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Probability and Manipulation: Evolution and Simulation in Applied Population Genetics

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Abstract

I define a concept of causal probability and apply it to questions about the role of probability in evolutionary processes. Causal probability is defined in terms of manipulation of patterns in empirical outcomes by manipulating properties that realize objective probabilities. The concept of causal probability allows us see how probabilities characterized by different interpretations of probability can share a similar causal character, and does so in such way as to allow new inferences about relationships between probabilities realized in different chance setups. I clarify relations between probabilities and properties defined in terms of them, and argue that certain widespread uses of computer simulations in evolutionary biology show that many probabilities relevant to evolutionary outcomes are causal probabilities. This supports the claim that higher-level properties such as biological fitness and processes such as natural selection are causal properties and processes, contrary to what some authors have argued.

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Notes

  1. I briefly sketched the idea of causal probability in (Abrams 2012a), and gave summary descriptions of the coalescent simulation example in Abrams (2012a, b).

  2. My conclusions are in some respects sharper than previous conclusions about natural selection and drift as causes based on arguments involving manipulability or counterfactuals (Reisman and Forber 2005; Forber and Reisman 2007; Shapiro and Sober 2007; Huneman 2012, 2014b).

  3. I’ll assume in Sect. 3.2 that none of these “realization” relationships between properties of chance setups and probabilities are causal relationships.

  4. A complex chance setup can include continuously varying probability distributions as a function of the outcome of an earlier trial if we treat each such distribution as a distinct chance setup.

  5. Except where noted, I’ll use “propensity” to mean single-case propensity, in its traditional philosophical sense originated by Popper (1959, 1973), which takes propensities to be dispositions of varying strengths satisfying probability axioms. This has become known as a “tendency” account of single-case propensity (Eagle 2004; Berkovitz 2015) from the use of this term in e.g. (Mellor 1971; Giere 1973). It’s the simplest notion of propensity, and I suspect that it’s what many philosophers of biology have had in mind when they’ve gone beyond vague generalities about propensity. Humphreys’ paradox (Salmon 1979; Humphreys 1985; Lyon 2014) is an argument that propensities are not probabilities, because if the propensity that A causes B is represented as a conditional probability, the application of Bayes theorem leads to paradoxes. Fetzer (1981), Miller (1994), and Berkovitz (2015) provide possible ways of dealing with Humphreys’ paradox; for this paper, I would adopt one of the strategies discussed by Berkovitz that treat propensities as probabilities. Eagle (2004) and Berkovitz (2015) survey a variety of criticisms of propensities; another criticism is provided in (Abrams 2007, 2012c).

  6. Debates about frequentist versus Bayesian statistics are not unknown in biology, however (Clark 2005).

  7. The emphasis on prediction suggests that Walsh et al. (2002) intend “expected” and “expectation” in a psychological rather than mathematical sense. This point would be clearer with the addition of the quotation's surrounding context.

  8. Except on some classical, logical, or epistemic interpretations, but such probabilities need not be related to patterns in the world (Abrams 2012d).

  9. Gelman and Nolan (2002) explain that it’s impossible for a coin to have a significant bias if it’s flipped in the air and doesn’t bounce before coming to rest. A coin that’s spun rapidly on a point on its edge and allowed to fall can be biased, however. Assume that Olivia tosses or spins her coins in such a way that the stated biases exist.

  10. Excluding, possibly, cases in which the limit is singular, a point due to Joseph Berkovitz (personal communication). Such cases are rare in most areas of science.

  11. Walsh (2007 pp. 283ff) gave an example illustrating what he called “statistical properties of populations” (p. 283)—summary properties defined in terms of the probabilities that generated those populations. Walsh argued that the sense in which natural selection and random drift explain evolutionary outcomes depends only on this kind of statistical property, and that this implies that selection and drift are not causes of evolution. Space constraints prevent me from discussing Walsh’s arguments, but I believe that the point made in this section has the potential to undercut them.

  12. Such outcomes could, of course, still have epistemic or logical or classical probabilities.

  13. Of course there could still be a determinate epistemic probability for heads.

  14. There is another kind of mixed interpretation case, in which two interpretations of probability are characteristic of a system, but there is no real connection between the two interpretations. One kind of case involves probability defined as frequency, where the processes that generate frequencies involve propensities. Such cases do not really raise new issues, and in the interest of space, I don’t discuss them here.

  15. This section refines and extends the notion of causal probability in (Abrams 2012a).

  16. Causal probability bears some similarities to to Lyon’s (2011) concept of “deterministic probability”, but Lyon’s goals are different from mine.

  17. The problem arises for infinite long-run propensity interpretations, too, but these interpretations have more serious problems (Hájek 2009).

  18. For example, if the single-trial probabilities are independent and identically distributed, and the probability of twelve is .05, then the probability that the frequency of twelve will be 1/36 in 72 trials is \(\left( {\begin{array}{c}72\\ 2\end{array}}\right) \times .05^2 \times (1-.05)^{70} \cong .176\) (e.g. Grimmett and Stirzacker 1992).

  19. Strevens (2011; cf. 2008) argues that his interpretation of probability, which is similar to Rosenthal’s and mine, does provide single-case probabilities.

  20. Berkovitz’s (2015) discussion shows how difficult it is to make this intuition more precise.

  21. “Probabilistic patterns” are “probabilistic” only in that they’re often generated by chance setups that realize probabilities; that they are isn’t essential.

  22. The fact that we use frequencies to guide our attributions of probabilities to dice doesn’t change the fact that the probabilities, if they are objective, usually bear the right relationships to frequencies.

  23. It’s better to imagine that Y is, say, vector-valued, rather than scalar-valued; probabilistic patterns might involve complex patterns of frequencies.

  24. Woodward’s (2003) statements M, IV, and IN on pages 59 and 98 summarize the main ideas used here.

  25. The probability-quantity-realizing properties may be complex, and the variables could be vector-valued or function-valued. For example, \(X_1\) might range over variations in distributions of densities within pairs of dice.

  26. For example, suppose that there are some values of \(X_2, \ldots , X_m\) for which there is a variable I with a value that (1) would cause \(X_1\) to take on a specific value, and this manipulation would also cause Y to take on a particular value, in such a way that (2) the value of \(X_1\) no longer depended on the values of other variables \(X_i\), and (3) I affected Y only via I’s effect on \(X_1\), where (4) I is probabilistically independent of any variable that could affect Y by causal paths not passing through \(X_1\) (cf. Woodward 2003, pp. 59 and 98).

  27. Do causal probabilities suffer from Humphreys’ paradox (Salmon 1979; Humphreys 1985) (cf. note 5)? Here’s a short answer in an already long paper: Yes, but only if we treat the causal probability of an outcome \(A_r\) given properties \(p_i\) of a chance setup as a normal conditional probability \(\mathsf {P}(A|\cap _i p_i)\). (cf. Miller 1994; Berkovitz 2015).

  28. However, I’m open the possibility that some conceptions of causation would allow Best System interpretations to count as causal. Note that Hoefer (2007) allows some probabilities to be defined in terms of “statistical nomological machines” (SNMs), devices which tend to generate outcomes with certain patterns of frequencies. If Hoefer attributed probabilities only to outcomes of SNMs, his theory might be a good candidate for a causal interpretation, but he also allows probabilities to depend more directly on frequencies.

  29. To take the an extreme case, suppose that \(S_0\) realized a probability near 1 that \(S_1\) was to be tokened, but usually produced the outcome that causes trials of \(S_2\). Then probabilities of outcomes of S would be near the probabilities realized by \(S_1\), but S’s frequencies of outcomes would be near the probabilities realized by \(S_2\).

  30. I don’t consider hypothetical infinite frequencies, whose problems are too deep (Hájek 2009).

  31. One might think that my argument implies that all causal probabilities ultimately depend on probabilities realized at the level of fundamental physics if they don’t bottom out in frequencies at some level. This conclusion depends on denying the possibility that there can be chance setups whose probabilities depend only on higher-level properties. However, CCS probabilities depend on higher-level properties, and long-run propensities can as well.

  32. This example is based on one described by Joseph Berkovitz (personal communication).

  33. These properties might include facts about what you believe about the two coins, for example.

  34. Given the arguments here, Matthen and Ariew’s (2002, 2009) willingness to attribute causal powers to individual organisms may be in tension with their view that natural selection is a not cause.

  35. The simple scientific examples common in philosophy of biology can give the impression that relationships between data, models, and conclusions in evolutionary biology depend only on simple properties, which in turn makes it easier to think that biologists’ claims about the occurrence of natural selection, drift, and so on are very abstract summaries of an underlying complexity, as statisticalists sometimes claim. The example I discuss, by contrast, is one in which biologists’ conclusions are justified by richly structured probabilistic patterns in the data, which reflect some of the underlying complexity of biological processes. Moreover, the example I use illustrates the fact that population genetics is not an isolated theoretical discipline that answers only to its own rules (some remarks in (Matthen 2009) may suggest this view), but is intimately connected to practical applications in many parts of biology.

  36. For evidence that this pattern is common, see e.g. (Hoban et al. 2012). Cursory examination of two arbitrarily chosen issues of the top journal Evolution (2013, volume 67, numbers 5 and 6) turns up at least six articles illustrating this pattern.

  37. Here “homozygosity” refers to similarity of patterns of alleles in homologous chromosomes that are randomly chosen from all individuals in a population or sample. Comparing randomly chosen chromosomes provides a way of measuring the frequency of sequences of alleles in a population.

  38. On the order of 1000 generations.

  39. I briefly summarize details in a supplementary document that will be available from the author’s website (http://members.logical.net/~marshall) or upon request.

  40. Figure 2 was generated according to a Wright-Fisher model with fixed population size, asexual reproduction, and no natural selection, using the gray-scale version of lescent (Abrams 2015).

  41. This isn’t a species tree.

  42. When we add recombination to this scheme, relations between lineages won’t fit into a tree structure, since segments of token chromosomes from one lineage can be combined with a segments from those in another lineage, producing a chromosome derived from different branches. However, we can still use knowledge of the probability of mutations per segment length and per number of generations to generate realistic distributions of patterns of SNPs on which to calculate iHS values (Hudson 1991; Wakeley 2009; Tavaré 2003).

  43. Gillespie (2004, section 2.6) provides an introduction to basic concepts of coalescent theory. Rosenberg and Nordborg (2002) and Tavaré (2003) do as well, and also introduce coalescent simulations. Good textbooks on coalescent theory include Hein et al. (2005) and Wakeley (2009).

  44. For example, because there is no natural selection in the basic coalescent model, individual organisms can be modeled as having no characteristics—they are bare identities—and random variables for individuals’ numbers of offspring are exchangeable. This allows one to calculate the probability that two organisms have the same parent in a simple manner from assumptions about population size, without taking into account differences in traits, reproductive success, etc.

  45. There are also other, more direct arguments for conclusions closely related to mine, e.g. (Abrams 2012a, b, 2013; Huneman 2012, 2013, 2014b; Millstein 2006; Northcott 2010; Ramsey 2006; Reisman and Forber 2005; Forber and Reisman 2007; Shapiro and Sober 2007; Sober 2013).

  46. I’m taking computer simulations to be similar to concrete models, such as animal models (e.g. mice) that are used in experiments designed to learn about human genetics. This view that simulations are in many respects like experiments is defended by Winsberg (2010). More specifically, Voight et al.’s (2006) use of simulations is an instance of the kind of “weak simulations” (Huneman 2014a) that are designed to test a hypothesis for which relevant data is difficult to collect. Huneman argues that weak simulations represent what they do partly by realizing causal processes isomorphic to real-world causal processes—as I claim, for Voight et al.’s simulations. (It might be necessary to extend Huneman’s account slightly for this case; his suggestion that weak simulations merely provide “candidate explanation[s]” (Huneman 2014a, p. 76) may give a misleading view of cases like this one, in which the theory and data on which the simulations are based is very well-established.)

  47. ms can be run by a command at a command prompt with parameters specified on the command line or in a file. Thus, for example, the chance setup for a particular simulation run might be the computer after the ms command and parameters have been typed in but before the scientist has hit the Enter key.

  48. Of course random number generators are deterministic, but for all we know, so is coin tossing, even if we include the physiological state of a coin tosser in the chance setup. It’s true that even very good random number generating algorithms will eventually repeat patterns after a very long series of trials (Park and Miller 1988). My assumption that some interpretation of probability applies to a program depending on the operation of a random number generator would be warranted only in those cases in which the random number generator is used appropriately within the limits of its capabilities. Readers who doubt that an interpretation of probability applies to algorithmic random number generators might take my arguments to apply to an alternative set of simulations in which random numbers are generated by a person tossing a coin.

  49. I also think Voight et al.’s assumptions are justified by inferences from past successes with similar methods, but I won’t argue for that point here.

  50. At the time that Voight et al. (2006) did their research, coalescent simulation software couldn’t represent both natural selection and population size changes. Voight et al. used the SelSim package (Spencer and Coop 2004) to simulate natural selection for the African population, which wasn’t thought to have experienced extreme size fluctuations.

  51. See Sect. 2.4 and e.g. (Brandon 1990; Millstein 2006; Abrams 2012b; Sober 2013; Matthen and Ariew 2002, 2009; Matthen 2010; Walsh 2007).

  52. See (Abrams 2007, 2009a, 2012b, 2014); see also (Ariew and Ernst 2009).

  53. Not an actual frequency distribution.

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Acknowledgments

The editors of this issue, Philippe Huneman and Joseph Berkovitz, both provided extremely helpful comments and conversation about the content of my paper at various stages of its development. Extra thanks are due to Jossi, who repeatedly gave very detailed comments, leading to a much deeper, more precise, and in the end very different paper. Denis Walsh and Thomas Reydon read and provided feedback on versions of this paper along the way. I also benefited from feedback from and conversations with Prasanta Bandyopadhyay, Anouk Barberousse, John Bickle, Kristin Boyce, Michael Bruno, Isabelle Drouet, Sara Franceschelli, Carl Hoefer, Paul Humphreys, William Kallfelz, Francoise Longy, Francesca Merlin, Juan Montana, Wayne Myrvold, John Norton, Samir Okasha, Daniel Parker, Charles Pence, Michael Strevens, Jackie Sullivan, Jos Uffink, Scott Weingart, and Jim Woodward. I received helpful comments from audience members at the Workshop on Probability in Physics and Biology, Institut d’Histoire et de Philosophie des Sciences et des Techniques, Université de Paris 1; the International Society for History, Philosophy, and Social Studies of Biology, Brisbane, Australia; the University of Pittsburgh’s Department of History and Philosophy of Science; Mississippi State University’s Department of Philosophy and Religion; and the European Philosophy of Science Association.

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  1. Department of Philosophy, University of Alabama at Birmingham, Birmingham, AL, USA

    Marshall Abrams

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Abrams, M. Probability and Manipulation: Evolution and Simulation in Applied Population Genetics. Erkenn 80 (Suppl 3), 519–549 (2015). https://doi.org/10.1007/s10670-015-9784-4

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