Difference between revisions of "Florencia Reali and Thomas L. Griffiths, Words as alleles: connecting language evolution with Bayesian learners to models of genetic drift, Proceedings of The Royal Society of London. Series B, Biological Sciences 2010"

Latest revision as of 12:01, 31 March 2011

Citation

Florencia Reali and Thomas L. Griffiths, Words as alleles: connecting language evolution with Bayesian learners to models of genetic drift, Proceedings of The Royal Society of London. Series B, Biological Sciences 2010

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Summary

This paper addresses the problem of language evolution by relating it to models of genetic drift in biological evolution. Although the mechanisms of biological and language evolution are very different: biological traits are transmitted via genes while language is transmitted via learning, the paper shows that these different mechanisms can arrive at the same results. Specifically, the paper demonstrates that transmission of frequency distribution over linguistic variants by Bayesian learners arrives at the same results as the Wright-Fisher model of genetic drift.

Description of the method

The focus of this paper is to model how language changes as a consequence of being passed from one learner to another. The learning problem considered is the estimation of the frequencies of a set of linguistic variants. Learning is modeled with learners using Bayes' Law to estimate probability distribution over the set of variants.

Assume that a learner is exposed to N occurrences of a linguistic token such as word, sound, or grammatical construction, partitioned over K different variants. Let the vectors $x=[x_{1},x_{2},...,x_{K}]and\theta =[\theta _{1},\theta _{2},...,\theta _{K}]$ denote the observed frequencies and the estimated probabilities of the K variants, respectively. The learner's expectations are expressed in a prior probability distribution $P(\theta )$ . After seeing the data $x$ , the learner updates the posterior probability of $\theta$ , $P(\theta |x)$ using Bayes' Law:

$P(\theta |x)={\frac {P(x|\theta )P(\theta )}{\int {P(x|\theta )P(\theta )d\theta '}}}$

Although the model is neutral over the linguistic variants with no variant being favored a priori over the others, learners can differ in their expectations about the amount of probabilistic variation in a language. For example, a learner may expect the language to be more deterministic thus assigning a very small probability to any unobserved variant, while another learner may assign it a much higher probability, indicating the willingness to consider the unobserved variants as part of the language. This is related to the process of smoothing in language modeling.

A way to capture such expectations while still maintaining neutrality between variants is to use, just like in smoothing, the K-dimensional Dirichlet distribution as priors. Specifically, if the prior $P(\theta )$ is a symmetric K-dimensional Dirichlet distribution with parameters $\alpha /K$ , the probability a learner will assign to the next observation being variant k after seeing $x_{k}$ instances of that variant from a total of N observation is equal to $(x_{k}+\alpha /K)/(N+\alpha )$ .

Thus, when $\alpha /K<1$ , learner will assign small probability to unseen variant, reducing the probabilistic variation of the language and favoring 'regularization' of languages towards deterministic rules. When $\alpha /K>1$ on the other hand, learner will assign higher probability to unseen variant.

Using this framework, how a language evolves can thus be formulated as such: The learner estimates $\theta$ from a sample of N tokens produced by a speaker before generating any utterances x himself by sampling from the distribution $P(x|\theta )$ associated with his estimate of $\theta$ . His generated utterances are then presented to the next learner and so on, thus forming a kind of iterated learning process:

Since the frequencies generated by a learner depend only on the frequencies generated by the previous learner, this iterated learning process of how language evolves is a Markov chain. It is possible to analyze the dynamics of the process (i.e. the dynamics of how a language changes) by computing a transition matrix indicating the probability of moving from one frequency value to another across generations. Its asymptotic consequences can be characterized by computing the stationary distribution to which the Markov chain converges.

Datasets used

Aside from using simulated data in its experiments, the paper also uses a corpus of child-directed speech. The experiments involve simulating the process of language evolution via iterated learning.

Experimental Results

The model developed in this paper is related to The Wright-Fisher model in biology. The Wright-Fisher model describes the behavior of alleles evolving in the absence of selection. Hence the Wright-Fisher model is 'neutral' over the alleles variants; just like the model in this paper is 'neutral' over the linguistic variants. In drawing the similarity between Wright-Fisher model and the model proposed, the paper can therefore use results from population genetics to characterize the dynamics and stationary distribution of the Markov chain defined by iterated learning, indicating the kind of languages that will emerge over time. In particular, the equivalence between these models can account for three basic regularities in the form and evolution of languages:

S-shaped curves in language change: When old linguistic variants are replaced by new ones, an s-shaped curve is typically observed in plots of frequency over time. Using the model proposed, the paper shows that such curve can emerge in the frequency plot over time provided that learners have priors favoring regularization, i.e. $\alpha /K<1$

Emergence of power-law distribution: One of the interesting properties of human languages is that word frequencies follow a power-law distribution. The paper shows that such phenomena can be produced via simulation with select parameters using the proposed 'neutral' model. Comparison of the distribution over frequencies produced by the simulation with that computed from the child-directed speech corpus shows that the model produces a power law with exponent $\gamma =1.74$ , providing a close match with the exponent estimated from the child-directed speech corpus ( $\gamma =1.7$ )

Frequency effects in lexical replacement rates: Another properties of human languages is that frequently used words are replaced much more slowly than less frequent ones. In other words, there is an inverse power-law relationship between frequency of use and replacement rate. Using simulation based on the proposed model, the paper is able to show that replacement rate follows an inverse power-law relationship with frequency.

Discussion

The novelty of the paper lies in its ability to draw similarity between simple iterated learning mechanism that uses Bayesian inference with model of genetic drift (Wright-Fisher model), hence providing justification of using the models of genetic drift to account for language changes over time in the absence of selection of its linguistic variants. By manipulating the values of the priors, the paper is able to model the different expectations of learners on the variability of the language while still maintaining neutrality between its variants. The similarity drawn between the model proposed and the Wright-Fisher model from biology is also interesting as it can possibly shed more lights into the nature of language evolution and how closely related it is to biological evolution.

The drawback of this paper is that it is basically simulation of unigram language modeling with smoothing, conducted iteratively over several 'generations'. Unfortunately, these 'generations' of iterated learning are not anchored to real time sequence or dataset. In the future, a mapping between 'generations' to actual time period can be explored.

Related Papers

Unlike this paper which defines a 'neutral' model of how languages evolve in the absence of selection at the level of linguistic variants: it models language evolution only as a result of being transmitted from one learner to another; other recent computational work has focused on the role of selective forces or directed mutation at the level of linguistic variants:

@@ Line 12: / Line 12: @@
 == Description of the method ==
-In HMM model to segment a document, the document is treated as a collection of mutually independent sets of words. Each set is probabilistically generated by a hidden topic variable in a time series. Transition probabilities determine the value of the next hidden topic variable in the series.
+The focus of this paper is to model how language changes as a consequence of being passed from one learner to another. The learning problem considered is the estimation of the frequencies of a set of linguistic variants. Learning is modeled with learners using [[UsesMethod::Bayes' Law]] to estimate probability distribution over the set of variants.
-[[File:HMM.png]]
+Assume that a learner is exposed to ''N'' occurrences of a linguistic token such as word, sound, or grammatical construction, partitioned over ''K'' different variants. Let the vectors <math>x = [x_1, x_2, ..., x_K] and  \theta = [\theta_1, \theta_2, ..., \theta_K]</math> denote the observed frequencies and the estimated probabilities of the ''K'' variants, respectively. The learner's expectations are expressed in a prior probability distribution <math>P(\theta)</math>. After seeing the data <math>x</math>, the learner updates the posterior probability of <math>\theta</math>, <math>P(\theta|x)</math> using [[UsesMethod::Bayes' Law]]:
-The generative process is as follows: choose a topic ''z'', then generate a set of ''L'' independent words ''w'' from a distribution over words associated with that topic. Then choose another topic from a distribution of allowed transitions between topics. Given an unsegmented document, the most likely sequence of topics that generate the observed ''L''-word sets in the document are computed (using [[UsesMethod::Viterbi]] algorithm). Topic breaks occur at points where the value two consecutive topics are different.
+<math>P(\theta|x) = \frac {P(x|\theta)P(\theta)} {\int {P(x|\theta)P(\theta) d\theta'}}</math>
-The drawback of HMM method is the [[UsesMethod::Naive Bayes]] assumption of conditional independence of words within each ''L''-word set given a topic:
+Although the model is neutral over the linguistic variants with no variant being favored a priori over the others, learners can differ in their expectations about the '''amount of probabilistic variation in a language'''. For example, a learner may expect the language to be more deterministic thus assigning a very small probability to any unobserved variant, while another learner may assign it a much higher probability, indicating the willingness to consider the unobserved variants as part of the language. This is related to the process of [[UsesMethod::smoothing]] in language modeling.
-<math>P(o_t|z)=\prod_{i=1}^L P(w_i|z)</math>
+A way to capture such expectations while still maintaining neutrality between variants is to use, just like in [[UsesMethod::smoothing]], the ''K''-dimensional [[UsesMethod::Dirichlet distribution]] as priors. Specifically, if the prior <math>P(\theta)</math> is a symmetric ''K''-dimensional Dirichlet distribution with parameters <math>\alpha/K</math>, the probability a learner will assign to the next observation being variant ''k'' after seeing <math>x_k</math> instances of that variant from a total of N observation is equal to <math>(x_k + \alpha/K)/(N + \alpha)</math>.
-This assumption works well as ''L'' becomes large. However, the larger ''L'' becomes, the less precise (coarser) is the segmentation.
+Thus, when <math>\alpha/K < 1</math>, learner will assign small probability to unseen variant, reducing the probabilistic variation of the language and favoring 'regularization' of languages towards deterministic rules. When <math>\alpha/K > 1</math> on the other hand, learner will assign higher probability to unseen variant.
-The aspect HMM segmentation model does away with this Naive Bayes assumption of conditional independence of words, by adding a probability distribution (an aspect model) over pairs of discrete random variables: in this case the pair consists of the ''L''-word window of observation and a word. The ''L''-word window of observation is not represented as a set of its words but simply a label which identifies it. It is associated with its corresponding set of words through each window-word pair. With this aspect model, the occurrence of a window of observation ''o'' and a word ''w'' are independent of each other given a hidden topic variable ''z'':
+Using this framework, how a language evolves can thus be formulated as such:
+The learner estimates <math>\theta</math> from a sample of ''N'' tokens produced by a speaker before generating any utterances ''x'' himself by sampling from the distribution <math>P(x|\theta)</math> associated with his estimate of <math>\theta</math>. His generated utterances are then presented to the next learner and so on, thus forming a kind of ''iterated learning'' process:
-<math>P(o,w,z)=P(o|z)P(w|z)P(z)</math>
+[[File:IteratedLearning.png]]
-[[File:aspectHMM.png]]
+Since the frequencies generated by a learner depend only on the frequencies generated by the previous learner, this ''iterated learning'' process of how language evolves is a [[UsesMethod::Markov chain]]. It is possible to analyze the dynamics of the process (i.e. the dynamics of how a language changes) by computing a transition matrix indicating the probability of moving from one frequency value to another across generations. Its asymptotic consequences can be characterized by computing the stationary distribution to which the Markov chain converges.
-The paper uses [[UsesMethod::Expectation Maximization]] and [[UsesMethod::Bayes' Law]] to estimate the parameters <math>P(z)</math>, <math>P(o|z)</math>, and <math>P(w|z)</math>.
-Given an unsegmented document, aspect HMM divides its words into observation windows of size ''L'' and running the Viterbi algorithm to find the most likely sequence of hidden topics that generate the given document. Segmentation breaks occur when the topic of one window is different from the next window.
 == Datasets used ==
-The aspect HMM segmentation model is applied on two corpora:
+Aside from using simulated data in its experiments, the paper also uses a corpus of [[UsesDataset::child-directed speech]]. The experiments involve simulating the process of language evolution via iterated learning.
-* A corpus of [[UsesDataset::SpeechBot]] transcripts from ''All Things Considered'' (ATC), a daily news program on National Public Radio. This dataset consists of 4,917 segments with 35,777 word types and about 4 million word tokens. The word error rate in this corpus is estimated to be in the 20% to 50% range.
-* A corpus of 3,830 articles from the New York Times (NYT) consisting of about 4 million word tokens and 70,792 word types.
-The aspect HMM is trained with 20 hidden topics in the experiments.
 == Experimental Results ==
-Three variants of the two corpora are used in the experiments:
+The model developed in this paper is related to The Wright-Fisher model in biology. The Wright-Fisher model describes the behavior of alleles evolving in the absence of selection. Hence the Wright-Fisher model is 'neutral' over the alleles variants; just like the model in this paper is 'neutral' over the linguistic variants. In drawing the similarity between Wright-Fisher model and the model proposed, the paper can therefore use results from population genetics to characterize the dynamics and stationary distribution of the Markov chain defined by iterated learning, indicating the kind of languages that will emerge over time. In particular, the equivalence between these models can account for three basic regularities in the form and evolution of languages:
-* a random sequences of segments from the ATC corpus
-* a random sequences of segments from the NYT corpus
-* actual aired sequences of ATC segments (in this audio transcripts, clear demarcations of segmentation breaks are not explicitly given; this is the primary problem that the paper is trying to tackle)
-The paper uses [[UsesMethod::co-occurrence agreement probability]] (CoAP) to quantitatively evaluate the segments produced by their model. In short the paper uses CoAP to measure how often a segmentation is correct with respect to two words that are ''k'' words apart in the document.
-A useful interpretation of the CoAP is through its compliment:
+* S-shaped curves in language change: When old linguistic variants are replaced by new ones, an s-shaped curve is typically observed in plots of frequency over time. Using the model proposed, the paper shows that such curve can emerge in the frequency plot over time provided that learners have priors favoring regularization, i.e. <math>\alpha/K < 1</math>
-<math>P(disagreement) = P(missed)P(seg) + (1 - P(seg))P(false)</math>
+[[File:s-shapedCurve.png]]
-where <math>P(seg)</math> is the a priori probability of a segment, <math>P(missed)</math> is the probability of missing a segment, and <math>P(false)</math> is the probability of hypothesizing a segment where there is no segment.
+* Emergence of power-law distribution: One of the interesting properties of human languages is that word frequencies follow a power-law distribution. The paper shows that such phenomena can be produced via simulation with select parameters using the proposed 'neutral' model. Comparison of the distribution over frequencies produced by the simulation with that computed from the [[UsesDataset::child-directed speech]] corpus shows that the model produces a power law with exponent <math>\gamma = 1.74</math>, providing a close match with the exponent estimated from the child-directed speech corpus (<math>\gamma = 1.7</math>)
-In the random sequences of segments, the model performs better (i.e. produces better segmentation) on NYT randomized segments than on ATC randomized segments; probably since NYT is a cleaner, more error-free corpus than ATC. The result on actual aired ATC sequence seems worse than either of the randomized test set.
+[[File:corpusData.png]]
-[[File:resultAHMM.png]]
+[[File:simulatedData.png]]
-When comparing the performance of aspect HMM (AHMM) to HMM model in segmenting NYT corpus, AHMM outperforms HMM segmentation for small window widths. As the window size increases HMM increasingly does well since all words are counted equally in increasingly larger windows, satisfying HMM's Naive Bayes assumption of mutual independence between words. However, as window size increases the precision of the segmenter also decreases due to coarser segmentation.
+* Frequency effects in lexical replacement rates: Another properties of human languages is that frequently used words are replaced much more slowly than less frequent ones. In other words, there is an inverse power-law relationship between frequency of use and replacement rate. Using simulation based on the proposed model, the paper is able to show that replacement rate follows an inverse power-law relationship with frequency.
-[[File:resultAHMM-HMM.png]]
+[[File:replacementRate.png]]
 == Discussion ==
-The novelty of the paper lies on its addition of aspect model to HMM model for segmenting documents. It removes HMM naive assumption that words are generated independently given the hidden topic variable. Instead, words from the selected hidden variable are generated via the aspect model rather than independently generated.
+The novelty of the paper lies in its ability to draw similarity between simple iterated learning mechanism that uses Bayesian inference with model of genetic drift (Wright-Fisher model), hence providing justification of using the models of genetic drift to account for language changes over time in the absence of selection of its linguistic variants. By manipulating the values of the priors, the paper is able to model the different expectations of learners on the variability of the language while still maintaining neutrality between its variants. The similarity drawn between the model proposed and the Wright-Fisher model from biology is also interesting as it can possibly shed more lights into the nature of language evolution and how closely related it is to biological evolution.
-However, one of the possible drawback of the paper is its very coarse approximation to the probability distribution over the observation windows ''o''. The Viterbi algorithm requires the observation probability <math>P(o_t|z)</math> for each time step. While the HMM uses its Naive Bayes assumption to compute this distribution, the AHMM can only compute conditional probabilities about observation windows which it was exposed to in training. In testing, the observation window <math>o_t</math> may not be something the model has seen before. The paper therefore uses an online approximation to EM to find <math>P(o_t|z)</math> that refines its probability approximation recursively as it sees more words in the observation window. Words in the beginning of the window are weighted more heavily than words towards the end of the window. Therefore, as window size increases, more words make less impact on the observation distribution and the segmenter does not perform as well.
+The drawback of this paper is that it is basically simulation of unigram language modeling with smoothing, conducted iteratively over several 'generations'. Unfortunately, these 'generations' of iterated learning are not anchored to real time sequence or dataset. In the future, a mapping between 'generations' to actual time period can be explored.
-Another possible drawback is that AHMM does not model topic breaks explicitly. Topic breaks are implicitly assumed when two adjacent windows have different topic variables. This lack of explicit modeling of topic breaks is possibly what causes the model's tendency to undersegment, as indicated by the high <math>P(missed)</math> values in the experiment. In future, direct modeling of topic breaks may be explored. The idea of an overlapping window may also be good to improve the precision of the segmentation. The idea to automatically assign labels on each segment (i.e. topic labeling) is also an interesting future direction.
+== Related Papers ==
-== Related Papers ==
+Unlike this paper which defines a 'neutral' model of how languages evolve in the absence of selection at the level of linguistic variants: it models language evolution only as a result of being transmitted from one learner to another; other recent computational work has focused on the role of selective forces or directed mutation at the level of linguistic variants:
-The main contribution of this paper is its addition of aspect model to HMM model for segmenting documents. Other earlier works for topic segmentation include:
+* [[RelatedPaper::Komarova, N. L. & Nowak, M. A. 2001 Natural selection of the critical period for language acquisition. Proc. R. Soc. Lond. B. 268, 1189 – 1196]]
-* [[RelatedPaper::Doug Beeferman, Adam Berger, and John Lafferty. Statistical models for text segmentation. Machine Learning, 1999]]
+* [[RelatedPaper::Christiansen, M. H. & Chater, N. 2008 Language as shaped by the brain. Behav. Brain Sci. 31, 489–558]]
-* [[RelatedPaper::Marti A. Hearst. Context and structure in automated full-text information access. University of California at Berkeley dissertation. Computer Science Division Technical Report, 1994]]
-* [[RelatedPaper::P. van Mulbregt, I. Carp, L. Gillick, S. Lowe, and J. Yamron. Text segmentation and topic tracking on broadcast news via a hidden markov model approach. 1998]]
-* [[RelatedPaper::Thomas Hofmann, Probabilistic Latent Semantic Indexing, SIGIR 2009]]

Difference between revisions of "Florencia Reali and Thomas L. Griffiths, Words as alleles: connecting language evolution with Bayesian learners to models of genetic drift, Proceedings of The Royal Society of London. Series B, Biological Sciences 2010"

Latest revision as of 12:01, 31 March 2011

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Summary

Description of the method

Datasets used

Experimental Results

Discussion

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