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standard error of a measurement that occurs in the following way:
Flip a fair coin.
- If heads, use measuring deviceAfor which the measurement is distributed
normally with variance one and expected value equal to the truth.
- If tails, use measuring deviceBwhich has zero measurement error.
What is therightstandard error ifBis chosen? Although I had not given the matter
a lot of thought before, it seemed obvious to me, a non-Bayesian, that the answer
would be zero. Thus it came as a surprise to learn that, on some accounts, a non-
Bayesian is “supposed” to give an answer of^1 + 20.^10 By way of contrast, the Bayesian
is described as someone who “naturally” avoids this inference, being “allowed” to
“condition” on whether the measurement was made with machineAorB.^11
As to the vehemence of the debate, LeCam (1977), a thoughtful non-Bayesian,
prefaced his (rare) published remarks on “metastatistics”^12 by observing:
Discussions about foundations are typically accompanied by much unnecessary
proselytism, name calling and personal animosities. Since they rarely contribute
to the advancement of the debated discipline one may be strongly tempted to
brush them aside in the direction of the appropriate philosophers. However,
there is always a ghost of a chance that some new development might be spurred
by the arguments. Also the possibly desirable side effects of the squabbles on
the teaching and on the standing of the debated disciplines cannot be entirely
ignored. This partly explains why the present author reluctantly agreed to add
to the extensive literature on the subject.
It is also a literature which (until recently) seemed almost entirely dominated
by “Bayesians” of various stripes – the iconoclastic Bayesian I.J. Good (Good,
1971) once enumerated 45,656 different varieties of Bayesianism. There are also
“objective” Bayesians and radical subjectivists. We might also choose to distinguish
between “full-dress Bayesians” (for whom estimation and testing is fully embed-
ded in a decision-theoretic framework) as well as a “Bayesian approach in mufti”
(Good and Gaskins, 1971). As I discuss below, this variety and depth results in part
from the view that probability and statistics are tools thatcanandshouldbe used
in a much broader variety of situations than dreamed of by the usual non-Bayesian
regression runner: “Probability is the very guide to life.” The non-Bayesian rarely
thinks of statistics as being anall-purposeway tothink(see Surprising Idea 3 in
section 3.2).
This is not to suggest that there isnonon-Bayesian philosophy involving statis-
tics. Most notably, Mayo (1996) has recently stepped in to present a broader view of
the philosophical underpinnings of non-Bayesian statistics that I find helpful, espe-
cially her notion of “severe testing.” And there is an older tradition as well: Peirce
(1878a, 1878b) and Venn (1888) are notable examples. The latter still remains an
exceptionally clear exposition of non-Bayesian ideas; the articles by Peirce inPopu-
lar Science Monthlyare insightful as well, but probably a slightly more difficult read.
Nonetheless, such examples are few and far between.
