Excursion 2 Tour I: Induction and Confirmation (Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars)
Tour Blurb. The roots of rival statistical accounts go back to the logical Problem of Induction. (2.1) The logical problem of induction is a matter of finding an argument to justify a type of argument (enumerative induction), so it is important to be clear on arguments, their soundness versus their validity. These are key concepts of fundamental importance to our journey. Given that any attempt to solve the logical problem of induction leads to circularity, philosophers turned instead to building logics that seemed to capture our intuitions about induction. This led to confirmation theory and some projects in today’s formal epistemology. There’s an analogy between contrasting views in philosophy and statistics: Carnapian confirmation is to Bayesian statistics, as Popperian falsification is to frequentist error statistics. Logics of confirmation take the form of probabilisms, either in the form of raising the probability of a hypothesis, or arriving at a posterior probability. (2.2) The contrast between these types of probabilisms, and the problems each is found to have in confirmation theory are directly relevant to the types of probabilisms in statistics. Notably, Harold Jeffreys’ non-subjective Bayesianism, and current spin-offs, share features with Carnapian inductive logics. We examine the problem of irrelevant conjunctions: that if x confirms H, it confirms (H & J) for any J. This also leads to what’s called the tacking paradox.
Quiz on 2.1 Soundness vs Validity in Deductive Logic. Let ~C be the denial of claim C. For each of the following argument, indicate whether it is valid and sound, valid but unsound, invalid. Continue reading
My new book, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars,” you might have discovered, includes Souvenirs throughout (A-Z). But there are some highlights within sections that might be missed in the excerpts I’m posting. One such “keepsake” is a quote from Fisher at the very end of Section 2.1.
These are some of the ﬁrst clues we’ll be collecting on a wide diﬀerence between statistical inference as a deductive logic of probability, and an inductive testing account sought by the error statistician. When it comes to inductive learning, we want our inferences to go beyond the data: we want lift-oﬀ. To my knowledge, Fisher is the only other writer on statistical inference, aside from Peirce, to emphasize this distinction.
In deductive reasoning all knowledge obtainable is already latent in the postulates. Rigour is needed to prevent the successive inferences growing less and less accurate as we proceed. The conclusions are never more accurate than the data. In inductive reasoning we are performing part of the process by which new knowledge is created. The conclusions normally grow more and more accurate as more data are included. It should never be true, though it is still often said, that the conclusions are no more accurate than the data on which they are based. (Fisher 1935b, p. 54)
How do you understand this remark of Fisher’s? (Please share your thoughts in the comments.) My interpretation, and its relation to the “lift-off” needed to warrant inductive inferences, is discussed in an earlier section, 1.2, posted here. Here’s part of that.
Where you are in the Journey*
Cox: [I]n some ﬁelds foundations do not seem very important, but we both think that foundations of statistical inference are important; why do you think that is?
Mayo: I think because they ask about fundamental questions of evidence, inference, and probability … we invariably cross into philosophical questions about empirical knowledge and inductive inference. (Cox and Mayo 2011, p. 103)
Contemporary philosophy of science presents us with some taboos: Thou shalt not try to ﬁnd solutions to problems of induction, falsiﬁcation, and demarcating science from pseudoscience. It’s impossible to understand rival statistical accounts, let alone get beyond the statistics wars, without ﬁrst exploring how these came to be “lost causes.” I am not talking of ancient history here: these problems were alive and well when I set out to do philosophy in the 1980s. I think we gave up on them too easily, and by the end of Excursion 2 you’ll see why. Excursion 2 takes us into the land of “Statistical Science and Philosophy of Science” (StatSci/PhilSci). Our Museum Guide gives a terse thumbnail sketch of Tour I. Here’s a useful excerpt:
Once the Problem of Induction was deemed to admit of no satisfactory, non-circular solutions (~1970s), philosophers of science turned to building formal logics of induction using the deductive calculus of probabilities, often called Conﬁrmation Logics or Theories. A leader of this Conﬁrmation Theory movement was Rudolf Carnap. A distinct program, led by Karl Popper, denies there is a logic of induction, and focuses on Testing and Falsiﬁcation of theories by data. At best a theory may be accepted or corroborated if it fails to be falsiﬁed by a severe test. The two programs have analogues to distinct methodologies in statistics: Conﬁrmation theory is to Bayesianism as Testing and Falsiﬁcation are to Fisher and Neyman–Pearson.