Priors: Silent or Active Partners of Bayesian Inference?
3.1 Priors in the Very Beginning
In Bayes' celebrated paper, posthumously submitted and added to by Price (Bayes and Price), there is no explicit statement of Bayes' theorem that highlights the role of the prior. Instead Bayes' argument focused on the need to evaluate credible intervals. Thus began the quiet but important role of priors in Bayesian inference.
In the appendix, co-author Price did argue why assumptions, equivalent to assuming a uniform prior distribution, would be appropriate, while allowing the possibility that priors could be informed by other information (Bayes and Price 1763): ‘What has been said seems sufficient to shew us what conclusions to draw from uniform experience. ...supposing our only data derived from experience, we shall find additional reason for thinking thus if we apply other principles, or have recourse to such considerations as reason, independently of experience, can suggest.' In contrast de la Place (1774) simply accepted a uniform prior as an ‘intuitively obvious axiom'. These two approaches marked the beginning of ‘efforts to discover the statistical holy grail: prior distributions reflecting ignorance’ (Fienberg 2006), and of a continuing debate (Section 3.1.2) on objective and informative priors (Sections 3.2 and 3.3).
From the premise ...