[linear algebra], [differential equations], [wrongthink]
I thought about writing a bit of prologue to this, providing some reflection on my emotional response to discovering behavioral genetics (there was a small existential crisis involved), but then I realized, Dude; this is Twitter. No one gives a fuck. And so I decided to jump right into the discussion. It starts with an ill-posed question:
How much of my personality (that is, that which makes me ME) is just due to what’s encoded in my DNA?
To be clear, we can’t possibly know. Behavioral genetics is geared toward estimating the percentage of trait variance due to genes in a population – not in an individual. More frustrating, the (say, narrow-sense) heritability \(h^2\), that many such papers estimate is subject to change simply because social conditions and/or influences become more varied or less varied. That is, these estimates are not immutable across time. I think it’s important to go a step further though, and point out that the above question is actually ill-posed, and it seems that most reasonable and informed persons discussing these topics tend to agree that indeed the question is rather ill-posed. More to the point, they tend to agree that this general question of “nature vs nurture” is so over-simplified that it shouldn’t even be asked because doing so inevitably increases misunderstanding rather than decreasing it. To people new to these ideas, it is rather confusing and frustrating that such questions are “not even wrong,” so I’d like to propose an analogous question which highlights the problem.
Consider a rectangle, with a length of 3 inches, and a width of 2 inches. What percentage of the area of this rectanble is due to length, and what percentage of the area is due to width?
I claim this question is nonsense. The area is the product of length and width rather than the sum, so it is meaningless to ask about percentage of area that’s due to length or width. I’d argue the question itself involves a category error (products versus sums), and I believe the general “nature vs nurture” question is similar. Indeed, genes and environment interact in massively complicated ways, so asking how much of a person’s personality is due to one or the other – or even which contributed more – is just a nonsense question. And I believe this holds more generally in regards to genes vs social conditioning. Our personality and behavioral traits are a complicated mix of our genes and our social environment, and we can’t say which is playing a bigger part.
Okay, so this the main thesis of this post, so I need to clarify my meaning so I don’t sound like I’m completely contradicting myself. Still, I think there is something interesting here, so let me try to explain with a simple model. Let’s consider a very simple society consisting of just two people which we’ll label \(x\) and \(y\). Suppose each has some level of a specific trait \(T\), and we’ll denote the respective trait levels \(T_x\) and \(T_y\) respectively. Let’s also suppose that the trait in one person is influenced by two things: that same person’s genetic contribution, and the other person’s trait level. For the sake of simplicity, suppose that the trait value \(T_x\) for person \(x\) is just the average of genetic contribution of \(x\), namely \(G_x\), and the trait value of person \(y\), namely \(T_y\). Writing out the equation, we have:
\[T_x = {\textstyle\frac{1}{2}}G_x + {\textstyle \frac{1}{2}} T_y.\]
Similarly for \(y\):
\[T_y = {\textstyle \frac{1}{2}}G_y + {\textstyle \frac{1}{2}}T_x.\]
The idea here is simply to allow that a person’s trait level is influenced by a combination of their own genetic predispositions, together with the traits of others. Yes, of course it can be more complicated, but for now let’s focus on this simple case. In particular, let’s use a bit of linear algebra to combine the two equations into one:
\[ \left( \begin{matrix} T_x \\ T_y \end{matrix} \right) = \frac{1}{2} \left( \begin{matrix} G_x \\ G_y \end{matrix} \right) + \left( \begin{matrix} 0 & {\textstyle \frac{1}{2}} \\ {\textstyle \frac{1}{2}} & 0 \end{matrix} \right) \left( \begin{matrix} T_x \\ T_y \end{matrix} \right). \]
Ah, but of course we can collect the \(T\) terms on one side, so let’s do that. \[ \left( \begin{matrix} 1 & {\textstyle -\frac{1}{2}} \\ {\textstyle -\frac{1}{2}} & 1 \end{matrix} \right) \left( \begin{matrix} T_x \\ T_y \end{matrix} \right) = \frac{1}{2} \left( \begin{matrix} G_x \\ G_y \end{matrix} \right). \]
Oh, but the matrix on the left hand side is non-singular, so we can compute its inverse as follows:
\[ \left( \begin{matrix} 1 & {\textstyle -\frac{1}{2}} \\ {\textstyle -\frac{1}{2}} & 1 \end{matrix} \right)^{-1} = \frac{4}{3} \left( \begin{matrix} 1 & {\textstyle \frac{1}{2}} \\ {\textstyle \frac{1}{2}} & 1 \end{matrix} \right). \]
So we can multiply both sides of our previous equation by that inverse to obtain
\[ \left( \begin{matrix} T_x \\ T_y \end{matrix} \right) = \frac{2}{3} \left( \begin{matrix} 1 & {\textstyle \frac{1}{2}} \\ {\textstyle \frac{1}{2}} & 1 \end{matrix} \right) \left( \begin{matrix} G_x \\ G_y \end{matrix} \right) \]
Or breaking it into two equations, we have
\[T_x = {\textstyle \frac{2}{3}}G_x + {\textstyle \frac{1}{3}}G_y\qquad \text{and} \qquad T_y = {\textstyle \frac{1}{3}}G_x + {\textstyle \frac{2}{3}}G_y. \]
Uhhh… Uh oh. This is clearly “problematic.” I mean, we started with this model in which a person’s trait level was just an average of 1) their own genetic influence with 2) the trait level of everyone else (just one other person in this case), and all we did was formulate this into an equation, and then it turned out we could solve it, so we did. And, as we can see, the trait value of person \(x\) and of person \(y\) is given explicitly as a function only of genetic influences. Put differently, the trait value of any given person is determined completely by the genetic contributions to that trait from everyone else in the population. That is: trait values of the population are determined completely by the genetic contributions in the population. In particular, the role of society (as some sort of explicitly influencing entity) doesn’t appear at all.
Okay, okay, this seems like the toy version of a toy model, and there are million and one ways to critique it, so let’s try to make it more robust. First, a two person society seems… well, let’s just say uninteresting, so suppose it’s \(n\) people instead, so we have people \(x_1, \ldots, x_n\) rather than just \(x\) and \(y\). For trait \(T\) the trait levels in the population will be given by \(T_1, \ldots, T_n\), and genetic contributions given by \(G_1, \ldots, G_n\). Also, suppose that each \(T_i\) equals a weighted average of the form \(p G_i + (1-p) \bar{T}_i\) where \(0 < p < 1\) and \(\bar{T}_i\) denotes the average trait level of everyone other than \(x_i\). In this case, the linear equation looks like the following:
\[ \left( \begin{matrix} T_1 \\ T_2 \\ \vdots\\ T_{n-1} \\ T_n \end{matrix} \right) = p \left( \begin{matrix} G_1 \\ G_2 \\ \cdots \\ G_{n-1} \\ G_n \end{matrix} \right) + \frac{1-p}{n-1} \left( \begin{matrix} 0 & 1 & \cdots & & 1 \\ 1 & 0 & 1 & & \\ \vdots & 1 &\ddots &\ddots & \vdots\\ & & \ddots& 0 & 1 \\ 1 & & \cdots & 1 & 0 \end{matrix} \right) \left( \begin{matrix} T_1 \\ T_2 \\ \vdots\\ T_{n-1} \\ T_n \end{matrix} \right). \]
It might not be clear from the equation, but the \(n\times n\) matrix above is the matrix which has a \(1\) in every entry except along the diagonal where the value is \(0\). This equation is still ugly to look at, so let’s rewrite it (with obvious substitutions) as the following:
\[ \vec{T} = p \vec{G} + (1-p) N \vec{T}. \]
Here I’m thinking of \(N\) has my social network of influence (\(N\) for network), which tells me how people influence other people. Anyway, if you are playing along at home, then you know the next step is to rewrite this as
\[ \big(Id - (1-p) N\big)\vec{T} = p \vec{G} \]
where \(Id\) is the identity matrix, and then hope that \(Id- (1-p)N\) is invertible. Fortunately the math gods are smiling today because as long as \(0 < p < 1\), this matrix is invertible. To prove this, first observe that this matrix has the form \(Id-A\) (with \(A= (1-p)N\)) and hence has inverse given by
\[ (Id-A)^{-1} = \sum_{k=0}^\infty A^k \]
which is defined whenever it converges, and it converges whenever \(\|A\| <1\); in our case we have \(A=(1-p)N\), so indeed \(\|A\|<1\) because \(0<p<1\) and because it can be shown (without too much difficulty) that \(\|N\| \leq 1\). But then once our inverse exists, we again have a formula the trait values in terms of the genetic contributions in the population. Indeed, it’s given by the following:
\[ \vec{T} = p \big(\sum_{k=0}^\infty (1-p)^k N^k \big) \vec{G}. \]
Again, the same idea resurfaces: the trait value of any one given person is completely determined by an aggregate of genetic influences of individual persons. There is no influence from “society” other than the structure of the social network that determines how the genetic aggregate is computed. And that’s what I mean when I say “But really it’s all just genes.” Naively formulated, it looks like “genes” and “society” both play important roles, but it seems that the influence of “society” is just some aggregate influence of “genes.”
Before declaring some sort of victory, let’s generalize a bit more. I think the next thing to modify is the matrix \(N\) which measures the network of social connections. I won’t be completely certain until I write a proof, but I’m pretty sure you can replace the given \(N\) with any matrix with the properties that
This would account for any weighted average of influence from other people. I may try to prove this, but at the moment I can’t seem to get the “lemma” environment to work properly in RMarkdown. Perhaps later.
Okay, the next biggest critique might well be that people’s traits change over time based on the influence of others. This is of course reasonable, but the fix is write down the differential equations which governs that change. My first attempt to model that would be to write the vector of trait values as a function of time: \(\vec{T}(t)\), so the change in trait values as a function of time is given by the derivative \(\vec{T}'(t)\). Then there should be a tendency for \(\vec{T}(t)\) to move toward our genetic predisposition, \(\vec{A}(t) = \vec{G} - \vec{T}(t)\), and a tendency to move toward the trait mean of the remaining population \(\vec{B}(t)=N\vec{T}(t)-\vec{T}(t)\). Suppose then that \(\vec{T}'(t)\) is the weighted sum of \(\vec{A}(t)\) and \(\vec{B}(t)\). Our differential equation then becomes:
\[\vec{T}'(t)=a\vec{A}(t) + b\vec{B}(t) = a\vec{G} +b N\vec{T}(t) - (a+b)\vec{T}(t).\]
Letting \(p=\frac{a}{a+b}\) and \(h=a+b\), we can rewrite this as
\[\vec{T}'(t)= ph \vec{G} +(1-p)h N\vec{T}(t) - h\vec{T}(t).\]
I’m fairly confident you don’t want to see me work this out, but this differential equation can be solved explicitly, and the rest points (where the trait values \(\vec{T}(t)\) end eventually end up at) are the exact same points found above, namely
\[ \vec{T} = p \big(\sum_{k=0}^\infty (1-p)^k N^k \big) \vec{G}. \]
Also of interest is that the solutions of this differential equation depend only on intial trait values and genetic contributions. And again this comports with the principle idea here: in some sense, it all just boils down to the population distribution of genetic influences, and everything else is determined from that. This is at least true of the steady state (i.e. rest point the dynamical system), but to another extent it’s also true of the trajectory the trait values take, although such trajectories depend on initial conditions as well.
Okay, there is a good chance I’ve nearly lost you, so if further discusions of model limitations and/or model generalizations sounds completely boring, then just skip down to the next section where I try to summarize my main points. I promise, I won’t think less of you. Actually, I already think more of you simply for making it this far.
Right, so the real world is way more complicated than the above model suggests; it’s imperative to make that clear. There are lots of interesting changes to the model to consider to make it more accurate. First and foremost I think is determining whether the system is first order or second order. I mean, it could totally be plausible that traits are governed by a differtential equation like some over-damped mass-spring system. I mean, why not? Or maybe the system is fully non-linear. For example, I could imagine if two people have traits which are very far apart, then maybe neither ends up changing because the other is simply too different. And two people who are different but closer together might move to balance much faster. It could be.
Another really interesting possibility would be to consider how that social network of influence \(N\) might change over time. Recall, this matrix records how much influence individuals have over which other individuals. Suppose that the further one’s trait is from their genetic predisposition, the more distress they feel as they are someone torn between their own desires and that of the rest of the people who influence them. This distress could be alleviated by changing the composition of people who have influence over said individual. There could be a tendencey to make more friends with people who are closer to one’s genetic predispositions, which would result in a changing \(N\). At first glance, this looks like a cool way to model the general polarization phenomena we see playing out in society. Could be rather intersting.
Hm. I should address my perhaps problematic use of the word “genes” throughout this post. Of course, what I should really say is that \(\vec{G}\) is supposed to represent immutable contributions to one’s trait values. Sure, this could be genetic influence, but it could also be random noise from the contstruction process of being built (or “growing up” as a normal person might say). Or it could be other environmental consequences that can’t be changed. This also would include technology changes which influence trait expression (for example, the effect of birth control on sexual freedom), which would change over time, but not due to social conditioning.
Oh, and another important point: Noise. Although I’ve clearly suggested that trait values are fundamentally deterministic, this completely ignores the role of chance events which can shape the trajectory, which almost certainly play an important role. Indeed, a more accurate title for the previous section would be “But really it’s all just genes and noise.” Unfortunately, I can’t discuss this improvement in any satisfactory way because my knowledge of stochastic differential equations is non-existent. To be revisited.
This ended up being far longer than I intended, and the point that was fairly clear in my head may have become completely lost in the math and text, so let me try to shore things up. We’re all familiar with the pseudo-strawman of the blank slate hypothesis, in which one believes that people’s traits are completely determined by social conditioning, with no immutable individuating contributions (like genetic influences). So with some hard work, and and open mind, one can take the eventual step towards what sounds like a much more reasonble position: People’s pyschological and behavioral traits are the result of some mix of both genetic/immutable influences and social conditioning influences. This of course seems to be Steven Pinker’s conclusion, as well as that of most behavioral geneticists and evolutionary pyschologists. And all these folks are completely correct in rejecting the extreme biological determinist position, which posits that individuals are shaped by their own genes alone.
However, the purpose of this post is to pick up where Pinker and others have left off, and suggest that personalities are shaped not by a mix of one’s genes and social conditioning, but rather just one’s genes and a mixture of everyone else’s genetic predispositions. That is, the role that society plays is more of an illusion than an actual conditioning force. More specifically, when we work at population level, and we formulate the relationship between traits and immutable and transmutable influences, a bit of compuation frequently seems to make the social forces disappear. Maybe I can narrow this further: I see no evidence that the steady-sate of a trait distribution is shaped by anything other than genetic influences at population level. In particular then, society itself does not shape anything, but rather society is that whih is shaped by genes.
Such a notion may well be heretical, but I’d like to see a plausible model which suggests otherwise. If such a model exists, we should be able to implement it for very small populations (say 2 - 12), and I’d like to get a better understanding. Until then though, it looks to me that at populations level, most traits should be expected to be completely governed by genes and noise.
-jwf