, even with Bayesian thinking, it is possible to rig a prior probability distribution, to produce a posterior probability distribution that has this de-politicizing property.

The prior probability distribution also has an effect on the posterior probability distribution through experiment selection. A toy problem can be constructed, in which there are two known theories, $T_1$ and $T_2$, a deterministic, as yet unknown, absolutely true theory $T_0$, and two possible experiments, $X$ and $Z$. $T_1$ predicts results for $X$, which agree with those from $T_0$ deterministically, and results for $Z$, which disagree with those from $T_0$ with high probability, while $T_2$ predicts results for $X$, which disagree with those from $T_0$ with high probability, and results for $Z$, which agree with those from $T_0$ deterministically. Using statistical decision theory, with the Shannon entropy as uncertainty function, to choose one experiment to undertake, it is found that whichever theory is assigned the higher prior probability is able to make the experiment that confirms it appear to be the most interesting experiment, and thereby to enhance its probability. This may be related to certain recent epistemological observations, outside of the standpoint epistemology tradition.