Error Statistics

C. S. Peirce: 10 Sept, 1839-19 April, 1914

Memory Lane* in Honor of C.S. Peirce’s Birthday:
(Part 3) of “Peircean Induction and the Error-Correcting Thesis”

Deborah G. Mayo
Transactions of the Charles S. Peirce Society 41(2) 2005: 299-319

(9/10) Peircean Induction and the Error-Correcting Thesis (Part I)

(9/10) (Part 2) Peircean Induction and the Error-Correcting Thesis

8. Random sampling and the uniformity of nature

We are now at the point to address the final move in warranting Peirce’s [self-correcting thesis] SCT. The severity or trustworthiness assessment, on which the error correcting capacity depends, requires an appropriate link (qualitative or quantitative) between the data and the data generating phenomenon, e.g., a reliable calibration of a scale in a qualitative case, or a probabilistic connection between the data and the population in a quantitative case. Establishing such a link, however, is regarded as assuming observed regularities will persist, or making some “uniformity of nature” assumption—the bugbear of attempts to justify induction.

But Peirce contrasts his position with those favored by followers of Mill, and “almost all logicians” of his day, who “commonly teach that the inductive conclusion approximates to the truth because of the uniformity of nature” (2.775). Inductive inference, as Peirce conceives it (i.e., severe testing) does not use the uniformity of nature as a premise. Rather, the justification is sought in the manner of obtaining data. Justifying induction is a matter of showing that there exist methods with good error probabilities. For this it suffices that randomness be met only approximately, that inductive methods check their own assumptions, and that they can often detect and correct departures from randomness.

… It has been objected that the sampling cannot be random in this sense. But this is an idea which flies far away from the plain facts. Thirty throws of a die constitute an approximately random sample of all the throws of that die; and that the randomness should be approximate is all that is required. (1.94)

Peirce backs up his defense with robustness arguments. For example, in an (attempted) Binomial induction, Peirce asks, “what will be the effect upon inductive inference of an imperfection in the strictly random character of the sampling” (2.728). What if, for example, a certain proportion of the population had twice the probability of being selected? He shows that “an imperfection of that kind in the random character of the sampling will only weaken the inductive conclusion, and render the concluded ratio less determinate, but will not necessarily destroy the force of the argument completely” (2.728). This is particularly so if the sample mean is near 0 or 1. In other words, violating experimental assumptions may be shown to weaken the trustworthiness or severity of the proceeding, but this may only mean we learn a little less.

Yet a further safeguard is at hand:

Nor must we lose sight of the constant tendency of the inductive process to correct itself. This is of its essence. This is the marvel of it. …even though doubts may be entertained whether one selection of instances is a random one, yet a different selection, made by a different method, will be likely to vary from the normal in a different way, and if the ratios derived from such different selections are nearly equal, they may be presumed to be near the truth. (2.729)

Here, the marvel is an inductive method’s ability to correct the attempt at random sampling. Still, Peirce cautions, we should not depend so much on the self-correcting virtue that we relax our efforts to get a random and independent sample. But if our effort is not successful, and neither is our method robust, we will probably discover it. “This consideration makes it extremely advantageous in all ampliative reasoning to fortify one method of investigation by another” (ibid.). Continue reading →