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Chapter 9. A Model of Cultural Evolution and Its Application to Language From “The Computational Nature of Language Learning and Evolution” Summarized by Seok Ho-Sik
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© 2009 SNU CSE Biointelligence Lab 2 Contents 9.1 Background 9.2 The Cavalli-Sforza and Feldman Theory 9.3 Instantiating the CF Model for Languages 9.3.1 One-Parameter Models 9.3.2 An Alternative Approach 9.3.3 Transforming NB Models into the CF Framework 9.4 CF Models for Some Simple Learning Algorithms 9.4.1 TLA and Its Evolution 9.4.2 Batch- and Cue-Based Learners 9.5 A Generalized NB Model for Neighborhood Effects 9.5.1 A Specific Choice of Neighborhood Mapping 9.6 A Note on Oblique Transmission 9.7 Conclusions
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9.1 Background Change in linguistic behavior of human populations must be a result of a change in the internal grammars that successive generations of humans employ. Q: Why do the grammars of successive generations differ from each other? Need to know: 1) how these grammars are acquired 2) how the grammars of succeeding generations are related to each other. Then, it is possible to predict the envelope of possible changes. The applicability of the CF model to the framework for linguistic change CF model: Cavalli-Sforza and Feldman model (cultural change model). Introducing on possible way in which principle and parameters approach is be amenable to CF framework. © 2009 SNU CSE Biointelligence Lab 3
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9.2 The Cavalli-Sforza and Feldman Theory (CF model) A theoretical model for cultural change over generation “Cultural” parameters Being transmitted from parents to children with certain probabilities. The mechanism of transmission is unknown – only the probabilities are known. An example of CF model Variables with binary value L & H. The proportion of L types in the population will evolve according to the following equation The evolution (change) of cultural traits essentially driven by the probabilities with which children acquire the traits given their parental types. © 2009 SNU CSE Biointelligence Lab 4 Table 9.1 The cultural types of parents and children related to each other by their proportions in the population. The values are for vertical transmission. b i : the probability with which a child of ith parental type will attain the trait L. p i : the probability of the ith parental type in the population. u t : the proportion of people having type L in the parental generation.
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9.3 Instantiating the CF Model for Languages 9.3.1 One-Parameter Models (1/2) In language change, the transmission probability depends upon the learning algorithm. Assumption 1. Children of parents with the same language receive examples only from the parental language. 2. Children of parent with different language receive examples from an equal mixture of both languages. 3. After k examples, children “mature”. The probability which the algorithm A hypothesizes grammar g 1 given a random i.i.d. draw of k examples according to probability distribution P. Statement 9.1 If the support of P is L 1 then and if the support of P is L 2 then © 2009 SNU CSE Biointelligence Lab 5
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9.3 Instantiating the CF Model for Languages 9.3.1 One-Parameter Models (2/2) It is possible to express the b i ’s in the CF model of cultural transmission in terms of the learning algorithm. © 2009 SNU CSE Biointelligence Lab 6 Table 9.2 The probability with which children attain each of the language types, L1 and L2 depends upon the parental linguistic types, the probability distribution P 1 and P 2 and the learning algorithm A.
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9.3 Instantiating the CF Model for Languages 9.3.2 An Alternative Approach Assumptions in previous chapters 1. The population can be divided into children and adults. 2. All children in the population are exposed to sentences from the same distribution 3. The distribution depends upon the distribution of speakers in the adult population. The evolution of s t over time ( s t : state of the population) Interpretation If the previous state was s t, then children are exposed to sentences drawn according to The probability with which the average child will attain a language is correspondingly provided by g. © 2009 SNU CSE Biointelligence Lab 7
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9.3 Instantiating the CF Model for Languages 9.3.3 Transforming NB Models into the CF framework NB: Niyogi and Berwick model NB update rule: CF update rule NB update rule CF update rule The differences in the evolutionary dynamics of CF- and NB-types models 1. The evolutionary dynamics of the CF model depends upon the value of f at exactly 3 points (0, ½, 1). 2. If f is linear, then the NB and CF update rules are exactly the same. If f is nonlinear, these update rules potentially differ. 3. The CF update is a quadratic iterated map and has one stable fixed point. 4. For some learning algorithms, there may be qualitatively similar evolutionary dynamics for NB and CF models. Essential one: NB assumes that all children receive input from the same distribution. Cavalli-Sforza and Feldman assume that children can be grouped into four classes depending on their parental types. © 2009 SNU CSE Biointelligence Lab 8
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9.4 CF Models for Some Simple Learning and Algorithms 9.4.1 TLA and Its Evolution (1/4) TLA (Triggering Learning Algorithm) working 1. Initialize: start with randomly chosen input grammar. 2. Receive next input sentence, s. 3. If s can be parsed under current hypothesis grammar, go to 2. 4. If s cannot be parsed under current hypothesis grammar, choose another grammar uniformly at random. 5. If s can be parsed by new grammar, retain new grammar, else go back to old grammar. Go to 2. For two grammars in competition under the assumptions of the NB model, a: -the probability that ambiguous sentences (parsable both g 1 and g 2 )are produced by L 1 speakers. b: k: # sentences a child receives from its linguistic environment before maturation. © 2009 SNU CSE Biointelligence Lab 9
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9.4 CF Models for Some Simple Learning and Algorithms 9.4.1 TLA and Its Evolution (2/4) Remarks 1. For k=2 and NB model, (i) for a=b, there is exponential growth to one fixed point of p* = ½. (ii) a≠b, there is logistic growth and if a ½. else if a>b, p*< ½ © 2009 SNU CSE Biointelligence Lab 10 Fig. 9.2: The fixed point for various choices of a and b for the CF model with k=2 Fig. 9.1: The fixed point for various choices of a and b for the NB model with k=2
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9.4 CF Models for Some Simple Learning and Algorithms 9.4.1 TLA and Its Evolution (3/4) For k=2 and CF model, (i) for a=b, there is exponential growth to one fixed point of p* = ½. (ii) a≠b, there is logistic growth and if a ½. else if a>b, p*< ½ © 2009 SNU CSE Biointelligence Lab 11 Fig. 9.3: The difference in the values of p*(a,b) for the NB model and the CF model p* NB – p* CF for various choices of a and with k=2.
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9.4 CF Models for Some Simple Learning and Algorithms 9.4.1 TLA and Its Evolution (4/4) For k , (i) for a=b, there is no change in the linguistic composition, (ii) for a>b s t tends to 0, (iii) for a<b s t tends to 1. Thus one of the languages drives the other out and the evolutionary change proceeds to completion. In real life, a = b is unlikely to be exactly true, therefore language contact between population is likely to drive one out of existence. © 2009 SNU CSE Biointelligence Lab 12
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9.4 CF Models for Some Simple Learning and Algorithms 9.4.2 Batch- and Cue-Based Learners (1/2) For batch learners (chapter 5), the system has two stable fixed points = * and = 1. There is one unstable fixed points between them case in which NB dynamics is bistable. CF model will have only one stable fixed point. CF update rule when s t = 1, 1 is a fixed point of the CF system. If f( ½ )>1 s =1 is a stable fixed point, otherwise s=1 is unstable. In contrast, the NB system is always stable at 1. © 2009 SNU CSE Biointelligence Lab 13
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9.4 CF Models for Some Simple Learning and Algorithms 9.4.2 Batch- and Cue-Based Learners (2/2) For cue-based learner NB dynamics In a regime, NB dynamics has only one stable fixed point =0. There is another regime where NB dynamics has two stable fixed points =0 and = * >0 CF dynamics CF dynamics has an fixed point at s = 0. Stable fixed point f( ½ ) ½ s = © 2009 SNU CSE Biointelligence Lab 14
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9.5 A Generalized NB Models for Neighborhood Effects CF model: sentences from different distribution. NB model: sentences from the same distribution. Generalization Key idea: heterogeneous communities speakers often tend to cluster in linguistically homogeneous neighborhoods. children’s input sentences are depending on their location in the neighborhood. h -mapping: mapping from neighborhood to -type proportion of L1 speakers that an -type child is exposed to. The percentage of speakers of L 1 f( ): a probability of attaining the grammar L 1 © 2009 SNU CSE Biointelligence Lab 15 P h ( ): distribution of
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9.5 A Generalized NB Models for Neighborhood Effects 9.5.1 A Specific Choice of Neighborhood Mapping For figure 9.6 Update rule 1. The linear map implies an exponential growth to a stable fixed point 2. s* =1 requires very unlikely no language is likely to be driven out of existence completely. In contrast, NB and CF models result in extinction if a ≠ b. Remarks 1. h is not fixed function. It changes from generation to generation. 2. The population of mature adults is always organized into two linguistically homogeneous neighborhood in every generation. © 2009 SNU CSE Biointelligence Lab 16
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9.6 A Note on Oblique Transmission Oblique transmission: the effect that members of the parental generation at large have on the transmission of cultural traits. Stage 1: children acquire on the basis of preliminary exposure to their parents. Stage 2: juvenile acquire a “mature” state. One computes the probability with which trait transitions occur. If 1, 2 are known compute the dynamics of H types in the population. © 2009 SNU CSE Biointelligence Lab 17
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