Back at Acme, our team of epistemologists are on the scene looking at Inspector 14’s record of length measurements for pole 453-01-120. Name this pole “p”. To simplify, suppose there are n physical measurements, the conditions for measurement were standardized, the measurement device was calibrated, errors are distributed normally, et cetera, et cetera:
1. The measured length-1 of p is 6.005 meters.
2. The measured length-2 of p is 6.003 meters.
n. The measured length-n of p is 5.099 meters.
Our assumptions about those n measurements license us to view them as a random sample from all possible measurements of p. This in effect is what it means to assume that errors are normally distributed. The negation of any of the other conditions (e.g., calibration, standardization) would function like defeaters for the randomization assumption, and block the inference I am about to make.
Because we have good reason to view those n measurements as randomly drawn, and no evidence that would suggest that n is a biased sample, we say the length of p is the mean of those n measurements plus or minus the product of 1.96 and 2 over the square root of n.
The example imagines that the mean of n is 6.003; the ‘plus or minus (1.96 x 2/root n)’ gives us the confidence interval, at 2 standard deviations, which we imagine to be [6.002, 6.004] meters. This says that the 95% confidence interval for the length of the flag pole p is [6.002, 6.004] meters. This expresses that 95% of the intervals calculated in this fashion will contain the true value of p; or, alternatively, that the probability that the true length of p is equal to a value within [6.002, 6.004] meters is 0.95.
Now to the philosophy of statistics:
The first point to notice is that we are talking about the actual length of the pole, which is a fixed-value (under standardized conditions, et cetera, et cetera) rather than a random variable. The actual length of the pole is either between 6.002 and 6.004 or it isn’t; there is no random variable to attach this 0.95 probability to. Rather, we say that we are confident to level 0.95 that the interval we calculated contains the length in meters of this pole. So, assuming here that 0.95 is a high enough confidence level, we accept flat out that the pole is between 6.002 and 6.004. And notice that this is what Inspector 14 does: he puts his “Inspected by 14” sticker on p and adds it to the stock of Model A’s; he does not put 95% of the sticker on the pole, nor hedge his bet by thinking that it is partially not in the stock of Model A’s. It’s an A, in his judgment, on his evidence.
Notice what happens when you switch to the Bayesian view, for you can tell a mathematically equivalent story in Bayesian dress. If you want to hedge, want to assign a probability to a proposition, then you’ve got to have a random variable. So, the Bayesian interpretation lets the length of the pole be a random variable! This doesn’t make sense directly, of course: a pole can be no more 95% a magnitude of length than a woman can be 95% pregnant. Either pole or woman has or has not the stated attribute. But, if you change the story around to talk about the belief about the pole, or the belief that a certain woman is pregnant, then you can view the interval as bounds on your (non-extreme) credal probability that the length of the pole (pregnancy status of the woman in question) is in that prescribed interval.
Now the jump from statistics to epistemology!
To effect this move, a Bayesian must assume a flat subjective prior probability distribution to get his mathematical representation of this story into equilibrium with the frequentist story. If you are a statistician doing this, this isn’t (or doesn’t need to be) such a big problem: you are interested in tools for modeling parameters, and you know about this equivalence between methods and can exploit whichever seems right for the task.
But, if you are an epistemologist making this move, you’ll likely be driven to do so by ideas about principles of rational belief fixation, and if you are making this move to explicate partial belief and use the Kolmogorov axioms as consistency constraints on such a notion, then you are doing so not as a technical ploy but rather because you think that issues of rationality are in play.
But for the epistemologist so-described the assumption of a flat prior takes on enormous weight, for it is not always reasonable to assume a flat prior, and the only reason that it is reasonable to do so here is because this is the assumption needed to get the Bayesian reconstruction to agree with the classical treatment of measurement! So, this reliance on a flat prior is crucial; the whole normative story hangs on it, but it is a thin reed from which to hang rational belief.
(I should stress that Ralph rejects probabilism, but he does accept a notion of partial belief; it remains to be seen whether there is a view of partial belief as opposed to full belief that escapes the general objection here.)
To tie this post up, a thought about the vaguely contemporary, mainstream view on the lottery paradox. In my view the move to avoid inconsistency by adopting partial belief pushes the key epistemic problems into the philosophy of science, where one finds ready assurances that the complaints I have just made are old and solved, or old and nearly solved, or at least old. From a review of the literature on the lottery paradox, the most striking feature I found in my reading was that at some point in the late 70s or early 80s, people largely stopped engaging the philosophy of statistics and instead focused on the thought experiments of the lottery and the preface. Epistemologists began writing on the particulars of these puzzles while assuming that the fundamental parameters of the puzzle were fixed, such as the interpretation of probability.
Christensen’s book is remarkable in this sense, because it is constructed entirely within this contemporary and restricted view of the paradoxes; it primarily engages the post 1980 “received view” literature.
There is, however, a recent lottery literature outside of this tradition, some of which is arguable outside of philosophy. Nevertheless, there is engaging work to consider. Joe Halpern addresses the lottery with a theory of first-order probability; the field of non-monotonic reasoning, going back to Hayes and McCarthy in 1969, has observed a tension between defeasible conditions for belief fixation and logical rules for manipulating those beliefs: David Makinson is more famous for the AGM belief revision paradigm than his Analysis paper; Eric Neufeld and Scott Goodwin rebut Pollock’s claim that there are fundamental differences between the lottery and the preface (which might make a good foil, Ralph, btw: Computational Intelligence, 14(3),1998); and, to self promote, I’ve proposed a limited system for 1-monotone capacities (JoLLI,2006) that gives a unified treatment of the lottery and preface, a statistical default logic (NMR2004) which treats the normality assumption for sample distributions as a default assumption, and a review of the lottery paradox in the 2007 Harper and Wheeler collection.