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Variance, Invariance and Statistical Explanation

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Abstract

The most compelling extant accounts of explanation casts all explanations as causal. Yet there are sciences, theoretical population biology in particular, that explain their phenomena by appeal to statistical, non-causal properties of ensembles. I develop a generalised account of explanation. An explanation serves two functions: metaphysical and cognitive. The metaphysical function is discharged by identifying a counterfactually robust invariance relation between explanans event and explanandum. The cognitive function is discharged by providing an appropriate description of this relation. I offer examples of explanations from portfolio theory and population genetics that meet this characterisation. In each case the invariance relation holds between a statistical property of an ensemble and a change in structure of the ensemble. In neither case, however, does the statistical property cause the outcome it explains. There are genuine statistical, non-causal scientific explanations.

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Notes

  1. Salient examples include Salmon (1984), Woodward (2003), Machamer et al. (2000), Craver (2007), Strevens (2004, 2008).

  2. Other examples abound, in economics, in the statistical interpretation of thermodynamics. But I shall restrict my attention here to theoretical population biology.

  3. A terminological clarification: by a statistical explanation I mean one in which the explanandum is (or involves) a statistic—a measure of the distribution of some property amongst the individuals of an ensemble. The terms ‘probabilistic’ and ‘statistical’ seem to be used interchangeably in much of the explanation literature, but they are not the same. All statistical explanations may be probabilistic, but not all probabilistic explanations are statistical.

  4. Sober (1984), Stephens (2004), Millstein (2006), Shapiro and Sober (2007) are among the proponents of the first view. Bouchard and Rosenberg (2004), Rosenberg (2006) appear to hold the second.

  5. Articulating the conditions under which the cognitive function is fulfilled is notoriously difficult. Hempel (1965), for example, tries to cash it out in a number of ways. He suggests that understanding the relation between explanans and explanandum gives us reason to believe that the phenomenon occurs (365–376). In other places he suggests that the relation gives us some kind of familiarity with the explanandum (430–433). See Lipton (2004). For my purposes it is enough to acknowledge that an explanation serves a cognitive function which may be left unfulfilled, even when the explanans condition has been identified.

  6. Woodward (2002, 2003) is sometimes explicit about this. In the passage quoted, the right kind of behaviour under intervention is merely sufficient for invariance.

  7. On certain reductive accounts of causation, this claim would turn out to be false. On some accounts, like that of Lewis (1973), change-relating invariance just is causation. On Lewis’ revised (2000) view, this is not so.

  8. She claims to have taken the idea that thick causal concepts explain from the modern mechanists.

  9. This being a philosopher’s investment portfolio, the only real payoff is conceptual. These are low-performing investments; each will be expected to show virtually no growth. The expected return on portfolio 1 is 99 % of the capital. The expected return on portfolio 2 is 101 % of the capital. The reason for the choice of these numbers will become clear in the next section.

  10. I’m not claiming here that no real change in the world could instantiate Variance Effect, or that sample variance cost could not explain a real occurrence. My only claim is that in this particular experiment the Variance Effect in question is not a real change in the world. I thank Philippe Huneman for help on this issue.

  11. The explanation can be found in Gillespie (1977). A lucid account of the effect of variance of reproductive output on fitness is offered by Orr (2007). The comparison with portfolio theory also comes from Orr.

  12. There are other such illustrations, see for example Beatty and Finsen (1989), Sober (2000). In these examples reproductive output varies only between generations and it is not stochastic.

  13. These figures are chosen to mimic the portfolio example, and also to meet the Gillespie condition of no expected population growth.

  14. The term ‘Gillespe Effect’ has been used in an unpublished manuscript by Michael Strevens. I thank Michael for helpful discussions in this topic.

  15. This is the sort of relation that leads many to claim that trait fitness is a cause of population change (Stephens 2004; Lewens 2009).

  16. In population ecology, the effect of variance in growth rate has been well studied. Here again, variation suppresses growth rate (Lewontin and Cohen 1969). The implications of variance in reproductive output for population genetics, life history strategies, (Lacey et al 1983) and foraging strategies (Stephens and Krebs 1986) have been extensively discussed.

  17. It may be more helpful to think of the distribution as a higher-order property, a property of the arrangement.

  18. Ironically, Shapiro and Sober (2007) appear to acknowledge this. They argue that one cannot conclude that the distribution is epiphenomenal and the arrangement causal because manipulations of this sort are inappropriate for determining the efficacy of putative causal properties A or B (or both) when there is a relation of supervenience between A and B.

  19. What follows is largely a rehearsal of an argument presented in Walsh (2010).

  20. STP is a necessary but not sufficient condition on a chance raising relation being causal.

  21. This argument is given in greater detail in Walsh (2010).

  22. This claim has been argued in various ways in a number of places (Matthen and Ariew 2002; Walsh et al. 2002; Walsh 2007, 2010).

  23. There are also, I contend, distinct modes of mathematical and teleological (Walsh 2013) explanations.

  24. I have argues elsewhere (Walsh 2006, forthcoming b) that the same sort of invariance relation holds between a goal and the various means for its attainment.

  25. I don’t disallow by fiat the possibility that sometimes these statistical relations are also causal. But I think we have seen that sometimes they are not.

  26. It isn’t entirely clear that Strevens’ proposed explanatory components map perfectly onto mine. This is an area for further investigation.

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Authors and Affiliations

  1. Department of Philosophy, IHPST, University of Toronto, Toronto, Canada

    D. M. Walsh

  2. Department of Ecology and Evolutionary Biology, University of Toronto, Toronto, Canada

    D. M. Walsh

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  1. D. M. Walsh
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Walsh, D.M. Variance, Invariance and Statistical Explanation. Erkenn 80 (Suppl 3), 469–489 (2015). https://doi.org/10.1007/s10670-014-9680-3

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