Natural selection from a functional causal modelling perspective

With a principled way to distinguish natural selection from drift, I now move to the literature on causal modelling. This literature is very technical and I do not intend here to use all the theoretical apparatus developed by Pearl (2000) and followers to quantitatively study causality. Rather I will only use the parts of his account necessary for my purpose, which mostly is the fact

that causal modelling makes use of causal graphs and will thus render more intuitive my account of natural selection. Causal modelling is in fact a technique that allows to translate statistical data into the language of causality by using the combination of a set of equations (in the simplest form linear equations) often called structural equations and a graphical representation. The combination of the set of equations and the graphical representation is fundamental to the technique, since equations in and of themselves only allow represent symmetrical relations between variables or parameters.31 _{Yet causal relations are asymmetric relationships between}

variable – X causes Y is not equivalent to Y causes X. By combining equations to a graph, asymmetric causal relationships causality can be represented quantitatively.

Another particularly interesting property of causal modelling and causal graphs, for our purpose, is that they can be interpreted from a counterfactual perspective. In philosophy of science Woodward (2003) is one proponent of this approach. Apart from Woodward (2003) and Pearl (2000), a good introduction to causal graphs and causal modelling, less technical than Pearl (2000) and more empirically minded than Woodward (2003), can be found in Shipley (2002). In this section, I present very briefly and incompletely the causal modelling approach and propose a simple example of causal graph in the context of evolutionary change to illustrate this approach. This simple example will be made more complex in Section 1.6 to show how the use of causal graphs can help teasing apart natural selection from drift in purely individual causal terms. In

31 _{I use indistinguishably ‘variable’ and ‘parameter’ to mean a property of organisms or event changing a property of}

organisms. I make this choice to avoid the confusion between ‘intrinsic-variable properties’ and a ‘variable’ in a model.

Section 1.7 I will defend the ILC view against its main rival, namely the PLC view since both approaches can be used in the causal modelling framework.32

To start off, one can remark that Judea Pearl, the leading expert of the causal modelling approach, has an interpretation of probabilities throughout his book in terms of epistemic limitations that matches with Bouchard and Rosenberg’s view on probabilities in evolutionary theory. Pearl’s commitment to this interpretation of probabilities is fundamental to this approach (see Pearl 2000, xiii, 26-27). This suggests, without being decisive, that the reasoning behind functional causal modelling is well suited to test the validity of Bouchard and Rosenberg claims about the statistical nature of natural selection and that if B&R and I are wrong about causal nature of selection and drift, a causal modelling approach to natural selection relying on individual properties should be unsuccessful.

To illustrate the basic reasoning behind causal modelling, let us postulate a very simple case of evolution similar to the most basic models of population genetics. Suppose a population in an environment (E) composed of two types of haploid organisms differing only with respect to the phenotype ‘colour’ genetically determined at a single locus. One type of organisms is ‘white’ (W) and the other ‘black’ (B). For simplicity suppose that these organisms reproduce in discrete generations. Imagine now the case where we observe that although the population was initially composed of 50% of each type, after one generation, it is now composed of 70% of W and 30% of B, because Ws have overall more offspring than Bs. Suppose now that we want to know what the causes of this evolutionary change are. The only information we have about the organisms forming the population, besides their reproductive outputs, is that E is not perfectly

32 _{Note that the statistical approach being a non-causal one, it cannot be used to separate natural selection from drift}

homogeneous that is, has different states with parameters at different values. Call this simple case EC for ‘evolutionary change’.

Suppose now that the measures we have are exact or precise enough for our purpose. In front of this difference in frequency there are broadly three non-mutually exclusive possible explanations of the evolutionary change in EC:

 (drift) The two types have the same (ecological) fitness which means that B’s traits and W’s do not causally lead to any difference in their reproductive output, but Bs and Ws happened to have on average differences in extrinsic and/or intrinsic-variable properties leading to different reproductive outputs;

 (correlated response) There is a difference in the B and W’s traits (intrinsic-invariable properties) which leads Ws to produce more offspring than Bs, but the phenotype colour is not causally responsible for this difference, only correlated to it. Besides, there is no other difference in intrinsic-variable properties leading to different reproductive outputs;

 (natural selection) the explanation is the same as the previous one, but in this case, the phenotype colour is causally responsible for the difference in reproductive output between the two types.

Without more information on at least one biological mechanism33 _{causally responsible}

for the type W to have more offspring that the type B and involving the phenotype ‘colour’, establishing whether the population evolved by natural selection on this trait is impossible unless

33 _{I use a definition of mechanism taken from Machamer, Darden, & Craver (2000, 3): “Mechanisms are entities and}

activities organized such that they are productive of regular changes from start or set-up to finish or termination conditions.”

one makes blind assumptions on the distributions of events undergone by the two types. This would be equivalent to assume that a correlation between “colour” and “reproductive output” implies causation. Yet, this is famously problematic and the three explanations presented above are prima facie all plausible explanations if one has for sole information, differences in reproductive outputs.

Figure 1.3. DAG representing the variables involved in the reproductive output of an organism in the case EC.

This can be visualised using a directed acyclic graph (DAG) which is the type of graphical representation widely used in the causal modelling literature (see Pearl 2000). Figure 1.3 is a DAG that represents the variables involved in the determination of the reproductive output of a given organism in the case EC.34 _{Each box represents a variable for a given organism in the population.}

34 _{Note however that it only represents the reproductive output of one given type of organism in EC. Yet}

evolutionary change can easily be computed by calculating the difference between the numbers of offspring produced by the two types as displayed on Figure 1.8.

Each arrow represents a causal35 _{relationship between two variables. The variable preceding}

directly the bottom of an arrow is called a parent of the variable at the top (if it precedes it indirectly it is called an ancestor) and the variable directly at the top of this same arrow is called a child of the variable at the bottom (or if indirectly, a descendent). A parent or ancestor is understood as a cause and a child or descendent as an effect in a relation between two variables.36 _{For example, in Figure}

1.3, the variable ‘survival’ which can either take the value 0 (dead) or 1 (alive) for each organism, is a parent (that is, causally influences) the variable ‘reproduction’ (also binary: “produces no offspring” or “produces some offspring”). Manipulating an ancestor will make a difference in the value of its descendent. The precise relation between ancestors and descendants can be calculated by a set of functions providing the relations between parents and offspring and matching the variables in the DAG. In the case of EC the equations (written in a functional form here) are:

 ‘𝑅𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛’ = 𝑓 (_{‘o𝑡ℎ𝑒𝑟 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑖𝑛𝑔 𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛’)} ‘𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙’,

 ‘𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙’ = 𝑓 (‘𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑖𝑛𝑔 𝑠𝑢𝑟𝑣𝑖𝑣𝑎𝑙’)

 ‘𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑓𝑓𝑠𝑝𝑟𝑖𝑛𝑔’ =

𝑓 (‘𝑜𝑡ℎ𝑒𝑟 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑖𝑛𝑔 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑓𝑓𝑠𝑝𝑟𝑖𝑛𝑔’,_{‘𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛’} )

Using the distribution for each type of organism on each observed variable, one can assess quantitatively, using this set of functions, the relative effect of manipulating one variable

35 _{Bear in mind that by ‘causal’ no distinction between causal and constitutive relationship is made here since all the}

relationships described are diachronic.

36 _{More abstractly, each variable child is a function (linear or otherwise) of its parents. This function is then}

on the other variables downstream the graph and thus obtain a measure of the causal relation between them.

The graph on Figure 1.3 shows that the number of offspring a given individual produces is the result of three exogenous variables, namely ‘parameters conditioning survival’, ‘other parameters conditioning reproduction’37 _{and ‘other parameters conditioning number of}

offspring’ which are variables with no parents represented in the graph. This means that those variables are given as inputs into the model. Between the two variables ‘parameters conditioning survival’ and ‘other parameters conditioning reproduction’ and the variable ‘number of offspring’, which represents the number of offspring produced by an organism, there are two endogenous

variables namely ‘survival’ and ‘reproduction’ that is, variables which are functions of exogenous

ones or of other endogenous variables.

One important remark to make at this point, is that although functional causal graphs represent the underlying causes of individual variable (events or properties such as the number of offspring produced by one individual organism in our example), causal inference using causal modelling is a statistical technique that requires data on ensembles of entities. This is potentially confusing since we want to establish whether natural selection as a cause should better be understood at the individual or population (ensemble) level. Yet, the causes of a phenomenon should not be confused with the means of epistemologically reaching them. Thus even uncontroversial cases of causal inference of individual level events, population level data are necessary. Another important remark is that a causal graph supposes that the sum of all the causal

37 _{‘Other parameters conditioning reproduction’ includes all the parameters necessary for an individual to reproduce}

influence of a child comes directly from its parents. Thus there is no missing causal influence in the model.

The type of graph represented in Figure 1.3 is called a DAG because it contains no cycle and has a direction interpreted as a direction in time. Thus there are no feedback loops in a DAG. This feature is important if one assumes that causality is an asymmetric relationship between two variables in time (time’s arrow) which is an assumption I will make throughout the rest of the chapter. Once this assumption is made, no genuine feedback loop is possible. This is because a descendant cannot causally influence its ancestors, for its ancestors only exists earlier in time (Shipley 2002).38 _{However, a causal descendant can influence another particular descendant of its}

ancestor that bears some structural or constitutive similarities with its ancestors, leading to the impression that a genuine feedback loop exists, when it is in fact a pseudo one. A graphical representation of this kind of causal process such as depicted in Figure 1.4 with 6 variables ‘A1’, ‘B1’, ‘A2’, ‘B2’, ‘A3’ and ‘B3’ makes this idea salient. Once time is taken into account, A at time t1, is different from A at time t2 (A at time t1 is a parent of A at time t2) and thus no feedback loop

from B1 to A1 is possible. Another important property of causal graphs is that a DAG represents a complete causal structure, which is to say that all the sources of dependence are explained by the causal links.

38 _{Feedback loops are also possible when one talks about type causes that are not bound to particular space-time}

locations. Yet, causal modelling is a method developed to deal with token causes and thus bound to particular space- time locations.

Figure 1.4. DAG of a pseudo feedback loop between A and B when time is taken into account.