The Direct Role of the Prior Probability

As mentioned above (section 2.1,) logical probability is [34] a view of inference similar to the Bayesian one, but in which a particular prior probability distribution is regarded as objectively correct, and all others as invalid. Modern Bayesianism instead regards [42,43] any particular initial prior probability distribution as an arbitrary and subjective choice¹². For example, when Patrick and Wallace [52] undertook a Bayesian comparison of theories of the design principles behind stone circles, they stated, as a corollary of a set of axioms, `A continuous measure for the ranking of hypotheses and their evidentiary support is the length of the message that describes an hypothesis statement and the evidence (data) optimally encoded according to that hypothesis.' They derived from this a prior probability distribution

$$P(T) \propto{} \exp{}(-N(T))\textrm{,}$$ (18)

where $N(T)$ is the length of a computer program implementing theory $T$. The explicit statement of axioms is an apparent acknowledgement of the subjectivity of this prior probability distribution.

For the author, this subjectivity of the initial prior probability distribution is the intuitive principle that, before any evidence is examined, there is no basis, on which to recommend as right, or criticise as wrong, particular beliefs, save for the warning (section 2.1.1) against prior probability distributions that prevent account from being taken of whatever evidence may arise, and the provisional acceptance (section 2.1.4) of a zero prior probability for self-contradictory theories.

For a Bayesian, logical probability, and the positivist tradition, which the author argues (chapter 2) is closely associated with logical probability, do not remove this arbitrary and subjective choice from the interpretation of evidence and the problem of induction, but instead conceal the choice and render the chosen prior probability distribution immutable. It is easy to slip into the use of an undeclared prior probability distribution: the author, in reporting an experiment [30] designed to measure the spin distribution of electrons from a magnetic surface, as an intermediate step to inferring the magnetic properties of that surface, adopted a uniform prior probability distribution, over the ratio of the rate of electron arrival at each of the detectors in the experiment to the rate of production of electrons for the incident beam, without explicitly stating this. As the author subsequently realized, this prior probability distribution was inappropriate for the situation, because for some of the measurements, the likelihood was significant in regions of negative ratio, for which it would have been reasonable to assign a prior probability much lower than that for positive ratios.

Rather than claiming supremacy for a particular choice of prior probability distribution, Bayesian inference allows [42,43] for the analysis of the same evidence, from the starting points of many prior probability distributions. The Bayesian conception of objectivity is [34] that a posterior probability is objective if similar posterior probabilities arise from a wide range of prior probability distributions. Here, another similarity with standpoint epistemology becomes clear, for standpoint epistemology and the new social movements hold that the interpretation of evidence is [23,28] affected¹³ by the socially-generated prior beliefs of the interpreter, that objectivity is [4] achieved through a convergence, termed ``intersubjectivity'' and achieved by evidence-gathering, between the views of those with different prior beliefs, and that a sufficiently grounded theory can [23] only be achieved by combining the sciences developed by those with different previous experiences, in particular by people of differing gender, i.e. that many interpretations of reality each have [69] an important contribution to make to a complete and balanced understanding of the world. The different previous experiences can (section 5.1) be treated as influences on prior probability distributions.

Once objectivity is conceived as a lack of dependence of the posterior probability on the prior probability distribution, it is possible to propose measures of the subjectivity of a posterior probability $P(T_j\vert E)$. One possible measure is

$$S(T_j\vert E) = \sum_{T_i\in{}S}\left(\left(\frac{\mathrm{d}P(T_j\vert E)}{\mathrm{d}P(T_i)}\right)^2\right)\protect$$

$$= \sum_{T_i\in{}S}\left(\left(\frac{\delta_{ij}P(E\vert T_j)}{P(E)}-\frac{P(T_j)P(E\vert T_j)P(E\vert T_i)}{(P(E))^2}\right)^2\right)\textrm{.}$$ (19)

It could be instructive to analyse the qualitative features of this, and of other measures of subjectivity.

In Harding's [28,29] codification of standpoint epistemology, there is an apparent attempt to reach beyond the domain in which Bayesian analysis is useful, by deciding, qualitatively, the relative weight to be assigned to each individual's subjective interpretation of evidence (prior probability distribution;) the view appears to be expressed that the greatest weight should be given to interpretations by people who have been excluded or marginalized by the academic enterprise. It could be argued that to give enhanced weight to the views of the previously marginalized is just as undemocratic as to give enhanced weight to the views of the previously privileged.

However, standpoint epistemology's emphasis on interpretations by the marginalized has a purpose: the experience of a marginalized life provides [69,48,29] evidence $E_m$, which is not provided by the experience of a privileged life, rendering [69,48,29] those who have lived on the margins especially qualified to interpret some subsequent evidence of academic interest, particularly, although not only, in the social sciences. For a Bayesian, the ideal treatment of $E_m$ is the use of Bayes' theorem (equation 7) to generate posterior probabilities $P(T\vert E_m)$. However, $E_m$ might be in the same position as $E_0$ in section 5.1, requiring the use of an approximation method in which, for subsequent inference, a prior probability distribution is generated which has been adapted, in an attempt to take account of $E_m$. It is plausible that the experiences of the marginalized are particularly subject to the difficulty of theories not having been developed in such a way as to provide likelihood values for them. Either the exact Bayesian inference, or the generation of an adapted prior probability distribution, can be performed equally well by others (the advantaged) as by those (the marginalized) who discovered the evidence, so long as the latter are prepared to share their evidence, and the former are prepared to acknowledge it. There is then no necessity to give enhanced weight to the views of the previously marginalized, on the basis of their social identity.

Harding [29] is at pains to answer the criticism that to give enhanced weight to the views of the previously marginalized is just as undemocratic as to give enhanced weight to the views of the previously privileged, and to this end, stresses that it is taking account of the extra evidence available to the marginalized which is important, and that the advantaged can do this alongside the marginalized, so long as the latter are prepared to share their evidence, and the former are prepared to acknowledge it. As in the Bayesian approach, there is no necessity to give enhanced weight to the views of the previously marginalized.