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ABSTRACT
This second part of my reply to Davidson (Citation2015) discusses mathematical and statistical matters using a simple model of ergodicity whose properties do not match many of those asserted by ENE. It responds to Davidson's counterarguments against pre-infinity indeterminacy, examines the works of Billingsley, Uffink, and Malinvaud as cited by Davidson, and then turns to important methodological issues. As in Part 1 of my reply, Davidson's rejoinder helps make the critique stronger, deeper, and more relevant.
Notes
1In this context, the shorthand expression “at infinity” used in my critique is to be understood as “at infinity that never arrives,” or “until infinity that is never reached.” No suggestions of final destinations are intended.
2These data sets then become community- or activity-specific. The further complications that this introduces in estimating the true probability distribution are not pursued here.
3Virtually any other series of outcomes could be chosen.
4Remember that some of them will have experienced economic failure/disappointment during this period.
5All unreferenced page numbers in this article are to PD's rejoinder in Davidson (Citation2015).
6On the latter, see, for example, O'Donnell (Citation1989, pp. 259–262; Citation1991, pp. 40–45).
7If time-independence prevails here, then ergodicity prevails, in which case there is no need for it to be assumed.
8This mistreatment of Billingsley's text is not an isolated instance. The same rearranged attribution and wrong conclusion is present in Davidson (Citation1988, p. 332n1) and Davidson (Citation1996 p. 35n8). See also the section on Uffink below.
9Note that sampling error, even if declining, implies a degree of uncertainty about the underlying probability distribution, this uncertainty being inconsistent with the proposition that the distribution can be certainly (reliably) calculated here and now.
10ISLM provides a good example; see O'Donnell and Rogers (Citation2016, pp. 358–363).
11This intuitive argument assumes ε is finite. It is more likely to be infinite, however, in which case the a priori probability is a small infinity divided by a very much larger infinity.
12In PD's references, the publication year is wrongly given as 1949, but since the other details are correct, I assume the 1966 first English edition is the relevant volume.
13Further, after retention in the second English edition of 1970, the brief discussion was dropped altogether from the third edition of 1980.
14On the latter two points, see Uffink, Citation2006, pp. 87, 98, 34, and n22. It is now my turn to thank the anonymous referee for bringing Uffink's paper to our attention.
15Also of interest here is that Laplace brings Keynes back to the concept of “unknown probabilities,” which I view as the strongest (but not the only) HAC foundation for the irreducible uncertainty in the GT and Quarterly Journal of Economics (QJE) paper. See O'Donnell (Citation1989, chs. 3, 12; Citation1991, ch. 1).
16PD's claims here are simply a recycled version of his earlier arguments in Davidson (Citation1996, pp. 29, 35n1).
17Consideration of both the full correspondence and wider contexts would also assist here.
18The potency or impotence of policy under both ergodicity and nonergodicity is a far more complicated subject than ENE's simplistic rendering.
19By contrast, HAC has no such structural defects. It begins, stays, and ends with humans, institutions, and social systems in the here and now, deploys modes of analysis appropriate to its subject matter, recognizes uncertainty in all forms, and emphasizes epistemology.