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Bias and population structure in the actuation of sound change

Why do human languages change at some times, and not others? We address this longstanding question from a computational perspective, focusing on the case of sound change. Sound change arises from the pronunciation variability ubiquitous in every speech community, but most such variability does not lead to change. Hence, an adequate model must allow for stability as well as change. Existing theories of sound change tend to emphasize factors at the level of individual learners promoting one outcome or the other, such as channel bias (which favors change) or inductive bias (which favors stability). Here, we consider how the interaction of these biases can lead to both stability and change in a population setting. We find that population structure itself can act as a source of stability, but that both stability and change are possible only when both types of bias are active, suggesting that it is possible to understand why sound change occurs at some times and not others as the population-level result of the interplay between forces promoting each outcome in individual speakers. In addition, if it is assumed that learners learn from two or more teachers, the transition from stability to change is marked by a phase transition, consistent with the abrupt transitions seen in many empirical cases of sound change. The predictions of multiple-teacher models thus match empirical cases of sound change better than the predictions of single-teacher models, underscoring the importance of modeling language change in a population setting.


Introduction
Language changes over time: words come and go, pronunciations shift, and the structure of sentences mutates, such that the 'same' language becomes unintelligible to speakers of earlier generations. While language change is far from deterministic, it is often strikingly systematic. Indeed, it is the regularity of sound-meaning correspondences between words in different languages (e.g. Latin pedis, pater, pisces vs. English foot, father, fish) that licenses hypotheses about a common ancestor. Documenting these sound changes helped to establish linguistics as a scientific discipline in the 18th-19th centuries (Jones, 1788), resulting in a rich knowledge of what types of sound changes have occurred in the world's languages (Kümmel, 2007;Paul, 1880).
For almost as long, linguists have asked why sound change occurs-in particular, why particular changes take place, or actuate, at the time and place they do-a question which has proven much harder to answer, known as the 'actuation problem' (Baker, 2008;Baker et al., 2011;Garrett and Johnson, 2013;Weinreich et al., 1968). One strand of research has emphasized the role of universal phonetic pressures or channel biases that introduce systematic, potentially asymmetric errors in transmission of a phonetic signal between teacher and learner (Blevins, 2004;Moreton, 2008;Ohala, 1993). A commonly cited example of a channel bias is coarticulation, which causes a speech sound to be produced differently depending on the preceding and following sounds. Sound changes such as Germanic i-umlaut, whereby low back vowels were fronted and raised when a high front vowel or glide occurred in the following syllable (e.g. Proto-Germanic gasti > West Germanic gesti 'guests', modern German Gäste), have been proposed to find their source in this kind of conditioned variation (Blevins, 2004;Iverson and Salmons, 2003;Ohala, 1993). This leads to a view of actuation as a two-stage process: first, an individual learner interprets a coarticulated variant as conventional (Ohala, 1981); then, via a process of cultural transmission, the change subsequently spreads throughout the speech community (Labov, 2010;Milroy, 1980). While intuitively plausible, important aspects of this model remain to be fully specified. First, if channel biases such as coarticulation are universally active, why are all languages not constantly changing (Baker, 2008;Baker et al., 2011;Weinreich et al., 1968)? It is clear that the presence of bias does not invariably result in change: for instance, even while umlaut was spreading throughout the West Germanic languages, it did not affect Gothic (Cercignani, 1980). An adequate model of sound change must therefore also account for the possibility, even ubiquity, of stable variation at the level of the speech community. One explanation for stability would be the existence of (possibly domain-general) inductive biases guiding human inferences, which may facilitate or inhibit the learning of certain types of structures or patterns (Briscoe, 2000;Griffiths and Kalish, 2007;Kalish et al., 2007;Reali and Griffiths, 2009). Inductive biases have been proposed that favour phonetically-motivated hypotheses about phonological patterns over phonetically arbitrary ones (e.g. substantive biases: Moreton, 2008;Steriade, 2008;Wilson, 2006) 1 or which promote the stability of existing phonetic category structures over the creation of new ones (e.g. categoricity biases: Pierrehumbert, 2001;Wedel, 2006). However, if such preferences are strong enough to counteract channel bias, then how can change ever occur? Finally, when change does diffuse throughout a speech community, it often occurs suddenly following a period of prolonged stability (Kroch, 1989;Labov, 2010). What types of constraints on transmission and learning might interact to produce this type of rapid shift from one stable state to another?
In this paper, we address these questions by modeling the acquisition and propagation of a phonetic parameter in a population setting. Our goal is a model that predicts both stability and change in the presence of biases promoting the other outcome, and in which small changes in the magnitude of bias produces a sudden and nonlinear change from one stable state to another. Because such questions about language change are difficult to address empirically, we approach this problem from the perspective of computational and mathematical modeling, drawing on a large body of previous work in this tradition (Blythe and Croft, 2012;Boyd and Richerson, 1985;Burkett and Griffiths, 2010;Cavalli-Sforza and Feldman, 1981;Dediu, 2009;Griffiths and Kalish, 2007;Griffiths et al., 2013; J. Kirby, 2013;S. Kirby et al., 2007;Komarova et al., 2001;Kroch, 1989;Niyogi, 2006;Niyogi andBerwick, 2009, 1997;Pierrehumbert, 2001;Reali and Griffiths, 2009;Smith, 2009;Smith et al., 2003;Sonderegger and Niyogi, 2010;Wedel, 2006). Our approach differs crucially from previous work in two respects. First, while models of language change often frame the learner's task as choosing between competing discrete variants (Baker et al., 2011;Kroch, 1989;Niyogi, 2006;Sonderegger and Niyogi, 2010;Wang et al., 2004;Yang, 2000), a key part of learning the sound pattern of a language is learning distributions over continuous phonetic parameters, such as vowel formants (Vallabha et al., 2007). Second, in most existing models that have considered continuous parameters, change only and always occurs in the presence of a channel bias (Baker, 2008;J. Kirby, 2013;Pierrehumbert, 2001;Wedel, 2006). Here, we propose a model in which both stability and change of a continuous parameter are possible in the presence of channel bias.
By stability, we are referring to the structure of the stationary distribution of the continuous parameter in the population. Might stability at the population level have its roots in the inductive biases of individual learners? This seems plausible given work on the dynamics of cultural transmission showing that the distribution of a cultural trait evolves linearly to a unique stationary state that reflects the structure of learners' prior S. Kirby et al., 2007;Reali and Griffiths, 2009). However, while this result holds for chains of single teacher-learners, in general the dynamics become nonlinear as the population structure becomes more complex (Burkett and Griffiths, 2010;Dediu, 2009;Niyogi and Berwick, 2009;Smith, 2009), opening up the possibility of very different outcomes-such as stability and change-from similar initial conditions (Niyogi and Berwick, 2009). In what follows, we thus consider population structures of increasing complexity, and assess our models on the basis of their ability to explain how a nonlinear transition from stable variation to sound change could occur.

Model
We consider a scenario in which each agent may (a) function as a learner, receiving input from other agents and applying a learning algorithm to this input in order to learn a probability distribution over how a continuous parameter is realised, and (b) function as a teacher, generating data from this distribution for other learners. Within this framework, there are many assumptions one could make about each of these actions. Here we consider variants on a simple supervised learning scenario, where all that needs to be learned is a distribution over a single phonetic dimension, parametrized by a single continuous parameter. For concreteness, our exposition follows the example of umlaut described in the Introduction, but the basic results are applicable to the learning of a single continuous parameter more generally.

Linguistic setting
We assume that speech sounds have been organised into discrete segments, and that the learner has access to the complete segmental inventory. We consider here a simple language with the lexicon Σ = {V 1 , V 2 , V 12 }, where V 12 represents V 1 in the context of V 2 . V 1 and V 2 can be thought of as the vowels /a/ and /i/ in isolation, and V 12 as /a/ in a context where it is coarticulated (raised) towards /i/.
Tokens are represented by their first formant (F1) value, an acoustic measure of vowel height (Hillenbrand et al., 1995). 2 . We assume that the F1 distributions for V 1 and V 2 are normal (N (µ a , σ 2 a ), N (µ i , σ 2 i )), are known to all learners, are the same for all learners, and do not change over time. 3 The distribution of V 12 is normal, with (fixed) variance as for V 1 and a mean we denote by c: We will sometimes refer to V 12 (or equivalently, c, which determines the distribution of V 12 ) as the contextual variant.
In addition, we assume that productions of V 12 are subject to a coarticulatory channel bias corresponding to the general tendency in speech production to over-or undershoot articulatory targets based on speech context (Lindblom, 1983;Pierrehumbert, 2001). We take this bias to be normally distributed with mean −λ (because V 2 has lower F1 than V 1 ) and variance ω 2 , and to be applied i.i.d. to each vowel token. Thus, the actual productions of V 12 by a teacher with contextual variant c follow the distribution

Learning and evolution
We assume agents are divided into discrete generations of size M . Each learner in generation t + 1 receives n examples of V 12 (distributed according to Eq. 2) from one or more teachers from generation t. The learner's task is to infer c by application of some learning algorithm. We assume that learners apply a learning algorithm which is 'rational', in the sense that they assume that their learning data was generated i.i.d. according to Eq. 1, and estimate the most probable value of c. Results are presented below for three learning algorithms (Naive learning models, Simple prior models, Complex prior models) corresponding to different assumptions about learners' inductive biases. Here, we specifically model the effect of a categoricity bias, operationalised as a prior over values of c.
For each case, we consider three population structures (Fig. 2), corresponding to the number of teachers m in generation t each learner in generation t + 1 receives her learning data from: m = 1, m = 2, and m = all (equivalently, m = M ). These three values are chosen as representative for understanding the dynamics when any number m of teachers is assumed, which we are interested in in light of previous computational studies highlighting the differences between single-and multiple-teacher scenarios in language evolution (Burkett and Griffiths, 2010;Dediu, 2009;Niyogi and Berwick, 2009;Smith, 2009). The single-teacher case corresponds most closely to the population structure considered in 'iterated learning' models of language evolution (e.g. Griffiths and Kalish, 2007;S. Kirby et al., 2007;Reali and Griffiths, 2009;Smith et al., 2003). 4 The all-teachers case corresponds to the population structure usually assumed in dynamical systems models of language change (e.g. Niyogi, 2006;Niyogi and Berwick, 1997;Sonderegger and Niyogi, 2010). The two teacher-case is representative of all m between 2 and M − 1, because the dynamics turn out to be extremely similar for any m > 1, as we show below. We assume throughout that the m teachers are chosen uniformly from teachers in the previous generation, with replacement. Considering the ensemble of M teachers in generation t, the state of the population at t can be characterized by the random variable C t , whose distribution describes how likely different values of c are. Similarly, the values of c learned by the M learners in generation t + 1 can be characterized by C t+1 . For simplicity, we assume that M is infinite. The evolution of the distribution of c is then deterministic, making its behavior more easily analyzed as a dynamical system. This and several other aspects of our modeling framework (e.g. discrete generations) are shared with previous dynamical systems models of language change considering discrete variants (Niyogi, 2006;Niyogi and Berwick, 1997).
Given a choice of learning algorithm, population structure, and channel bias, we seek to characterize the evolution of the distribution of c, and determine to what extent it satisfies our modeling goals: stability in the presence of channel bias, change in the presence of categoricity bias, and a nonlinear shift from one stable state (where the distribution of c does not change over time) to another. We are especially interested in two types of stable state -stable contextual variation, where the mean value of c in the population is nearer to µ a than to µ i , and stable umlaut, where this mean is near µ i . Fig. 1 exemplifies what the distribution of c in the population over time could look like in both cases, as well as one possible way in which a nonlinear shift from stable contextual variation to stable umlaut could occur as system parameters are varied. 5 In the remainder of the paper, we first consider the simplest case, where learners have no prior on values of c (Naive learning models)); we then consider the effects of introducing different types of categoricity bias into the learning algorithm (Simple prior models and  Figure 2: Three types of population structure are considered in our models: (a) Singleteacher scenario. Each learner in generation t + 1 receives all her learning data from a single randomly-chosen teacher in generation t. (b) Multiple-teacher scenario (two teachers). Each data point comes from one of two teachers with equal probability. (c) Multiple-teacher scenario (M teachers). Each data point comes from a random teacher with equal probability. In (b)-(c), teachers are chosen uniformly at random from teachers in generation t (with replacement). In all cases, lines of descent may be pruned, i.e. some teachers may not provide data to any learners in the following generation.
Complex prior models), and conclude by discussing our results.
For each class of model (naive, simple prior, complex prior), we are interested in the evolution of the distribution of c, for which there is no general analytic solution. For the naive learning models and simple prior models, we consider how the mean and variance of this distribution change over time, which can be derived analytically using techniques familiar from the cultural evolution literature (Boyd and Richerson, 1985;Griffiths et al., 2013) and dynamical systems models of language change in discrete variants; derivations for all analytic results are are given in Appendix B-C. For the complex prior models, we proceed by simulation.

Naive learning models
We first consider maximum-likelihood (ML) learners, who are 'naive' in the sense of having no prior over c, and simply choose the value of c under which the likelihood of the data (according to Eq. 1) is highest.
In the case where each learner in generation t + 1 receives all n examples from a single teacher, the evolution of the mean and variance of c are: Thus, when there is coarticulation (λ > 0), the mean of the contextual variant decreases in every generation by an amount equal to the mean amount of channel bias; if there is no coarticulation (λ = 0), the mean stays the same over time. Regardless of the value of λ, however, the inter-speaker variability in the realization of the contextual variant increases without bound over time.
Next, consider the case where each learner receives all examples from two teachers. The evolution of the mean of the contextual variant (c) in this case is again described by Eq. 3, while the variance now rapidly converges to a fixed point, regardless of the initial distribution: In the single-teacher (m = 1) setting, the variance decreases without bound over time, while for two or more teachers, it rapidly stabilizes.
Thus, the mean value of c decreases without bound over time (λ > 0) or stays constant (λ = 0), while the variance quickly stabilizes, in contrast to the single-teacher case.
In fact, it can be shown that the dynamics are similar for any case where m > 1: the mean of c is described by Eq. 3, while its variance moves towards a fixed point. The larger m is, the smaller this stable variance is (Appendix B.2). In the limiting case where m = M (Fig. 2c), the variance converges to: The evolution of the variance in the single-teacher, two-teacher, and all-teacher cases are illustrated in Fig. 3. Summary In the naive learning models, if speakers do not coarticulate, the mean realization of V 12 in the population remains constant over time, regardless of the number of teachers. This is empirically inadequate, as it predicts change from stable contextual variation to stable umlaut to be impossible. In the presence of any channel bias (λ > 0), the mean of c in the population steadily increases over time, again regardless of the number of teachers. In this case, change from stable contextual variation to stable umlaut is not possible, because stability is not possible, in the sense of a distribution of c which does not change over time. This problem is even worse for the single-teacher model, where the variance of C t in the population is predicted to steadily increase. As far as we are aware, a permanently unstable and unstructured distribution of population-level variation in phonetic realization is uncharacteristic of speech communities.

Simple prior models
The main problem with the naive learning models change becomes inevitable once channel bias is introduced. This problem has led to criticism of theories of sound change that rely on the accumulation of incremental change (Baker, 2008;Baker et al., 2011;Weinreich et al., 1968). However, the inevitability of change in these models is not simply a function of the channel bias itself, but also because there is no force acting to counteract the bias. Perhaps the simplest type of countervailing force would be to assume that learners have a prior categoricity bias over c against values away from µ a . In particular, consider a simple gaussian prior centered at µ a with variance τ 2 (Fig. 4A), a type that has previously been considered in work on the evolution of a continuous parameter (Griffiths et al., 2013: we compare this study's results to ours in C.1).
Learners receive n examples in the same way as in the naive learning models, but now their knowledge about contextual variation is probabilistic: a given learner begins with a prior distribution on how likely different amounts of contextual variation are a priori, which is updated to a posterior distribution based on her data, assuming that the distribution of the data given c is given by Eq. 1. She then takes the maximum aposteriori (MAP) estimate as her point estimateĉ of the contextual variant. 6 As for the naive learning models, we consider the evolution of the mean and variance of C t .
Regardless of the number of teachers, the mean of C t rapidly moves towards a fixed point, namely: Thus, the stronger the prior bias against contextual variation there is (smaller τ ), the smaller the eventual mean degree of contextual variation in the population, but increasing the strength of the channel bias (larger λ) has the opposite effect (Fig. 4B). As in the naive learning models where m ≥ 2, the variance of C t always rapidly moves towards a fixed point for all types of population structure. The formulas for these fixed points depend on σ a , ω, τ , and n. To get a sense of their essential properties, we write them in a form which assumes n 0: where K = (1 + ω 2 /σ 2 a ) and O( 1 n 2 ) denotes a constant divided by n 2 . While the variance always stabilizes over time, even for the single-teacher case, comparing Eqs. 8-10 shows that (for large enough n) just as for the naive learning models, the larger the number of teachers, the smaller the eventual amount of population-level variability in c.

Summary
The qualitative evolution of c in simple prior models is the same regardless of the magnitude of channel bias (including when λ = 0): both the mean and variance of the realization of V 12 in the population always move to a stable value. In the limit of large n, the stable variance shows an important qualitative difference that depends on the number of teachers: while convergence to a form reflecting prior is seen in the single-teacher scenario, the stable value in scenarios with two or more teachers does not directly reflect the prior (τ is not a term in Eqns. 9-10, cf. Griffiths et al., 2013). The simple prior models allow for stable contextual variation at a value that depends on the relative strengths of the channel and categoricity biases. However, these models are in some sense too stable: because stability depends on particular values of the system parameters, in order for a change to 'go to completion' (i.e., to stable umlaut) the system parameters would need to be continually changing in each generation-implying that each generation coarticulates more than the previous, has a weaker categoricity bias, or both. While this is ultimately an empirical question, it seems to us useful to start from the assumption that the effects of purportedly universal biases do not change steadily over time. In this sense, the simple prior models are inadequate in that there is no threshold in the system parameters triggering rapid movement to stable umlaut.

Complex prior models
The simple prior is indeed a type of categoricity bias, but one that is asymmetrically biased entirely toward one of the two pre-existing categories. Here, we consider the ramifications of relaxing this assumption, assuming instead that learners have a complex prior which weights values of c near both µ a or µ i higher than values in between: The strength of this prior is controlled by a: as a → 0, values of c near µ a and µ i are maximally preferred (Fig. 7A).
We assume the learner takes the MAP estimateĉ over the range [µ i , µ a ]. Unlike in previous models, the mean and variance of C t cannot be determined analytically, and we thus proceeded by simulation to determine the evolution of the distribution of C t over time in this case. Technical details of these simulations are given in D.1; here we describe the basic setup of the simulations, and their results.  The simulations described below consider the evolution of a population that starts with a mean realization of V 12 similar to V 1 (C 1 ∼ N (µ a − 10, σ 2 a )), in order to determine whether both stable contextual variability and change to stable umlaut are possible in this model. Of interest is how the strength of the prior (a) and the coarticulatory channel bias (λ) affect the evolution of the distribution from this starting point, which we examine for the same three population structures as in previous models. We first examine the evolution of the distribution of C t over time (which we refer to as the trajectory of C t ) for particular values of a, λ, and m (Trajectories of C t ), then examine how the final mean of the distribution of C t depends on these three parameters (Final mean of C t ).

Trajectories of C t : examples
We show some qualitatively different ways in which the distribution of C t can evolve, by examining the trajectories of C t beginning from C 1 ∼ N (µ a − 10, 10), for particular values of a, λ, and m, stopping each simulation when t = 1000. (It is visually clear from the results of these simulations, shown in Figs. 5-6, that the distribution of C t is no longer  • Case 1: strong prior, weak channel bias (top row: a = 0.001, λ = 0.25): For a sufficiently strong prior relative to the strength of the channel bias, contextual variation is stable over time (for 1, 2, all teachers).The stable variance of the distribution is much larger for m = 1 than for m > 1, and is slightly larger for m = 2 than for m = M .
• Case 2: strong prior, strong channel bias (middle row: a = 0.001, λ = 4): For a sufficiently strong channel bias relative to the prior strength, change to stable umlaut rapidly occurs (for 1, 2, all teachers). The transition is slightly faster for m > 1 than for m = 1.
• Case 3: weak prior, weak channel bias (bottom row: a = 0.5, l = 0): In the single-teacher case, the variance rapidly spreads, and all values of c become roughly equiprobable. For more than one teacher, the mean changes little and the variance rapidly stabilizes, with the value of the stable variance is slightly larger for m = 2 than for m = M . These behaviors are similar to the analogous naive learning models, as expected given that a sufficiently weak prior is effectively flat.
In Cases 1-3, the evolution of C t looks qualitatively similar for m = 1, m = 2, and m = M , with a significantly larger variance of C t at each time point for the m = 1 case. However, there is also a range of (a, λ) parameter space where the evolution of C t looks qualitatively different depending on the number of teachers. Fig. 6 shows two ways in which this can happen: The final mean of c depends on the number of teachers (1 vs. 2+), and changes nonlinearly as λ and a are changed. In particular, for 2+ teachers there is a bifurcation: once λ is large enough relative to a, rapid change to stable umlaut occurs. m = 1 and m > 1. In the single-teacher case, the distribution of C t reflects the (strong) prior, in the sense that some individuals have values of c near µ a (contextual variation) and some have values of c near µ i (umlaut), with a gap in between. That is, change to umlaut has 'gone through' for some individuals, but not others. In contrast, in the multiple teacher cases, the distribution of C t becomes tightly clustered around the population mean (which is nearer to µ a than to µ i ).
• Case 5: medium prior, medium channel bias (bottom row: a = 0.01, λ = 1.3): In this case, the channel bias is kept at the same value, but the prior is weakened sufficiently that change to stable umlaut eventually occurs, regardless of the number of teachers. However, the trajectory of C t looks qualitatively different depending on the number of teachers. For m = 1, the population contains two types of individuals-those with values of c near µ a , and those with values of c near µ i -and the proportion of the second type becomes greater over time, until the whole population has c near µ i . For m > 1, individuals have values of c tightly clustered around the population mean, which steadily changes from near µ a to near µ i over time.
In Cases 4-5, it is again the case (as in Cases 1-3) that the two-teacher and M -teacher cases look very similar, with a slightly larger variance of C t when m = 2.
Of the trajectories considered above, Cases 1-2 are particularly important: they show that both stable contextual variation and stable umlaut are possible, as a and λ are varied. In particular, it is possible to get change to stable umlaut in the presence of a strong categoricity bias-which was not possible in the simple prior model-as well as stable contextual variation near µ a in the presence of channel bias. These outcomes are two of our modeling goals. We now consider how the final state of the population depends on prior strength and channel bias, as a and λ are varied between these limiting cases, to get a sense of whether the complex prior model meets our final modeling goal: a threshold in the system parameters (a and λ) which triggers rapid movement to stable umlaut. Fig. 7B shows the final mean of c in the population as λ and a are varied. In the singleteacher case (panel 1), stable contextual variation is possible only for the strongest priors or when λ = 0. As the strength of the prior is relaxed, the population mean comes to rest either in an intermediate state, or near µ i (i.e. stable umlaut). The distribution of C t in an intermediate state often corresponds to Case 4 above: individual learner's means are not tightly clustered around the population mean, but reflect the prior in the sense that some individuals are stable near one endpoint (c = µ a ) and some near the other (c = µ i ), corresponding to an empirical population in which a change has gone through for some speakers but not for others.

Final mean of C t as a function of system parameters
In multiple-teacher scenarios (panels 3-4), the results are quite different. There is a range of values of prior strength and channel bias which give stable contextual variation. However, for a given a, as λ is increased past a critical value, there is a rapid shift of the population to a stable state where most learners have umlaut (c ≈ µ i ). That is, there is a bifurcation where the strength of coarticulation has overcome the stabilizing affect of the prior. When this happens, the population mean rapidly moves towards the other category mean and stabilises. Panels 3-4 also illustrate the tradeoff between categoricity and channel biases: for a stronger prior, the critical value of λ increases (i.e., the degree of coarticulation needed to overcome the prior is greater).
Summary The complex prior model for multiple teachers meets all three of our modeling goals: stability of contextual variation in the face of coarticulation; stability of umlaut in the presence of categoricity bias; and rapid change in the population from stable contextual variation to stable umlaut as system parameters (a, λ) are varied around certain values.

Discussion
This paper has explored how assumptions about channel bias, categoricity bias, and population structure translated into population-level dynamics of a continuous parameter, evaluating models by their ability to meet two goals reflecting empirical cases of sound change: (1) the possibility of stable contextual variation and change to stable umlaut, in the presence of forces promoting the other outcome, and (2) a nonlinear transition from stable variation to sound change as a function of system parameters.
The first goal was met by all models where both a bias promoting change and a bias promoting stability were present: in both simple and complex prior settings, stable contextual variation can be maintained even in the presence of channel bias, and change to stable umlaut can occur even in the presence of categoricity bias. This is an important result, for two reasons related to the prevalence of both stable variation and sound change in the world's languages. First, it shows that it is possible to develop a model of sound change involving channel bias that does not overapply (Baker, 2008;Baker et al., 2011;Weinreich et al., 1968). Second, it shows that the distribution of a continuous parameter in the population does not necessarily come to reflect the structure of learners' hypothesis space, when other forces (such as channel bias) are present. Convergence to the prior has been emphasized in the cultural evolution literature Griffiths et al., 2013;S. Kirby et al., 2007;Reali and Griffiths, 2009), and would not allow for the possibility of both stable variation and change to stable umlaut, when the prior reflects both possibilities. Instead, both stability and change are possible in a model where biases promoting each outcome are both present.
Our second goal concerned how stability gives way to change as a function of the relative strength of these biases. Models in which learners were equipped with a complex prior showed a bifurcation: change from one stable state (contextual variation) to another (umlaut) occurred suddenly as the relative strength of the biases favoring each stable state is varied past a critical value, at which point 'actuation' can be said to have occurred. Bifurcations in linguistic populations have been suggested as a key mechanism underlying the actuation of linguistic change, but to our knowledge have previously only been shown to occur in models of change involving discrete variants Niyogi, 2006;Niyogi and Berwick, 2009;Sonderegger and Niyogi, 2010). Our demonstration that bifurcations are possible in a population of learners of a distribution of a continuous parameter supports the hypothesis that bifurcations play a key role in the actuation of language change more generally, and suggests that the ongoing empirical quantification of forces corresponding to channel and categoricity biases will be crucial to a detailed account of sound change actuation (Moreton, 2008;Sonderegger and Yu, 2010;Wilson, 2006).
Turning to the role of population structure, we observed significant differences between single-and multiple-teacher settings. These differences are important given the prevalence of the single-teacher assumption in much of the sound change modeling literature (J. Kirby, 2013;Pierrehumbert, 2001;Wedel, 2006), and echo similar differences found in previous work on the evolution of discrete linguistic traits (Burkett and Griffiths, 2010;Dediu, 2009;Niyogi and Berwick, 2009;Smith, 2009). For naive learners, single-teacher scenarios result in ever-increasing population variance. In the simple prior cases, convergence to a form reflecting the prior was seen in single-teacher settings (Griffiths et al., 2013), but not in multiple-teacher settings. For single teachers in the complex prior setting, the prior was reflected not in terms of individual's distribution of the learned phonetic parameter, but in terms of the population-level mixture: rather than a majority of individuals learning a phonetic parameter with a value intermediate between the prior endpoints, individuals tended to learn a value at one endpoint or the other, with the population consisting of a mixture of such individuals. This last result contrasts sharply with abundant sociolinguistic evidence showing that the distribution of linguistic traits in individuals tends to mirror that of their speech community (Fruehwald, 2013;Labov, 2010). Conversely, the results from multiple-teacher settings are consistent with the finding that social network ties can act as a conservative force promoting entrenchment (Milroy, 1980). Overall, our results in single-versus multiple-teacher settings suggest that in addition to categoricity bias, population structure itself can play a role in promoting stability of existing phonetic categories.
While assuming one versus multiple teachers greatly affected the dynamics, it is important to point out our potentially unintuitive finding that models assuming any number of teachers greater than one resulted in very similar dynamics. Thus, exactly how population structure affects the distribution of a linguistic parameter over time requires further study. Given the crucial role that social networks play in the propagation of language change (Labov, 2010;Milroy, 1980), we are currently extending this framework to handle different population structures with more complex teacher-learner relations, including socially stratified variation. Future work should also consider different types of biases promoting stability and change, such as asymmetries in the extent of contact between members and in the social weighting of groups and variants. These are some of many ways in which our current framework can be extended to better match the complex reality of sound change. However, even in the relatively simply model presented here, we have shown that a solution to the actuation problem is possible: understanding why a language changes, or fails to change, requires attention not only those forces promoting change, but their interplay with the forces constraining it.

Appendix A Model
We first review the general setting presented in Model in the main paper. We assume the terminology and notation introduced there, with some additions to be used in derivations (summarized in Table 1). Each generation at time t consists of M agents, who act as teachers for M learners in generation t + 1. Each learner receives n examples, drawn from m teachers, with values y = (y 1 , . . . , y n ). A new set of m teachers from generation t is drawn (with replacement) for each learner in generation t + 1. For a given learner, which teacher the ith example comes from is chosen randomly (each teacher has probability 1/m), and k j denotes the number of examples received from the jth teacher,ȳ j the mean F1 of the examples received from the jth teacher, andȳ the mean F1 of all n examples.
The distribution of V 12 for an agent with contextual variant c is The random variable C t corresponds to the contextual variant used by members of generation t. We use lower-case c to refer to draws from this random variable, at times with subscripts (c j will refer to the contextual variant for the jth teacher) or a hat (ĉ will refer to an individual learner's estimate of the contextual variant). All productions of V 12 are subject to a channel bias with distribution N (λ, ω 2 ). Thus, F1 for a teacher with contextual variant c is distributed as We assume that M is very large (M → ∞), in which case the evolution of the distribution of C t is deterministic, and can be described by a dynamical system. Analyzing the dynamical system under different assumptions lets us understand how different assumptions about bias and population structure affect the population-level distribution of the continuous phonetic parameter over time, analogously to existing dynamical systems models of language change which consider discrete linguistic variants (e.g. Mitchener, 2003;Niyogi, 2006;Niyogi and Berwick, 1997;Nowak et al., 2001).
It is worth briefly contrasting this setting with that considered in 'iterated learning' (IL) models which are common in the language evolution literature, where each generation consists of a single member (M = 1) (e.g. Griffiths and Kalish, 2007;Griffiths et al., 2013;S. Kirby et al., 2007;Reali and Griffiths, 2009;Smith et al., 2003). In IL models, the state of the population is a stochastic process: it consists of a single value of c at each time point, and can be described as a discrete-time Markov chain c t . IL studies generally examine the evolution of this Markov chain: what would the distribution of values of c t be if the chain were iterated a large number of times? 7 At time t in any given iteration, there is only one value of c. In contrast, in our infinite-population setting we are examining the evolution of C t , i.e. the distribution of c in the population at time t. In other words, we are interested not in how a single parameter evolves (stochastically) over time, but in how the distribution of this parameter in a population evolves (deterministically) over time. A more detailed presentation of the difference between iterated learning and the 'social learning' setting where M = ∞ is given by Niyogi and Berwick (2009).

Appendix B Naive learning models
Here we derive all analytical results referred to in Naive learning models in the main paper.
We first consider maximum-likelihood (ML) learners who are "naive" in the sense of having no prior over estimates of c. This setting is closely related to 'blending inheritance' models of cultural evolution of a quantitative character presented by Boyd and Richerson (1985, 71ff), which we make use of below.

B.1 Naive learning models: single teacher
Each learner in generation t+1 is associated with a value c (one draw from C t , representing the single teacher's contextual variant), which is used to generate n training examples for that learner. Let Y 1 , . . . , Y n be the random variables corresponding to these examples, which take on values y = (y 1 , . . . , y n ), and letȳ be the mean of this sample. Each example is normally distributed, following Eq. 2. Because Y 1 , . . . , Y n are independent and normally distributed, their mean is also normally distributed, with the same mean and reduced variance: Givenȳ, the learner's ML estimate of the contextual variant, assuming the data was generated by Eq. 12, isĉ =ȳ. Thus, using Eq. 14, the distribution over values ofĉ the learner could acquire given c is: that is,ĉ is a noisy version of c, decreased by the mean channel bias (λ). We are interested in the evolution of the distribution of c: that is, the distribution of C t+1 as a function of the distribution of C t . It is not in general possible to analytically derive what f C t+1 (c) is for an arbitrary f C t (c). However, we can get a sense of the evolution of the distribution of c by examining how its mean and variance change over time.
To do so, first consider the case where λ = 0. The learner's estimate of c can then be written asĉ where c i = c and i ∼ N (0, σ 2 a + ω 2 ). In this form, our setting can be related directly to the classic 'blending inheritance' model of a quantitative character (Boyd and Richerson, 1985, 71ff), where: • A child in generation t + 1 takes the mean value of the character from n cultural parents (the c i ).
• Her observation of the ith cultural parent is distorted by a noise term ( i ).
• The distribution of C t is the distribution over cultural parents in generation t.
Having made this equivalence, Eqs. 3.21 and 3.22 of Boyd and Richerson (1985) (rewritten using our notation) give the evolution of the mean and variance of C t : Var In our case, Cov( i , j ) = 0 (because i and j are independent) and Cor(c i , c j ) = 1 (because all the c i have the same value). After some algebra, Eq. 18 thus simplifies to Eq. 17 and Eq. 19 describe the evolution of the mean and variance of C t when λ = 0. In the case where λ > 0, the learner's estimate ofĉ changes by subtracting the constant λ from the right-hand side of Eq. 16, which entails subtracting λ from the right-hand side of Eq. 17 and 0 from the right-hand side of Eq. 19. 8 The evolution of the mean and variance of C t are thus which are Eqs. 3-4 in the main paper. We note that although results from Boyd and Richerson (1985) were used to derive these evolution equations, the result that the variance of c increases without bound over time (Eq. 21, illustrated in Fig. 3, panel 1) contrasts with their well-known finding that blending inheritance in general reduces variance of a quantitative trait over time, as emphasized in their discussion (p. 75). However, stable or increasing variance are possible for particular cases of Boyd and Richerson's model, such as the case considered here where each learner has a single cultural parent and there is noise in estimating the parent's cultural model.

B.2 Naive learning models: multiple teachers
We now consider the case where each learner in generation t + 1 receives n examples from m > 1 teachers in generation t. That is, values c 1 , . . . , c m , corresponding to the m teachers, are drawn i.i.d. from C t , and the teacher who generates each example is chosen randomly (with replacement). We assume that n > 1. 9 Let k j denote the number of examples drawn from the jth teacher (k 1 + · · · + k m = n) , and let k = (k 1 , . . . , k m ). Thus, k follows a multinomial distribution with n trials and event probabilities p 1 , p 2 , . . . , p m = 1/m. Without loss of generality, we can assume that examples 1, . . . , k 1 come from teacher 1, examples k 1 +1, . . . , k 1 +k 2 come from teacher 2, and so on. Letȳ j denote the mean of the examples from the jth teacher. The learner's ML estimate,ĉ =ȳ, can then be rewritten as:ĉ =ȳ = 1 n (y 1 + · · · + y n ) Note that conditional on k j , eachȳ j can be thought of as the learner's ML estimate of c j using k j examples from a single teacher. Thus, by the same logic used to derive Eqs. 17, 19, we have Becauseĉ is drawn from C t+1 , taking the expectation of Eq. 22 and substituting in Eq. 23 gives: 9 If n = 1, the multiple-teacher case is the same as the single-teacher case already considered.
Similarly, taking the variance of Eq. 22 gives: Using the expressions for E[k j ] and Var[k j ] for a multinomial distribution (where p j is the probability of the j th outcome): Substituting into Eq. 27 gives The evolution equations of the mean and variance are then Thus, the mean of c always decreases without bound, as in the single-teacher case (Eq. 20), regardless of the number of teachers or the number of examples.
Turning to the variance, define B = nm n+m−1 . Because m > 1 and n > 1 (by assumption): The variance evolution equation is an iterated map of the form where K 1 and B are constants. Because |B| > 1, the map has a unique fixed point α * which it converges to from any starting point (Hirsch et al., 2004). In particular, letting Var[C 1 ] be the variance of c in generation 1, we can rewrite Eq. 30 as where is the fixed point. Thus, for multiple teachers, the variance quickly converges to a fixed point α * , with its distance from α * decreasing geometrically (Eq. 34, illustrated in Fig. 3, panels 2-3). The value of the stable variance decreases as the number of examples (n) or the number of teachers (m) is increased. For example, for the two-teacher case (m = 2), Eq. 34 gives which is Eq. 5 in the main paper. For the case where learners learn from all teachers (m = M ), in the limit considered in our setting where M → ∞, Eq. 34 gives which is Eq. 6 in the main paper.
posterior distribution of c is gaussian as well. Using Eq. 3.49-3.51 from Bishop (2006), the posterior can be shown to be: 11 The learner must pick a point estimate of the contextual variant to use for generating training data for the next generation. The two common ways of obtaining a point estimate from a posterior distribution are taking the maximum a-posteriori value or the expected value. These are equivalent for Eq. 39 (because the mean and mode of a normal distribution are identical), namely:ĉ =ȳ where we abbreviate the denominator of Eq. 40 as Using the same notation as above (Table 1), we now determine the evolution of the mean and variance of C t for a population of simple prior learners whose estimate of c is given by Eq. 40. To reduce the number of cases which need to be considered below, we assume that n > 1, σ a > 0, and τ > 0: that is, each learner receives more than one example, there is some variability among a speaker's productions of V 12 , and the categoricity prior is not infinitely strong.

C.1 Simple prior models: single teacher
For the case where m = 1, the distribution ofȳ is still given by Eq. 14, where c is the value of the contextual variant used by the single teacher. Because µ a and D are constants, using Eq. 40, the distribution ofĉ is then Examining Eq. 15, we see that if X denotes the estimate ofĉ (given the teacher's value of c) in the single-teacher naive learner case, thenĉ for the current case is simply X translated and divided by constants: (X + (D − 1)µ a )/D. Thus, Eq. 20 can be used to find the evolution of the mean: and Eq. 21 can be used to find the evolution of the variance: 11 In particular, by making these substitutions into Eq. 3.49-3.51: w = ( c), S 0 = (τ 2 ), β = 1/σ 2 a , Φ = ( n times 1, . . . , 1) T , m 0 = µ a . Now, note that the assumption that σ a , τ > 0 means that D > 1, so that both Eq. 43 and Eq. 44 are iterated maps of the form in Eq. 33 with |B| > 1. These maps have unique stable fixed points, thus, both the mean and the variance of C t rapidly converge to fixed points from any starting values. Solving for the fixed points gives: 12 Eq. 45 is Eq. 7 in the main paper. The mean of the contextual variant in the population converges to the value favored by the prior (µ a ), minus an offset which depends on λ, n, τ , σ a in intuitive directions: stronger net channel bias (λ) over the n examples results in lower c, while stronger categoricity bias relative to the amount of production variability (τ /σ a ) results in c nearer to µ a . We discuss the expression for the stable variance below, along with the equivalent expression for the multiple-teacher case.

C.1.1 Comparison with previous work
Because our single-teacher simple prior scenario is particularly close to one of the iterated learning scenarios considered by Griffiths et al. (2013), it is worth comparing our results to theirs to see to what extent they diverge. 13 Individual learners in their 'category defined on a single dimension' setting (pp. 954-956) learn in essentially the same way as our singleteacher simple prior learners, except that no production bias is applied. In addition, each generation consists of one teacher/learner (M = 1), compared to our M = ∞. Thus, the value of c at each time point is a Markov chain, which we write as c t . In our notation, Griffiths et al. show that (p. 966,Eq. 11) Although each generation in an iterated learning model consists of only one agent, there is a natural interpretation of c t in a population context (where M = ∞) as describing the distribution of C t in a population of teacher/learners, each of whom learn from exactly one agent (in generation t − 1) and teach exactly one agent (in generation t) (as pointed out by Griffiths et al., 2013;Niyogi and Berwick, 2009). (In other words, the population consists of an infinite number of iterated-learning chains run in parallel.) This setting is slightly different from our single-teacher case (Fig 1, left panel in the main text), where two members of generation t could have the same teacher (and some members of generation t − 1 might never serve as teachers). How does this slight difference affect the dynamics? We can compare the stable state of the distribution of C t in the two cases by setting λ = 0, ω = 0 in Eq. 45-46, and taking the limit t → ∞ in Eq. 47: Var[C t ] → τ 2 (2 + σ 2 a nτ 2 ) −1 (our setting) τ 2 (1 + σ 2 a nτ 2 ) (iterated learning) Thus, the distribution of the coarticulation parameter comes to reflect the prior in both cases: the mean converges to the mean of the prior (in both models), while the variance converges to a value related to τ 2 (the width of the prior), but which is smaller in our model than in the iterated learning model, by at least a factor of 2 (depending on the values of τ , σ a , and n). The long-term dynamics are therefore similar in the two models, but slightly different.

C.2 Simple prior models: multiple teachers
When m > 1, we can proceed similarly to the naive-learner multiple teacher case, defining k j ,ȳ j , etc. in the same way. The learner's point estimate ofĉ is still given by Eq. 40, which can be used to rewriteĉ as: As in Naive learning models: single teacher,ȳ j can be thought of as the ML estimate made by a naive learner (in generation t + 1) based on drawing k j examples from a single teacher in generation t. Also, note that D does not depend on any k j . Becauseĉ is drawn from C t+1 , taking the expectation of both sides of Eq. 50 gives: Thus, the evolution of the mean in the multiple-teacher case (Eq. 51) is the same as in the single-teacher case (Eq. 43). In particular, the mean converges to the value given in Eq. 45, which gives Eq. 7 in the main paper. Similarly, taking the variance of Eq. 50 gives The underlined term is the same as Eq. 26 in the naive learner multiple-teacher case, and its derivation proceeds identically from that point on (up to Eq. 29), to give Var[C t+1 ] = σ 2 a + ω 2 nD 2 + Var[C t ] n + m − 1 nmD 2 (54) Recall that for multiple teachers (m > 1), provided that n > 1 (which is true, by assumption), we have that |(n + m − 1)/nm| < 1 (Eq. 32). Thus, because D ≥ 1 as well (Eq. 41), the evolution equation for the variance (Eq. 54) is an iterated map of the form in Eq. 33, with |B| > 1, which has a unique stable fixed point. Solving for it gives: Comparing Eq. 55 with Eq. 44, we see that the stable variance decreases monotonically as the number of teachers (m) is decreased, when all other parameters are held constant.
(This fact is referred to in Naive learning models in the main paper.) In particular, the stable variances for the three values of m considered in the main paper (1, 2, ∞) are: Var[C t+1 ] → τ 2 (1 + ω 2 These expressions for the stable variance are hard to understand intuitively. We can get a sense of their behavior by taking n to be large, in accordance with the intuition that each learner will receive many examples of a given phonetic category. Taking the Taylor expansions of Eqs. 56-58 in terms of 1/n gives: where O( 1 n 2 ) denotes a constant divided by n 2 . (These are Eqs. 8-10 in the main paper.) Thus, there are two important differences between the form of the stable variance for m = 1 and m > 1: 14 • First, the stable variance for m = 1 always reflects the prior (in the sense that the expression involves τ ), for any n, while the stable variance for m > 1 does not (τ only enters into second-order terms).
• Second, the stable variance for m > 1 goes to 0 as n is increased, while the stable variance for m = 1 goes to a constant value which reflects the prior. Thus, a population of simple prior learners who receive many examples would eventually show no variability in their contextual variants (values of c) for two or more teachers, while the same population with single-teacher learning would show variability in their contextual variants.
for values of a ∈ [0.001, 0.05] and λ ∈ [0, 2.0], for the single-teacher, two-teacher, and M -teacher cases (m = 1, 2, M ). Each run began by assigning a value of c to the M learners in generation 1, drawn according to a starting distribution. Because we are primarily interested in the evolution of a population which begins with the contextual variant uniformly pronounced similarly to V 1 , we always used C 1 ∼ N (µ a − 10, 10) as the starting distribution. For each of the M learners in generation t, where t > 2, m teachers were drawn at random from generation t − 1, and used to generate n = 100 examples (with the teacher for each example chosen randomly from the m teachers, with replacement). The learner's MAP estimateĉ for this data was found by maximizing Eq. 63 over values of c ∈ [µ i , µ a ], using the unidimensional optimize() function in R, which uses "a combination of golden section search and successive parabolic interpolation" (R Core Team, 2014). 15 For the two and M -teacher cases, simulations were run until t = 2500, at which point the distribution of C t had always reached a stable state (by visual inspection). For the single-teacher case, which converged much more slowly, simulations were run until the mean, 5th percentile, and 95th percentile of the distribution of C t had (each) not changed by more than 2 in 500 generations. If this criterion was not met by t = 10000, the simulation was stopped. At this point the distribution of C t had reached a stable state for runs corresponding to the dark red and dark blue regions of Fig. 7B Panel 1, though not necessarily for runs corresponding to the region in between.
15 optimize is guaranteed to find the global maximum only if Eq. 63 is unimodal over the interval c ∈ [µ i , µ a ]; otherwise, it is only guaranteed to find a local maximum. Whether Eq. 63 is unimodal over this interval in general depends on the values of the data ( y) and the system parameters (a, σ a , etc.). Eq. 63 can be shown to be concave on [µ i , µ a ] for any y if the condition a > 4σ 2 a n(µa−µi) 2 holds, which is the case for almost all simulation runs considered here (those with a > 0.0025). Note that concavity is not a necessary condition for optimize to find the global optimum, which it seems to nearly always do anyway in our setting. We satisfied ourselves that optimize getting stuck in local maxima was not a problem by comparing the results with those obtained by using grid search instead, for a subset of the runs.