Two Dogmas of Strong Objective Bayesianism

Abstract

We introduce a distinction, unnoticed in the literature, between four varieties of objective Bayesianism. What we call ‘strong objective Bayesianism’ is characterized by two claims, that all scientific inference is ‘logical’ and that, given the same background information two agents will ascribe a unique probability to their priors. We think that neither of these claims can be sustained; in this sense, they are ‘dogmatic’. The first fails to recognize that some scientific inference, in particular that concerning evidential relations, is not (in the appropriate sense) logical, the second fails to provide a non‐question‐begging account of ‘same background information’. We urge that a suitably objective Bayesian account of scientific inference does not require either of the claims. Finally, we argue that Bayesianism needs to be fine‐grained in the same way that Bayesians fine‐grain their beliefs.

Acknowledgements

We would like to acknowledge the help given by us by John Bennett, José Bernardo, Robert Boik, Dan Flory, Jack Gilchrist, Megan Higgs, Tasneem Sattar, Teddy Seidenfeld, and C. Andy Tsao for their comments related to several parts of the paper. An earlier version of the paper was presented at the ‘Objective Bayesianism’ conference held in the London School of Economics and Political Science in 2005. Both authors very much appreciate the comments they received there. They are especially thankful both to two anonymous referees for their insightful comments regarding the contents of the paper and to the editor of this journal for his helpful suggestions. We also thank Jan‐Willem Romeijn, John Welch, and Gregory Wheeler for their encouragement regarding the paper. Special thanks owe to John Bennett for his several e‐mails containing suggestions about the paper. The research for the paper has been financially supported both by our university’s NASA’s Astrobiology Center (Grant number 4w1781) with which both of us are associated and the Research and Creativity grant from our university.

Notes

[1] Williamson (2005) is presently the most vocal proponent of this position. See also Williamson (2007, 2009).

[2] Here we borrow the term ‘necessitarian’ from Levi (1980) who credits Savage with this coinage. For Levi’s position on objective necessitarianism see Levi (1980), especially chapters 15–17.

[3] Jaynes also expressed the same idea in his earlier writing: ‘the most elementary requirement of consistency demands that two persons with the same relevant prior information should assign the same prior probabilities’ (Jaynes 1968). Interestingly, Howson (Howson and Urbach 1993), who is a subjective Bayesian, accepts the first thesis of the strong objective Bayesian, but rejects her second thesis.

[4] In this respect we agree with Fisher’s judgement that ‘[a]lthough some uncertain inferences can be rigorously expressed in terms of mathematical probability, it does not follow that mathematical probability is an adequate concept for the rigorous expression of uncertain inferences of every kind’ (Fisher 1956, 40; emphasis added).

[5] Or robots, perforce ‘rational’ on Jaynes’s account.

[6] Our discussion is not intended to imply that there is no difference among these three objectivists. Although both Rosenkrantz (1977, ix and 254) and Williamson (2005, ch. 11) discuss why the probability theory is one significant tool for understanding objective Bayesianism, they are not as emphatic as Jaynes is to take the probability logic as an extension of deductive logic and then contend that this is the only way to approach scientific inference. However, what matters for our purpose is a general agreement among strong objective Bayesians about these two theses earlier attributed to them. We owe this point to one of the referees of this journal.

[7] The information matrix is the negative expected value of the second derivative of the log‐likelihood function with respect to the parameter.

[8] Some of these ideas go back to Hartigan (1983). For details of a recent, very general mathematical proof of this fact, see Berger, Bernardo, and Sun (2009).

[9] Jeffreys (1961) has sometimes been considered an objective Bayesian. See Bandyopadhyay (2007) for a discussion of Jeffreys’s views on scientific inference.

[10] For a nice historical discussion on the subjective and objective distinction in the probability theory dating back to 18 and 19th centuries please consult Zabell, 2010.

[11] A Bayesian of our kind should very well be able to construe an evidence relation to be independent of what the agent believes about those two competing hypothesis. The present‐day subjective‐Bayesian‐dominated philosophy of science is, however, slow to appreciate this point. For a subjective Bayesian, since the evidence relation is defined in terms of a ratio of two probabilities, those probabilities have to be understood in terms of an agent’s degree of belief. Therefore, according to her, the evidence relation must be subjective. However the situation need not be like that. A subjective Bayesian‐flavoured textbook on econometrics explains this point well and discusses how an objective Bayesian could approach the likelihood function which is at the core of our evidential account: ‘[t]he likelihood function was simply the data generating process (DGP) viewed as a function of the unknown parameters, and it was thought of having an “objective existence” in the sense of a meaningful repetitive experiment. The parameters themselves, according to this view, also had an objective existence independent of the DGP. … It is possible to adopt the so‐called objective Bayesian viewpoint in which the parameters and the likelihood maintain their objective existence …’ (Poirier 1995, 288–289).

[12] For some recent review of the literature on Bayes factor see Ghosh, Delampady, and Samanta (2006, chapter 6). This book also contains some interesting discussion and evaluation of objective Bayesianism.

[13] This way of reconstructing the example owes a great deal to the help we received from James Hawthorne, Teddy Seidenfeld, Peter Gerdes, and Robert Boik. These names are arranged in the order we received their help.

[14] We owe this point to one of the referees. We thank the editor for suggesting how to formulate the point of disagreement some Bayesians might have with us at this point.

[15] Our comments on entropy cannot properly be generalized to continuous variables. However, the purpose of this example addressing entropy needs to be taken as an illustrative case for explaining the intuition behind strong objectivism, and not in the spirit whether this example is generalizable. As our discussion of the example explaining entropy holds in discrete cases, this example serves its intuitive function.

[16] We owe this point of clarification to Robert Boik.

[17] Both the title of this section and the title of the paper have been motivated by Quine’s celebrated article ‘Two Dogmas of Empiricism’ (Quine, 1953).

[18] One of the referees has raised worries about this question.

[19] One possible objection to our version of Bayesianism is that the rejection of the both strong objectivist’s theses can’t satisfactorily be combined with quasi‐Bayesianism. According to the objection, in the case of the confirmation/evidence accounts, we assumed simple statistical hypotheses, whereas for the possibility of allowing infinitely many possible posterior probabilities for competing hypotheses we assumed statistical hypotheses to be complex. As we know, values of simple statistical hypotheses are independent of an agent’s degree of belief, but values of complex hypotheses are dependent on an agent’s degree of belief. Thus, the objector concludes our version is inadequate to be unified under one Bayesian framework. Our response is that both points, (i) assuming simple statistical hypotheses for the confirmation/evidence accounts and (ii) taking complex statistical hypotheses to be dependent on prior beliefs, are consistently accommodated within a Bayesian framework while keeping two issues distinct. Quasi‐Bayesianism acts like a Swiss army knife that has various functions. When we are called upon to perform a specific function we apply a specific aspect of our version to address it. Our version is far from the idea of ‘one size fits all’ philosophy. Whether other statistical schools have that flexibility is beyond the scope of the present paper.

[20] This belief is current among both Bayesian and non‐Bayesian philosophers: see Swinburne (2002) and Sober (2002). Although Swinburne is a Bayesian and Sober is a non‐Bayesian, they share the view about a sharp division into subjective and objective camps. For a sample of this rampant oversimplification among statisticians, look at any of the statistical literature on objective Bayesianism listed in the references.

[21] Interestingly, Good (1983) has some similarities with what we have been arguing here. Like Good, we are arguing that there are several varieties of Bayesianism. However, unlike him, we don’t have any specific number regarding how many such varieties exist or are possible.