154 Metastatistics for the Non-Bayesian Regression Runner
but unable to demonstrate?” (As cited in Hacking, 1967. The published versions seems
to have omitted some of the text.) See Hacking (1967) for a very useful discussion of the
issue.
- For instance, although it is easy to see how to partition a set ofevents, it is not always
possible to see how to partition a set ofpropositions.
- See Einstein (1920) for a thought-provoking discussion of Euclidean geometry as mathe-
matical statements versus Euclidean geometry as statements about things in the world. As
to Feller’s observation about alternatives to Newton’s law of attractions, see Cartwright
(1984) for a provocative discussion of how even “the laws of physics lie” and physi-
cists often fruitfully use different and mutually inconsistent models that makes a related
point.
- For a marvelous introductory exposition see Ch. 21 of Hacking (2001). For a more com-
plete discussion that includes a bit more mathematical formalism and may be congenial
to economists, see 2.1 of Poirier (1995).
- There is much debate about the utility ofdefiningprobability as the limiting behavior of a
sequence. This debate is intimately related to the debate about commending an estimator
because, in repeated applications and in the long run, it would do well. The canonical
problem is the “single case” exception (Hacking, 2001, Ch. 22) in which we are asked to
consider a situation where, for example, there are two decks of cards: one, “the redder
pack,” has 25 red cards and 1 black card. The other “blacker pack” has 25 black cards and
1 red card. You are presented with two gambles. In the first gamble, you “win” if a red
card is drawn randomly from the “redder” pack and lose otherwise (P(Win)= 25 /26). In
the second gamble, you “win” if a red card is drawn randomly from the “blacker” pack
(P(Win)= 1 /26). If you “win” you will be transported to “eternal felicity” and, if you
lose, you will be “consigned to everlasting woe.” As Hacking (and C.S. Peirce) and most
people would choose the first gamble and hope, butnotbecause of the long run; if we
are wrong, there will be no comfort from the fact that wewould have been right most of
the time! Peirce’s “evasion of the problem of induction” is to argue that we should not
limit ourselves to merely “individualistic” considerations. “[Our interests] must not stop
at our own fate, but must embrace the whole community. This community, again, must
not be limited but extend to all races of beings with whom we can come in to immediate
or mediate intellectual relation. It must reach, however vaguely, beyond this geological
epoch, beyond all bounds. He would not sacrifice his own to save the whole world is,
as it seems to me, illogical in all his inferences collectively. Logic is rooted in the social
principle” (Peirce, 1878b, pp. 610–11).
- Such a condition rules out, for example, the deterministic series (H,T,H,T...H,T)
discussed above.
- See Gillies (2000) for a nice discussion.
- It is thus easier to understand Poirier’s emphasis in the quotation above on whether the
probability is “in nature” or “in themselves”: that “to the extent that individuals agree
on a class of events, they share an objective frequency. The objectivity, however, is in
themselves, not in nature.”
- Again I have ignored “logical” probabilities, which are a class of epistemic probabili-
ties which incorporate the notion of evidence. In this view, a probability is a “rational
degree-of-belief” about a proposition or a measure of the degree of “credibility” of a
proposition.
- For three mutually exclusive analyses of what Bayes meant, and whether or not he suc-
ceeded in proving what he set out to establish, see Hacking (1965, Ch. 12). On whether
Bayes’ understanding is consistent with subsequent Bayesian interpreters (beginning with
the rediscovery by Laplace, 1795), see Stigler (1982).
- We have omitted one detail in this exposition, which is that the expression we are
required to evaluate is:
L(θ|N,h)f(θ)
∫ 1
0 L(θ|N,h)f(θ)
.
