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{
"paper_id": "I11-1034",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T07:32:31.086425Z"
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"title": "S 3 -Statistical Sam . dhi Splitting",
"authors": [
{
"first": "Abhiram",
"middle": [],
"last": "Natarajan",
"suffix": "",
"affiliation": {
"laboratory": "Brown Laboratory for Linguistic Information Processing Brown University",
"institution": "",
"location": {
"country": "USA"
}
},
"email": "abhiram.nat@gmail.com"
},
{
"first": "Eugene",
"middle": [],
"last": "Charniak",
"suffix": "",
"affiliation": {
"laboratory": "Brown Laboratory for Linguistic Information Processing Brown University",
"institution": "",
"location": {
"country": "USA"
}
},
"email": ""
}
],
"year": "",
"venue": null,
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"abstract": "The problem of Sam. dhi-Splitting is central to computational processing of Sanskrit texts. Currently the best-known algorithm for this task, given a chunk, generates all possible splits and chooses the Maximum-a-Posteriori estimate as the final answer. Our contributions to the task of Sam. dhi-Splitting are twofold. Firstly, we improve upon the current algorithm by proposing a principled modification of the posterior probability function to achieve better results. Secondly, we propose an algorithm based on Bayesian Word-Segmentation methods. We find that the unsupervised version of our algorithm achieves a better precision than the current algorithm with the original probabilistic model. We then present a supervised version of our algorithm that outperforms all previous methods/models.",
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"text": "The problem of Sam. dhi-Splitting is central to computational processing of Sanskrit texts. Currently the best-known algorithm for this task, given a chunk, generates all possible splits and chooses the Maximum-a-Posteriori estimate as the final answer. Our contributions to the task of Sam. dhi-Splitting are twofold. Firstly, we improve upon the current algorithm by proposing a principled modification of the posterior probability function to achieve better results. Secondly, we propose an algorithm based on Bayesian Word-Segmentation methods. We find that the unsupervised version of our algorithm achieves a better precision than the current algorithm with the original probabilistic model. We then present a supervised version of our algorithm that outperforms all previous methods/models.",
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"section": "Abstract",
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"text": "Sanskrit, considered to be among the oldest languages in the world, is elaborate in its oral specifications. It possesses a set of euphonic rules called Sam . dhi 1 rules, which when applied, cause phonological changes at word or morph boundaries. These rules are enumerated in P\u0101n . ini's As . t .\u0101 dhy\u0101y\u012b, and have been devised with the aim of achieving epigrammatic brevity. There are two kinds of Sam . dhi:",
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"section": "Introduction",
"sec_num": "1"
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"text": "1. Internal Sam . dhi: Euphonic transformation at morph boundaries; W 1 x + yW 2 \u21d2 W 1 zW 2 , where x and y are the final and initial segments of W 1 and W 2 , respectively, and z respresents the smoothed phonetic transformation of x and y together 1 Literally means \"Putting Together\". E.g: d\u012bpena udvejayati \u21d2 d\u012bpenodvejayati 2. External Sam . dhi: Phonetic change at word boundaries;",
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"section": "Introduction",
"sec_num": "1"
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{
"text": "W 1 x + W 2 \u21d2 W 1 x + W 2 ,",
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"section": "Introduction",
"sec_num": "1"
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"text": "where the final segment of the first word x changes to x E.g: utthitah . vidy\u0101dharah . \u21d2 utthito vidy\u0101dharah .",
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"section": "Introduction",
"sec_num": "1"
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"text": "We shall refer to the process of making the euphonic transformations as Sam . dhising and the text formed as a result as Sam . dhied text. The process of undoing the euphonic transformations shall be referred to as Analysis or Sam . dhi-Splitting 2 , and the text formed as a result will be referred to as Analysed text.",
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"section": "Introduction",
"sec_num": "1"
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"text": "It is easy to see that the task of Sam . dhi-Splitting is ambiguous. For instance, any of {(ca,api), (c\u0101,api), (ca,\u0101pi), (c\u0101,\u0101pi)} could have combined to form c\u0101pi.",
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"section": "Introduction",
"sec_num": "1"
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"text": "Contextual knowledge is necessary for perfect analysis of Sanskrit text (Hellwig, 2009) , and current methods have not evolved enough to be able to supply context. However the field of Statistical Machine Learning, by making reasonable assumptions, allows us to provide approximations of these processes. Previous work (Beesley, 1998; Hyman, 2007) indicates that Finite State Transducers (FST) could be used to generate morphologically valid splits. Mittal (2010) considers the FST approach as well as an approach based on Optimality Theory (Prince and Smolensky, 1993) , by defining a posterior probability function to choose among all morphologically valid splits of a chunk (OT1). More recently, Kumar et al. (2010) report findings using a different posterior probability function with the same Optimality Theory approach (OT2).",
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"start": 72,
"end": 87,
"text": "(Hellwig, 2009)",
"ref_id": "BIBREF8"
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"start": 319,
"end": 334,
"text": "(Beesley, 1998;",
"ref_id": "BIBREF3"
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{
"start": 335,
"end": 347,
"text": "Hyman, 2007)",
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"start": 541,
"end": 569,
"text": "(Prince and Smolensky, 1993)",
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"text": "Kumar et al. (2010)",
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"text": "Firstly, we derive our own posterior probability function, which is different from OT1 and OT2.",
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"text": "We observe better results. Secondly, and more importantly, we present a Sam . dhi-Splitting technique based on the Bayesian Word-Segmentation methods presented by Goldwater et al. (2006a) . These methods in turn are based on the Dirichlet process. The Dirichlet process is a continuous multivariate distribution used in nonparametric Bayesian Statistics, often as a convenient prior distribution. Gibbs sampling is used to sample from the posterior distribution of analysed texts given the Sam . dhied text.",
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"start": 163,
"end": 187,
"text": "Goldwater et al. (2006a)",
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"section": "Introduction",
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"text": "The paper is organised as follows. In Section 2, we discuss the Optimality Theory approach in detail. In Section 3, we introduce our framework for Sam . dhi-Splitting. We present both the unsupervised and supervised versions of our algorithm in this section. Section 4 then provides specific details of our implementations and Section 5 contrasts the results obtained by all the methods/models considered. Section 6 concludes the paper.",
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"section": "Introduction",
"sec_num": "1"
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"text": "The literature on Sam . dhi-Splitting is too huge for us to do justice. We shall stick to the best published results so far, which have been obtained by the OT method (Kumar et al., 2010; Mittal, 2010) . Procedure 1 is a pseudocode representation of the OT method.",
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"start": 167,
"end": 187,
"text": "(Kumar et al., 2010;",
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"start": 188,
"end": 201,
"text": "Mittal, 2010)",
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"section": "Current Methods",
"sec_num": "2"
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"text": "Procedure 1 : Sam . dhi-Splitting using OT 1: for each chunk \u2208 D do 2:",
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"text": "S \u2190 getAllPossibleSplits(chunk) 3: S \u2190 {x : x \u2208 S, x is morphologically valid} 4: l \u2190 arg max s {P (s) : s \u2208 S } 5: print l",
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"section": "Current Methods",
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"text": "The procedure is quite straightforward. Each candidate chunk is recursively broken (Line 2) to generate all possible splits. Each split is passed through a morphological analyser and the splits that contain one or more invalid morphs are discarded (Line 3). Finally, the Maximum a posteriori (MAP) estimate is chosen as the answer (Line 4). P (s) is the posterior probability function, which we derive as follows. Consider a morphologically valid split s =< c 1 , . . . , c m >, where c 1 . . . c m are its constituents, which was obtained after applying rules r =< r 1 , . . . , r m\u22121 > 3 on a chunk C. We would need to find the most probable split, which is given by arg max s\u2208S P (s|C). Using the noisy channel model framework (Shannon, 1948) , we get arg max s\u2208S P (s|C) = arg max s\u2208S P (C|s) \u00d7 P (s) 1We know that s is a result of applying r on C at specific points. Now, P (C|s) reads as the conditional probability of obtaining C given that r was applied at exactly those points. Clearly, this is 1. Thus the MAP estimate would be a split that maximises the probability of the prior",
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"start": 730,
"end": 745,
"text": "(Shannon, 1948)",
"ref_id": "BIBREF17"
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{
"text": "EQUATION",
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{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "l = arg max s\u2208S P (s)",
"eq_num": "(2)"
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],
"section": "Current Methods",
"sec_num": "2"
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"text": "Note that we use the notationP instead of P because we can never know the true value of P (s), but can only estimate it from our training set. Let us now turn our attention toP (s). We read P (s) as the probability of generating the morphs < c 1 , . . . , c m >. Thus we get:",
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"sec_num": "2"
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{
"text": "EQUATION",
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{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P (s) =P (c 1 ) \u00d7P (c 2 | c 1 ) \u00d7P (c 3 | c 1 , c 2 ) \u00d7 . . . = m j=1P (c j ) [Assuming Independence]",
"eq_num": "(3)"
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],
"section": "Current Methods",
"sec_num": "2"
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"text": "Equation 3 describes a generative model in that it shows us how we can generate the morphs < c 1 , . . . , c m >. Mittal (2010) ",
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{
"start": 114,
"end": 127,
"text": "Mittal (2010)",
"ref_id": "BIBREF14"
}
],
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"section": "Current Methods",
"sec_num": "2"
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"text": "definesP (s) as m\u22121 i=1 P (c i ) +P (c i+1 ) \u00d7P (r i ) m (4) Kumar et al. (2010) defineP (s) as m i=1P (c i ) \u00d7 m\u22121 j=1P (r j ) m (5)",
"cite_spans": [],
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"section": "Current Methods",
"sec_num": "2"
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"text": "In each of these models, the maximum likelihood estimators forP (c i ) andP (r j ) are used, i.e. P (c i ) is set to the relative frequency of c i in the training set, andP (r j ) to the relative frequency of the usage of r j in the training set.",
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"section": "Current Methods",
"sec_num": "2"
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"text": "Note that unlike Equation 4 and Equation 5, Equation 3 does not have rule probabilities. This can be explained by keeping in mind that Equation 3 describes a generative model. If we generated morphs and knew that they had to be combined to form a chunk, the rules are uniquely determined. In Section 5, we shall compare the results obtained by using each of Equations 3, 4 and 5 on standard datasets.",
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"section": "Current Methods",
"sec_num": "2"
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"text": "3 Sam . dhi-Splitting based on the Dirichlet Process",
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"section": "Current Methods",
"sec_num": "2"
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"text": "Our method is similar to the two-stage modelling framework described by Goldwater et al. (2006a; 2006b) . The framework has two components, a morph generator which generates morphs likely to be found in a lexicon, from some probability distribution, and an adaptor which determines how often each of these morphs occur. Section 3.1 and Section 3.2 describe the unsupervised version of our algorithm while Section 3.3 describes the supervised version.",
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"start": 72,
"end": 96,
"text": "Goldwater et al. (2006a;",
"ref_id": "BIBREF5"
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"text": "2006b)",
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"sec_num": "2"
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"text": "In the realm of the framework, a sam . dhied chunk C = c 1 c 2 . . . c m can be viewed to be created as follows:",
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"section": "Foundations",
"sec_num": "3.1"
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"text": "1. A generator P s generates a super-set sequence of morphs 4 : M = M 1 . . . M n from a probability distribution P s .",
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"text": "2. The adaptor P \u03b3 generates a sequence of integers: Z = z 1 . . . z m , each of which are identifiers of one particular item from M. This means that 1 \u2264 z i \u2264 n, and",
"cite_spans": [],
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"sec_num": "3.1"
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"text": "z i = x \u21d2 c i = M x .",
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"sec_num": "3.1"
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"text": "Once the morphs of a chunk are generated, Sam . dhi rules are applied between them to form C. Henceforth T woStage(P \u03b3 , P s ) shall be used to denote a two-stage framework with P \u03b3 as the adaptor and P s as the generator. Let us move to the Chinese Restaurant Process, which we use as our adaptor.",
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"section": "Foundations",
"sec_num": "3.1"
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"text": "Having studied many corpora of natural language utterances, Zipf (1932) made a famous empirical observation that word frequencies follow a powerlaw distribution, i.e., the frequency of a word is inversely proportional to its frequency rank. This means that the most frequent word in a corpus occurs twice as often as the second most frequent word, thrice as often as the third most frequent word, and so on. Mathematically speaking, if w r is the r-ranked word in a corpus, then",
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{
"start": 60,
"end": 71,
"text": "Zipf (1932)",
"ref_id": "BIBREF18"
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],
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"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
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"text": "f (w r ) \u221d 1 r c (6)",
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"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "where f (w r ) denotes the frequency of occurrence of w r , and c \u2248 1. The veracity of this law has hence been verified by a study on present-day English (Kucera and Francis, 1967) . Typically, power-law distributions are produced by stochastic processes in which outcomes accrue probability based on the probability they already have. Such processes are called preferential attachment processes. Let us turn our attention to the Chinese Restaurant Process (Aldous et al., 1985) (CRP), which is one such process. The process is best explained by specifying how to draw a sample from it. Consider a restaurant with an infinite number of tables, each with infinite capacity. Customers enter the restaurant, one at a time, and seat themselves at a table of their choice. They choose an occupied table with probability proportional to the number of people already present at the table, or an unoccupied table with probability proportional to some real-valued scalar parameter \u03b1. The first customer always sits at the first table. Then onwards, the i th customer sits at a table T i , and T i follows the distribution:",
"cite_spans": [
{
"start": 154,
"end": 180,
"text": "(Kucera and Francis, 1967)",
"ref_id": "BIBREF12"
},
{
"start": 457,
"end": 478,
"text": "(Aldous et al., 1985)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "P (T i = k | Previous Customers) = \uf8f1 \uf8f2 \uf8f3 n k i\u22121+\u03b1 1 \u2264 k \u2264 N i\u22121 \u03b1 i\u22121+\u03b1 k = N i\u22121 + 1 (7)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "where n k is the number of people occupying table k and N i\u22121 is total number of tables occupied by the previous (i \u2212 1) customers.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "When the CRP is combined with a morph generator, it can be used to generate a power-law distribution over morphs. Such a model can be described as T woStage(CRP (\u03b1), P s ). We could view this as restaurant where each table represents a morph. Every customer is also labelled with a morph and can either sit only at tables which are labelled with the same morph, or at an unoccupied table. A customer at a table represents one occurence of the morph that is represented by the table. To get the probability distribution of the i th morph, we must sum over all the tables labelled with that morph: 5",
"cite_spans": [],
"ref_spans": [
{
"start": 382,
"end": 484,
"text": "table. A customer at a table represents one occurence of the morph that is represented by the table.",
"ref_id": null
}
],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "P (c i = c | c \u2212i )",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "= P (assign customer to any of the c tables)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "+ P (assign customer to a new table) = n c \u2212i c i \u2212 1 + \u03b1 + P s (c) \u00d7 \u03b1 i \u2212 1 + \u03b1 = n c \u2212i c + \u03b1 \u00d7 P s (c) i \u2212 1 + \u03b1 (8)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "where",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "c \u2212i = c 1 . . . c i\u22121 and n c \u2212i c",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "represents the number of occurrences of c in c \u2212i .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "Let us examine Equation 8. It is in accordance with the principle of preferential attachment. This is because the probability of generating a morph that has already been generated increases as more instances of the morph are observed. Also note that the sparseness of the distribution generated increases with an increase in the value of the parameter \u03b1. This is explained by the fact that \u03b1 \u00d7 P s (c) is constant and it reduces w.r.t. the first summand over time. What this means is that the probability of generating a novel morph decreases (but never dissapears fully) as more data is observed.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "Let us now turn our attention to the generator. For simplicity, we choose a unigram phoneme distribution for P s . If a morph c i contains the phonemes a 1 . .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": ". a d P s (c i ) = p \u03a6 (1 \u2212 p \u03a6 ) d\u22121 d j=1 P (a j )",
"eq_num": "(9)"
}
],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "where p \u03a6 is the probability of a Sam . dhi rule being applied at any juncture. The reasoning for the equation is quite straightforward -the Sam . dhi rule is not applied (d \u2212 1) times, and applied at the d th phoneme. The unsupervised version of the algorithm uses an uniform distribution over phonemes.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Chinese Restaurant Process",
"sec_num": "3.1.1"
},
{
"text": "Given a coin with an unknown bias, say p, what would be a suitable distribution that reflects our expectations about p? It would be the Beta distribution. The Dirichlet distribution is a generalisation of the Beta distribution, in that it may have more than just two dimensions. The Dirichlet process is an infinite dimensional version of the Dirichlet distribution. Each sample from a Dirichlet process returns a distribution over an infinite set of random variables. Let us consider a Dirichlet process with its parameters as \u03b1 and P s . A sample from this process, say G, will consist of a set of all possible morphs and their corresponding probabilities. Now, we can generate each morph c i in the corpus from the distribution G. This can be represented as follows:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "c i \u223c G G \u223c DP (\u03b1, P s )",
"eq_num": "(10)"
}
],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "Following the argument in Goldwater (2006) , it is not hard to find that the model described by T woStage(CRP (\u03b1), P s ) is equivalent to the model represented by Equation 10.",
"cite_spans": [
{
"start": 26,
"end": 42,
"text": "Goldwater (2006)",
"ref_id": "BIBREF7"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "We must also keep in mind that, analogous to Equation 8, the probability of i th morph c i = c is given by",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P (c i = c | c \u2212i ) = n c \u2212i c + \u03b1 \u00d7 P s (c) i \u2212 1 + \u03b1",
"eq_num": "(11)"
}
],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "where c \u2212i = c 1 . . . c i\u22121 , and n",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "c \u2212i c",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "denotes the number of occurrences of c in c \u2212i . Going ahead a little, we realise that our input corpus is a set of sentences, where each sentence contains a set of Sam . dhied chunks, where each chunk is in turn formed by Sam . dhising separate morphs. The program does not need to separate the chunks from each other, all it needs to do is to separate a chunk into its constituent morphs. Referring to Equation 3, we recall our generative model view of how a Sam . dhied chunk is generated. We describe the process by the following PCFG: 6",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "P (r i = r branch | r \u2212i ): S \u2192 S C P (r i = r end | r \u2212i ): S \u2192 C P s (c i = c | c \u2212i ): C \u2192 c",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "We shall derive an equation for P (r i = r branch | r \u2212i ) in the next section.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Dirichlet Process Model",
"sec_num": "3.1.2"
},
{
"text": "As suggested in Gilks et al. (1996) , we use Gibbs Sampling to sample from the posterior distrubition of Sam . dhi Analyses. One iteration of the algorithm causes the program control to consider every possible splitting point in the data and form two hypotheses, say H 1 and H 2 . These hypotheses contain the same structure, except at the point of consideration. If the point of consideration is part of a chunk c, H 2 splits c into two individual morphs (say c 1 and c 2 ) while H 1 does not. Let H \u2229 denote the structure common to both H 1 and H 2 .",
"cite_spans": [
{
"start": 16,
"end": 35,
"text": "Gilks et al. (1996)",
"ref_id": "BIBREF4"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "Given the way Gibbs sampling works, we know that we only need the relative probabilities of H 1 and H 2 . Also, we must recall that the order of arrival of customers in the metaphorical Chinese Restaurant does not affect seating probability, as shown by Aldous et al. (1985) . This means that we could consider H \u2229 to have already been generated. From Equation 11 we get",
"cite_spans": [
{
"start": 254,
"end": 274,
"text": "Aldous et al. (1985)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P H 1 |H \u2229 = P c | H \u2229 = n H \u2229 c + \u03b1P s (c) |H \u2229 | + \u03b1",
"eq_num": "(12)"
}
],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "We also derive",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P H 2 |H \u2229 = P r branch \u222a c 1 \u222a c 2 | H \u2229 = P r branch | H \u2229 \u00d7 n H \u2229 c 1 + \u03b1P s (c 1 ) |H \u2229 | + \u03b1 \u00d7 n H \u2229 c 2 + I(c 1 = c 2 ) + \u03b1P s (c 2 ) |H \u2229 | + 1 + \u03b1",
"eq_num": "(13)"
}
],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "The extra '1' in the denominator of the third item of the product is because after c 1 , we have an additional morph. Also, I is the indicator function, which is 1 when the expression inside its parathesis is true. Let us now examine P (r branch | H \u2229 ). Clearly, each time we can choose only one among {r branch , r end }, so we have P (r branch ) + P (r end ) = 1. The number of r rules applied at the point of consideration would be (|H \u2229 | + 1) -one each for the |H \u2229 | morphs, and an extra one for the chunk under consideration. We apply a Beta( \u03c1 2 , \u03c1 2 ) prior to get: 7",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P r branch | H \u2229 = n H \u2229 r branch + \u03c1 2 |H \u2229 | + 1 + \u03c1",
"eq_num": "(14)"
}
],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "7 Henceforth, please substitute this expression in Equation 13",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "The algorithm works as follows -we begin with a random analysis of the corpus. After that, using Equations 12 and 13, we sample every potential boundary point. An iteration is said to have completed when the sampler samples over the entire corpus. We run multiple iterations of this process, until convergence.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "Simulated annealing (Aarts and Korst, 1989 ) is used to facilitate choosing of low probability transitions early in the algorithm, lest it gets stuck at a local maximum.",
"cite_spans": [
{
"start": 20,
"end": 42,
"text": "(Aarts and Korst, 1989",
"ref_id": "BIBREF0"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Gibbs Sampling for Sam . dhi Analysis",
"sec_num": "3.2"
},
{
"text": "As mentioned earlier, we use Gibbs Sampling to sample from the posterior distribution. Let us recall the relevant equations -Equations 12 and 13. In the supervised version of our algorithm, we use available training data (say D train ) to our advantage. Firstly, we define",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "H \u2229 = H \u2229 \u222a D train .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "Also, instead of assuming a uniform distribution over phonemes, we infer phoneme probabilities from D train :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P s (c i ) = p \u03a6 (1 \u2212 p \u03a6 ) d\u22121 d j=1P (a j )",
"eq_num": "(15)"
}
],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "Let \u03a0 denote the set of all phonemes found in D train . Given a phoneme a x , we setP (a x ) to be the maximum likelihood estimate:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "P (a x ) = n ax + \u03b1 0 n * + \u03b1 0 |\u03a0| (16)",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "where n ax is the number of times a x occurs in D train , n * = p\u2208\u03a0 n p , and \u03b1 0 is the unigram smoothing constant.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "Thus we get",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P H 1 |H \u2229 = n H \u2229 c + \u03b1P s (c) |H \u2229 | + \u03b1",
"eq_num": "(17)"
}
],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "EQUATION",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [
{
"start": 0,
"end": 8,
"text": "EQUATION",
"ref_id": "EQREF",
"raw_str": "P H 2 |H \u2229 = n H \u2229 r branch + \u03c1 2 |H \u2229 | + 1 + \u03c1 \u00d7 n H \u2229 c 1 + \u03b1P s (c 1 ) |H \u2229 | + \u03b1 \u00d7 n H \u2229 c 2 + I(c 1 = c 2 ) + \u03b1P s (c 2 ) |H \u2229 | + 1 + \u03b1",
"eq_num": "(18)"
}
],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "The supervised version of the algorithm now works exactly the same way as the unsupervised version, except that Gibbs sampling is done using Equations 17 and 18.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Supervised Version",
"sec_num": "3.3"
},
{
"text": "As a part of the Consortium project in India, a corpus of nearly 25000 parallel strings of Sam . dhied and analysed text has been formed (say D 1 ). OT trains on the 25000 string dataset and tests on a separate dataset of 2148 Sam . dhied Chunks (say D 2 ). We see that the test set used in OT We also use the morphological analyser (Akshar Bharati, 2006) provided by the apertium group.",
"cite_spans": [
{
"start": 291,
"end": 293,
"text": "OT",
"ref_id": null
},
{
"start": 341,
"end": 355,
"text": "Bharati, 2006)",
"ref_id": "BIBREF1"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Implementation Details",
"sec_num": "4"
},
{
"text": "As for the parameters of the model, we fixed \u03c1 = 2. We found that a decrease in p \u03a6 improved recall, but caused long words to get concatenated. At the same time higher values of p \u03a6 tended to cause over segmentation. It was also observed that higher \u03b1 values resulted in higher lexicon recall, once again only up to a point. Although it was not possible to arrive a single best value for either \u03b1 or p \u03a6 ; we fixed them to be p \u03a6 = 0.5 and \u03b1 = 20. The sampler is annealed with the following temperature schedule -1 \u03b3 = 0.1 to 1.0, in steps of 0.1.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Implementation Details",
"sec_num": "4"
},
{
"text": "For evaluation, standard statistical measures of Precision, Recall and F-Score were used. Precision indicates how many among the items found are correct; Recall indicates how many among the correct items are found. F-score is the harmonic mean of the Precision and the Recall.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Implementation Details",
"sec_num": "4"
},
{
"text": "Our implementations work with the Sanskrit Library Phonetic Basic format (SLP1). Since the morphological analyser uses the This explains why our implementations of OT achieve lower scores than what is reported in their paper. Readers must note that our implementation of OT, when trained on D1 and tested on D2, achieved nearly the same accuracy reported in their paper. Thus we believe our implementation of OT is accurate.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Implementation Details",
"sec_num": "4"
},
{
"text": "Tirupati format (WX), we use the transcoding tools on the Sanskrit Library Website (Scharf and Hyman, 2010). One may refer Huet (2009) for more details about the formats.",
"cite_spans": [
{
"start": 123,
"end": 134,
"text": "Huet (2009)",
"ref_id": "BIBREF9"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Implementation Details",
"sec_num": "4"
},
{
"text": "Let OT1, OT2, OT3 denote versions of Procedure 1, using Equations 4, 5, 3 as the posterior probability functions, respectively. Precision is calculated as the percentage of correct segmentations among all the segmentations made by the algorithm. Recall is calculated as the percentage of correct segmentations made by the algorithm among all the correct segmentations that need to be made. F-Score is the harmonic mean of the Precision and Recall. As we can see, using Equation 3 as the posterior probability function while obtaining the MAP estimate, we improve the previous best F-Score by nearly 3.5%. We shall now examine the results obtained by four different versions of the framework described in Section 3. Let these versions be denoted by NC1 . . . NC4:",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "5"
},
{
"text": "1. NC1 denotes the Two-Stage Framework in its most basic state. The algorithm possesses no linguistic knowledge whatsoever.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "5"
},
{
"text": "2. Kessler (1994) shows that each Sanskrit syllable must have only one Sonority peak. NC2 uses this knowledge to effect. The method is largely similar to NC1 except for the way in which sampling is done. We only sample when we are sure that the segments we generate do not violate Sonority Hierarchy. Refer Appendix A for more details about sonority hierarchy.",
"cite_spans": [
{
"start": 3,
"end": 17,
"text": "Kessler (1994)",
"ref_id": "BIBREF11"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "5"
},
{
"text": "3. NC3 uses a morphological analyser. Sampling is only done when we know that the segment we are leaving behind is morphologically valid.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Results",
"sec_num": "5"
},
{
"text": "4. NC4 uses a morphological analyser as well as training data. Sampling is done using Equations 17 and 18. Results obtained by these versions are shown in Table 2 . NC4 improves upon the F-Score achieved by OT2 by nearly 6%.",
"cite_spans": [],
"ref_spans": [
{
"start": 155,
"end": 162,
"text": "Table 2",
"ref_id": "TABREF3"
}
],
"eq_spans": [],
"section": "Results",
"sec_num": "5"
},
{
"text": "In this paper, we have presented an algorithm for Sam . dhi-Splitting which draws on expedients of Bayesian statistics. It is highly flexible in that it works even in the absence of (a) morphological analyser (b) training data. Also, one must remember that an entire corpus, as it is, can be fed to the algorithm, as opposed to OT which would require each Sam . dhied chunk separately. Finally, the fact that the algorithm runs in polynomial time, as opposed to OT which takes exponential time, further asserts the efficacy of the method.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Conclusion",
"sec_num": "6"
},
{
"text": "We shall use these terms interchangeably.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "Rule ri is applied between ci and ci+1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "In the case of Internal Sam . dhi, we consider morph boundaries and in the case of External Sam . dhi, we consider word boundaries. The theoretical foundations hold good for both cases. Thus the reader must always keep in mind that the concepts presented in this section apply at word boundaries too.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "Note that two different tables may represent the same morph.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "Probabilistic Context-Free Grammar",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
}
],
"back_matter": [
{
"text": "Sonority hierarchy is as follows:A syllable is said possess a sonority violation if it has more than one sonority peak (e.g. 'rjA'). However, at the beginning of a syllable, we must rate Spirants at the same level as Unvoiced Stops (e.g. 'sTApayati' is valid).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "A Sonority Hierarchy",
"sec_num": null
}
],
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"TABREF0": {
"content": "<table><tr><td>only</td></tr><tr><td>contains examples of Internal Sam . dhi. Thus, we merge the test set with the original 25000 string</td></tr><tr><td>dataset (D = D 1 \u222a D 2 ) and generate random samples of training data and test data. We use 3 4 of the data for training (D train ) and 1 4 for test-ing (D</td></tr></table>",
"type_str": "table",
"text": "test ). Note that D test can have instances of Internal Sam . dhi, External Sam . dhi as well as no Sam . dhi 8 at all. The scores we report are averaged over 5 random samples of D train and D test from D.",
"html": null,
"num": null
},
"TABREF1": {
"content": "<table><tr><td>shows the scores</td></tr></table>",
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"text": "",
"html": null,
"num": null
},
"TABREF3": {
"content": "<table/>",
"type_str": "table",
"text": "Accuracies obtained by different versions of NC; NC4 Outperforms NC1...3, as well OT1...3 in Precision and F-Score",
"html": null,
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}
}
}
} |