Online research in philosophy

In order to make scientific results relevant to practical decision making, it is often necessary to transfer a result obtained in one set of circumstances—an animal model, a computer simulation, an economic experiment—to another that may differ in relevant respects—for example, to humans, the global climate, or an auction. Such inferences, which we can call extrapolations, are a type of argument by analogy. This essay sketches a new approach to analogical inference that utilizes chain graphs, which resemble directed acyclic graphs (DAGs) except in allowing that nodes may be connected by lines as well as arrows. This chain graph approach generalizes the account of extrapolation I provided in my (2008) book and leads to new insights that integrate the contributions of the other participants of this symposium. More specifically, this approach explicates the role of “fingerprints,” or distinctive markers, as a strategy for avoiding an underdetermination problem having to do with spurious analogies. Moreover, it shows how the extrapolator’s circle, one of the central challenges for extrapolation highlighted in my book, is closely tied to distinctive markers and the Markov condition as it applies to chain graphs. Finally, the approach suggests additional ways in which investigations of a model can provide information about a target that are illustrated by examples concerning nanomaterials in sunscreens and Wendy Parker’s discussion of fingerprints in climate science.

We present a novel algorithm for the fast computation of PageRank, a hyperlink-based estimate of the “importance” of Web pages. The original PageRank algorithm uses the Power Method to compute successive iterates that converge to the principal eigenvector of the Markov matrix representing the Web link graph. The algorithm presented here, called Quadratic Extrapolation, accelerates the convergence of the Power Method by periodically subtracting off estimates of the nonprincipal eigenvectors from the current iterate of the Power Method. In Quadratic Extrapolation, we take advantage of the fact that the first eigenvalue of a Markov matrix is known to be 1 to compute the nonprincipal eigenvectors using successive iterates of the Power Method. Empirically, we show that using Quadratic Extrapolation speeds up PageRank computation by 50- 300% on a Web graph of 80 million nodes, with minimal overhead.

Zusammenfassung Unser Wissen von Zukunft beruht weder auf Erfahrung noch auf Extrapolation aus Erfahrung, sondern ist apriorisch. Da es also apriorisches Wissen von Zeit gibt, kann eine apriorische Lösung des Induktionsproblems sinnvoll versucht werden.

We present a novel technique for speeding up the computation of PageRank, a hyperlink-based estimate of the “importance” of Web pages, based on the ideas presented in [7]. The original PageRank algorithm uses the Power Method to compute successive iterates that converge to the principal eigenvector of the Markov matrix representing the Web link graph. The algorithm presented here, called Power Extrapolation, accelerates the convergence of the Power Method by subtracting off the error along several nonprincipal eigenvectors from the current iterate of the Power Method, making use of known nonprincipal eigenvalues of the Web hyperlink matrix. Empirically, we show that using Power Extrapolation speeds up PageRank computation by 30% on a Web graph of 80 million nodes in realistic scenarios over the standard power method, in a way that is simple to understand and implement.

Method by subtracting off the error along several nonprincipal eigenvectors from the current iterate of the Power Method, making use of known nonprincipal eigenvalues of the Web hyperlink matrix. Empirically, we show that using Power Extrapolation speeds up PageRank computation by 30% on a Web graph of 80 million nodes in realistic scenarios over the standard power method, in a way that is simple to understand and implement.

Inferences like these are known as extrapolations.

(Chapter 5 of Across the Boundaries, forthcoming, from Oxford University Press) This chapter argues that previous accounts of extrapolation, either by reference to capacities or mechanisms, do not adequately address the challenges confronting extrapolation. It then begins the account of how the mechanisms-approach can be developed so as to do better. The central concept in this account is what I term comparative process tracing.

Any account of extrapolation from animal models to humans must confront two basic challenges: explain how extrapolation can be justified even when there are causally relevant differences between model and target, and explain how the suitability of a model can be established given only limited information about the target. We argue that existing approaches to extrapolation—either in terms of capacities or mechanisms—do not adequately address these challenges. However, we propose a further elaboration of the mechanisms approach that provides a better treatment of this issue. The central concept in our proposal is what we term comparative process tracing.