Inferring Learning Strategies from Cultural Frequency Data

Abstract

Social learning has been identified as one of the fundamentals of culture and therefore the understanding of why and how individuals use social information presents one of the big questions in cultural evolution. To date much of the theoretical work on social learning has been done in isolation of data. Evolutionary models often provide important insight into which social learning strategies are expected to have evolved but cannot tell us which strategies human populations actually use. In this chapter we explore how much information about the underlying learning strategies can be extracted by analysing the temporal occurrence or usage patterns of different cultural variants in a population. We review the previous methodology that has attempted to infer the underlying social learning processes from such data, showing that they may apply statistical methods with insufficient power to draw reliable inferences. We then introduce a generative inference framework that allows robust inferences on the social learning processes that underlie cultural frequency data. Using developments in population genetics—in the form of generative simulation modelling and approximate Bayesian computation—as our model, we demonstrate the strength of this method with an example based on simulated data.

Appendix

In this chapter we assume that asocial and social learning strategies cause the cultural variants to change in frequency and describe those changes using a reaction–diffusion framework of the form

$$ \begin{aligned}[b]\frac{\partial {u}_i}{\partial t}\left(t,x\right)&=d\Delta {u}_i\left(t,x\right)-\nu {u}_i\left(t,x\right)+\xi \left(\mathrm{asocial}\ \mathrm{learning}\right) \\ &\quad +\left(1-\xi \right)\left(\mathrm{social}\ \mathrm{learning}\right),\ i=1,\dots, k\\ \frac{\partial K}{\partial t}\left(t,x\right)&=d\Delta K\left(t,x\right)+\left(\lambda -\nu \right)\\ &\quad \times K\left(t,x\right)\left(1-K\left(t,x\right)\right).\raisetag{16pt}\end{aligned} $$

(7.2)

Thereby the variable u _i describes the frequency of variant i at time t in the population, or in other words the fraction of the population that has adopted variant i. The variable K denotes the population size at location x and in the following we assume the population size to be the same for all locations x. It follows from the second equation in model (7.2) that $K\left(t,x\right)\le 1,\ \forall t$ and further, it holds ${\displaystyle {\sum}_{i=1}^k{u}_i\left(t,x\right)\le K\left(t,x\right)}$. The diffusion coefficient d describes the scale of spatial interactions, λ and ν the birth and death rates, respectively and ξ the reliance of the population on asocial learning. For sake of simplicity we stated the non-spatial version in the main text. All dynamics describes below hold in a similar way for this model.

In more detail, learning in various forms can increase or decrease the frequency of variant i. Asocial learning is based on the judgement about the benefit of specific variants in observed environmental conditions and consequently has two error sources: misjudgement of the current environmental condition and misjudgement of the adaptation levels of the different variants. Despite the conceptual differences both error sources lead to the same outcome in the modelling framework: a variant i is chosen for which holds ${\mu}_i\ne e$. Therefore the inaccuracy of asocial learning is modelled by assuming that asocial learning is based on $\overline{e}=e+\omega$ with $\omega \sim \mathcal{N}\left(0,{\sigma}_{\omega}^2\right)$. However, besides being error-prone asocial learning can lead to the introduction of new variants and its dynamic is modelled by (for sake of shortness we write $\overline{e}=\overline{e}\left(t,x\right)$)

$$ \begin{aligned}[b]&{P}_i\left({a}_i\left(\overline{e}\right)\right)\left(K\left(t,x\right)-{\displaystyle {\sum}_{j=1}^k{u}_j}\right) \\ &\quad +\displaystyle {\sum}_{j=1,j\ne i}^k\left({P}_{ji}\left({a}_i\left(\overline{e}\right),{a}_j\left(\overline{e}\right)\right){u}_j\left(t,x\right)\right. \\ &\quad \left.-{P}_{ij}\left({a}_i\left(\overline{e}\right),{a}_j\left(\overline{e}\right)\right){u}_i\left(t,x\right)\right)\end{aligned} $$

(7.3)

The parameter P _i describes the rate at which the fraction of the population which has not yet adopted a variants (described by the difference between the population size K(t, x) at time t and the sum of the fractions of the population which have adopted one of the k variants, $K\left(t,x\right)-{\displaystyle {\sum}_{j=1}^k{u}_j\left(t,x\right)}$) learns variant i asocially. P _i depends on the adaption level a _i (ē) meaning that asocial learning is not completely random: the higher the adaptation level in the estimated environment ē the higher is the adoption rate. Further, we allowed for the switching of variants which describes the process that individuals who already have adopted a cultural variant can switch to adopting a different variant. The coefficient P _ij models the rate at which the fraction of the population which has adopted variant i switches to variant j due to the evaluation of environmental cues. Again it holds the larger the difference ${a}_j\left(\overline{e}\right)-{a}_i\left(\overline{e}\right)$ between the estimated adaption levels the higher is the switching rate. Contrary to asocial learning, social learning is based on social cues and therefore can only lead to learning of variants which are already present in the considered location. In the considered framework we only considered two social learning strategies: direct biased social learning and conformist social learning. Direct biased social learning is modelled by (for sake of shortness we write $e=e\left(t,x\right)$)

$$ \begin{aligned}[b]&{r}_i\left({a}_i(e)\right){u}_i\left(t,x\right)\left(1-\frac{u_i\left(t,x\right)}{K\left(t,x\right)-\displaystyle {\sum}_{j=1,j\ne i}^k{u}_j\left(t,x\right)}\right) \\ &\quad +\displaystyle {\sum}_{j=1,j\ne i}^k\left({c}_{ji}\left({a}_i(e),{a}_j(e)\right)\right. \\ &\quad \left.-{c}_{ij}\left({a}_i(e),{a}_j(e)\right)\right){u}_i\left(t,x\right){u}_j\left(t,x\right).\raisetag{12pt}\end{aligned} $$

(7.4)

Similarly to the dynamic of asocial learning the first term

$$ {r}_i\left({a}_i(e)\right){u}_i\left(t,x\right)\left(1-\frac{u_i\left(t,x\right)}{K\left(t,x\right)-{\displaystyle {\sum}_{j=1,j\ne i}^k}{u}_j\left(t,x\right)}\right) $$

models the adoption of variant i by the population which has not adopted any variants yet. However contrary to asocial learning, this term is frequency-dependent. It is a logistic growth process with adoption rate (or intrinsic rate of increase) r _i and broadly speaking describes cultural reproduction. Per definition, the population size K(t, x) at location x is the upper limit of the total fraction of adopters in the population at this location (given by ${\displaystyle {\sum}_{j=1}^k{u}_j\left(t,x\right)}$), regardless of the adopted variant. Consequently, the upper limit for the fraction of the population that has adopted variants i is given by $K\left(t,x\right)={\displaystyle {\sum}_{j=1}^k{u}_j\left(t,x\right)}$ (i.e. we assume that our cultural variants compete for a common pool of adopters). The adoption rate r _i is assumed to be proportional to the adaptation level a _i in the currently experienced environmental condition e. It holds: The higher the adaptation level the higher is the adoption rate. The second term

$$ {\displaystyle {\sum}_{j=1,j\ne i}^k\left({c}_{ji}\left({a}_i(e),{a}_j(e)\right)\right.}\\{\left.-{c}_{ij}{a}_i(e),{a}_j(e)\right){u}_i\left(t,x\right){u}_j\left(t,x\right)} $$

describes the switching dynamic between the fractions of the population which has already adopted a variant. Again we assumed that individuals who have already adopted a variant have the chance to switch to another variant and therefore the different cultural variants compete with each other for use. These interactions between the variants are described by the terms c _ij (a _i (e), a _j (e))u _i (t, x)u _j (t, x) which model the switch process from variant i to variant j. The strength of this process is determined by the rate c _ij and it holds: The higher the difference ${a}_j(e)-{a}_i(e)$ of the adaptation levels of both variants the higher is the switching rate. In order to include conformist social learning we allowed these model parameters to be frequency-dependent. We assumed

$$ \begin{aligned}\tilde{r}_i&=\left(1-b\right){r}_i\left({a}_i(e)\right)+b\left({u}_i\left(t,x\right)-{c}_bK\left(t,x\right)\right)\ \mathrm{and}\\ {\tilde{c}}_{ij}&={\left[\left(1-b\right){c}_{ij}\left({a}_i(e),{a}_j(e)\right)+b\left({u}_j\left(t,x\right)\right.\right.}\\ &\quad\quad {\left.\left.-{c}_bK\left(t,x\right)\right)\right]}^{+}\end{aligned} $$

where b controls the reliance on adaptation information and frequency information, respectively. For b = 0 we obtain direct biased learning while b > 0 supports variants with a frequency higher than the commonness threshold c _b K(t, x). In this case the difference $\left({u}_i\left(t,x\right)-{c}_bK\left(t,x\right)\right)$ is positive and the adoption rate ${\tilde{r}}_i$ is increased. Contrary if $\left({u}_i\left(t,x\right)-{c}_bK\left(t,x\right)\right)$ is negative (and therefore variant i has a relatively small frequency) the adoption rate ${\tilde{r}}_i$ is decreased. A similar dynamic applies to the switching rate ${\tilde{c}}_{ij}$. If the frequency of variant j (the target of the switch process) exceeds the commonness threshold c _b K(t, x) then the rate ${\tilde{c}}_{ij}$ with which variant i is substituted by variant j is increased. The symbol ${\left[.\right]}^{+}$ denotes the positive part of any real number (e.g. ${\left[3.4\right]}^{+}=3.4$ but ${\left[-3.4\right]}^{+}=0$) ensures that there is no reversal of the switch direction.

We note that when considering a single cultural variant the dynamic of asocial learning (7.3) results in r-shaped adoption curve while the dynamic of asocial learning (7.4) results in a S-shaped curve whereby the existence of a conformist tendency (b > 0) produces long tails at the beginning and an accelerated adoption behaviour when the commonness threshold is exceeded. System (7.2) can be solved using the Finite-Element method and we obtain the time course of the frequencies u _i . of each cultural variant that are expected under the assumed learning hypothesis and environmental change.