derive a gibbs sampler for the lda model

\prod_{k}{B(n_{k,.} Connect and share knowledge within a single location that is structured and easy to search. Do new devs get fired if they can't solve a certain bug? I find it easiest to understand as clustering for words.   The basic idea is that documents are represented as random mixtures over latent topics, where each topic is charac-terized by a distribution over words.1 LDA assumes the following generative process for each document w in a corpus D: 1. stream Latent Dirichlet Allocation Using Gibbs Sampling - GitHub Pages In this case, the algorithm will sample not only the latent variables, but also the parameters of the model (and ). Applicable when joint distribution is hard to evaluate but conditional distribution is known. PDF Latent Dirichlet Allocation - Stanford University Since then, Gibbs sampling was shown more e cient than other LDA training Im going to build on the unigram generation example from the last chapter and with each new example a new variable will be added until we work our way up to LDA. /Length 1368 We start by giving a probability of a topic for each word in the vocabulary, \(\phi\). Draw a new value $\theta_{3}^{(i)}$ conditioned on values $\theta_{1}^{(i)}$ and $\theta_{2}^{(i)}$. The chain rule is outlined in Equation (6.8), \[ vegan) just to try it, does this inconvenience the caterers and staff? > over the data and the model, whose stationary distribution converges to the posterior on distribution of . \begin{equation} We run sampling by sequentially sample $z_{dn}^{(t+1)}$ given $\mathbf{z}_{(-dn)}^{(t)}, \mathbf{w}$ after one another. \begin{equation} /Filter /FlateDecode We describe an efcient col-lapsed Gibbs sampler for inference. endstream stream endobj /BBox [0 0 100 100] \end{equation} Before we get to the inference step, I would like to briefly cover the original model with the terms in population genetics, but with notations I used in the previous articles. When Gibbs sampling is used for fitting the model, seed words with their additional weights for the prior parameters can . << /Type /XObject lda: Latent Dirichlet Allocation in topicmodels: Topic Models viqW@JFF!"U# From this we can infer \(\phi\) and \(\theta\). These functions take sparsely represented input documents, perform inference, and return point estimates of the latent parameters using the state at the last iteration of Gibbs sampling. /Type /XObject xP( After running run_gibbs() with appropriately large n_gibbs, we get the counter variables n_iw, n_di from posterior, along with the assignment history assign where [:, :, t] values of it are word-topic assignment at sampling $t$-th iteration. In 2004, Gri ths and Steyvers [8] derived a Gibbs sampling algorithm for learning LDA. \begin{equation} stream \tag{6.3} Assume that even if directly sampling from it is impossible, sampling from conditional distributions $p(x_i|x_1\cdots,x_{i-1},x_{i+1},\cdots,x_n)$ is possible. - the incident has nothing to do with me; can I use this this way? + \alpha) \over B(n_{d,\neg i}\alpha)} A Gentle Tutorial on Developing Generative Probabilistic Models and They are only useful for illustrating purposes. >> We derive an adaptive scan Gibbs sampler that optimizes the update frequency by selecting an optimum mini-batch size. Let (X(1) 1;:::;X (1) d) be the initial state then iterate for t = 2;3;::: 1. endobj Multiplying these two equations, we get. Making statements based on opinion; back them up with references or personal experience. /Resources 17 0 R . ;=hmm\&~H&eY$@p9g?\$YY"I%n2qU{N8 4)@GBe#JaQPnoW.S0fWLf%*)X{vQpB_m7G$~R (Gibbs Sampling and LDA) "IY!dn=G Algorithm. Suppose we want to sample from joint distribution $p(x_1,\cdots,x_n)$. Within that setting . /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 100.00128] /Coords [0.0 0 100.00128 0] /Function << /FunctionType 3 /Domain [0.0 100.00128] /Functions [ << /FunctionType 2 /Domain [0.0 100.00128] /C0 [0 0 0] /C1 [0 0 0] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [0 0 0] /C1 [1 1 1] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [1 1 1] /C1 [1 1 1] /N 1 >> ] /Bounds [ 25.00032 75.00096] /Encode [0 1 0 1 0 1] >> /Extend [false false] >> >> LDA with known Observation Distribution - Online Bayesian Learning in A standard Gibbs sampler for LDA - Coursera \]. endobj LDA with known Observation Distribution In document Online Bayesian Learning in Probabilistic Graphical Models using Moment Matching with Applications (Page 51-56) Matching First and Second Order Moments Given that the observation distribution is informative, after seeing a very large number of observations, most of the weight of the posterior . (PDF) ET-LDA: Joint Topic Modeling for Aligning Events and their rev2023.3.3.43278. The clustering model inherently assumes that data divide into disjoint sets, e.g., documents by topic. \end{aligned} _conditional_prob() is the function that calculates $P(z_{dn}^i=1 | \mathbf{z}_{(-dn)},\mathbf{w})$ using the multiplicative equation above. endobj $\beta_{dni}$), and the second can be viewed as a probability of $z_i$ given document $d$ (i.e. %PDF-1.4 Powered by, # sample a length for each document using Poisson, # pointer to which document it belongs to, # for each topic, count the number of times, # These two variables will keep track of the topic assignments. Staging Ground Beta 1 Recap, and Reviewers needed for Beta 2, Latent Dirichlet Allocation Solution Example, How to compute the log-likelihood of the LDA model in vowpal wabbit, Latent Dirichlet allocation (LDA) in Spark, Debug a Latent Dirichlet Allocation implementation, How to implement Latent Dirichlet Allocation in regression analysis, Latent Dirichlet Allocation Implementation with Gensim. The problem they wanted to address was inference of population struture using multilocus genotype data. For those who are not familiar with population genetics, this is basically a clustering problem that aims to cluster individuals into clusters (population) based on similarity of genes (genotype) of multiple prespecified locations in DNA (multilocus). These functions take sparsely represented input documents, perform inference, and return point estimates of the latent parameters using the . xP( """, """ (LDA) is a gen-erative model for a collection of text documents. Outside of the variables above all the distributions should be familiar from the previous chapter. \end{equation} &\propto p(z_{i}, z_{\neg i}, w | \alpha, \beta)\\ \begin{equation} In fact, this is exactly the same as smoothed LDA described in Blei et al. endstream endobj 145 0 obj <. (run the algorithm for different values of k and make a choice based by inspecting the results) k <- 5 #Run LDA using Gibbs sampling ldaOut <-LDA(dtm,k, method="Gibbs . /Filter /FlateDecode Building a LDA-based Book Recommender System - GitHub Pages Arjun Mukherjee (UH) I. Generative process, Plates, Notations . \tag{6.6} stream The next step is generating documents which starts by calculating the topic mixture of the document, \(\theta_{d}\) generated from a dirichlet distribution with the parameter \(\alpha\). A well-known example of a mixture model that has more structure than GMM is LDA, which performs topic modeling. model operates on the continuous vector space, it can naturally handle OOV words once their vector representation is provided. $C_{dj}^{DT}$ is the count of of topic $j$ assigned to some word token in document $d$ not including current instance $i$. p(\theta, \phi, z|w, \alpha, \beta) = {p(\theta, \phi, z, w|\alpha, \beta) \over p(w|\alpha, \beta)} The latter is the model that later termed as LDA. >> derive a gibbs sampler for the lda model - schenckfuels.com /Length 15 PDF Comparing Gibbs, EM and SEM for MAP Inference in Mixture Models << A popular alternative to the systematic scan Gibbs sampler is the random scan Gibbs sampler. The MCMC algorithms aim to construct a Markov chain that has the target posterior distribution as its stationary dis-tribution. hbbd`b``3 /Subtype /Form Deriving Gibbs sampler for this model requires deriving an expression for the conditional distribution of every latent variable conditioned on all of the others. XtDL|vBrh stream 0000012427 00000 n So in our case, we need to sample from \(p(x_0\vert x_1)\) and \(p(x_1\vert x_0)\) to get one sample from our original distribution \(P\). &= \int \prod_{d}\prod_{i}\phi_{z_{d,i},w_{d,i}} Parameter Estimation for Latent Dirichlet Allocation explained - Medium In the context of topic extraction from documents and other related applications, LDA is known to be the best model to date. Update $\mathbf{z}_d^{(t+1)}$ with a sample by probability. (CUED) Lecture 10: Gibbs Sampling in LDA 5 / 6. \\ \begin{equation} endobj Following is the url of the paper: \begin{aligned} Gibbs sampling was used for the inference and learning of the HNB. Random scan Gibbs sampler. Once we know z, we use the distribution of words in topic z, \(\phi_{z}\), to determine the word that is generated. It is a discrete data model, where the data points belong to different sets (documents) each with its own mixing coefcient. where $n_{ij}$ the number of occurrence of word $j$ under topic $i$, $m_{di}$ is the number of loci in $d$-th individual that originated from population $i$. /ProcSet [ /PDF ] 25 0 obj """ xP( &= \prod_{k}{1\over B(\beta)} \int \prod_{w}\phi_{k,w}^{B_{w} + PDF C19 : Lecture 4 : A Gibbs Sampler for Gaussian Mixture Models /Shading << /Sh << /ShadingType 3 /ColorSpace /DeviceRGB /Domain [0.0 50.00064] /Coords [50.00064 50.00064 0.0 50.00064 50.00064 50.00064] /Function << /FunctionType 3 /Domain [0.0 50.00064] /Functions [ << /FunctionType 2 /Domain [0.0 50.00064] /C0 [1 1 1] /C1 [1 1 1] /N 1 >> << /FunctionType 2 /Domain [0.0 50.00064] /C0 [1 1 1] /C1 [0 0 0] /N 1 >> << /FunctionType 2 /Domain [0.0 50.00064] /C0 [0 0 0] /C1 [0 0 0] /N 1 >> ] /Bounds [ 20.00024 25.00032] /Encode [0 1 0 1 0 1] >> /Extend [true false] >> >> Gibbs Sampler Derivation for Latent Dirichlet Allocation (Blei et al., 2003) Lecture Notes . Fitting a generative model means nding the best set of those latent variables in order to explain the observed data. In population genetics setup, our notations are as follows: Generative process of genotype of $d$-th individual $\mathbf{w}_{d}$ with $k$ predefined populations described on the paper is a little different than that of Blei et al. /Matrix [1 0 0 1 0 0] LDA is know as a generative model. xuO0+>ck7lClWXBb4>=C bfn\!R"Bf8LP1Ffpf[wW$L.-j{]}q'k'wD(@i`#Ps)yv_!| +vgT*UgBc3^g3O _He:4KyAFyY'5N|0N7WQWoj-1 \int p(z|\theta)p(\theta|\alpha)d \theta &= \int \prod_{i}{\theta_{d_{i},z_{i}}{1\over B(\alpha)}}\prod_{k}\theta_{d,k}^{\alpha k}\theta_{d} \\ \begin{aligned} The main contributions of our paper are as fol-lows: We propose LCTM that infers topics via document-level co-occurrence patterns of latent concepts , and derive a collapsed Gibbs sampler for approximate inference. + \alpha) \over B(\alpha)} Multinomial logit . After sampling $\mathbf{z}|\mathbf{w}$ with Gibbs sampling, we recover $\theta$ and $\beta$ with. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Below we continue to solve for the first term of equation (6.4) utilizing the conjugate prior relationship between the multinomial and Dirichlet distribution. Gibbs sampling 2-Step 2-Step Gibbs sampler for normal hierarchical model Here is a 2-step Gibbs sampler: 1.Sample = ( 1;:::; G) p( j ). endobj endobj Read the README which lays out the MATLAB variables used. endobj Kruschke's book begins with a fun example of a politician visiting a chain of islands to canvas support - being callow, the politician uses a simple rule to determine which island to visit next. D[E#a]H*;+now derive a gibbs sampler for the lda model - naacphouston.org endstream Consider the following model: 2 Gamma( , ) 2 . << /S /GoTo /D [6 0 R /Fit ] >> QYj-[X]QV#Ux:KweQ)myf*J> @z5 qa_4OB+uKlBtJ@'{XjP"c[4fSh/nkbG#yY'IsYN JR6U=~Q[4tjL"**MQQzbH"'=Xm`A0 "+FO$ N2$u /Length 15 Building on the document generating model in chapter two, lets try to create documents that have words drawn from more than one topic. The intent of this section is not aimed at delving into different methods of parameter estimation for \(\alpha\) and \(\beta\), but to give a general understanding of how those values effect your model. /Type /XObject xP( This time we will also be taking a look at the code used to generate the example documents as well as the inference code. Interdependent Gibbs Samplers | DeepAI Video created by University of Washington for the course "Machine Learning: Clustering & Retrieval". Current popular inferential methods to fit the LDA model are based on variational Bayesian inference, collapsed Gibbs sampling, or a combination of these. Let. /Subtype /Form In previous sections we have outlined how the \(alpha\) parameters effect a Dirichlet distribution, but now it is time to connect the dots to how this effects our documents. A latent Dirichlet allocation (LDA) model is a machine learning technique to identify latent topics from text corpora within a Bayesian hierarchical framework. Find centralized, trusted content and collaborate around the technologies you use most. 0000013825 00000 n (b) Write down a collapsed Gibbs sampler for the LDA model, where you integrate out the topic probabilities m. R::rmultinom(1, p_new.begin(), n_topics, topic_sample.begin()); n_doc_topic_count(cs_doc,new_topic) = n_doc_topic_count(cs_doc,new_topic) + 1; n_topic_term_count(new_topic , cs_word) = n_topic_term_count(new_topic , cs_word) + 1; n_topic_sum[new_topic] = n_topic_sum[new_topic] + 1; # colnames(n_topic_term_count) <- unique(current_state$word), # get word, topic, and document counts (used during inference process), # rewrite this function and normalize by row so that they sum to 1, # names(theta_table)[4:6] <- paste0(estimated_topic_names, ' estimated'), # theta_table <- theta_table[, c(4,1,5,2,6,3)], 'True and Estimated Word Distribution for Each Topic', , . Do roots of these polynomials approach the negative of the Euler-Mascheroni constant? Is it possible to create a concave light? {\Gamma(n_{k,w} + \beta_{w}) /ProcSet [ /PDF ] /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 100.00128] /Coords [0 0.0 0 100.00128] /Function << /FunctionType 3 /Domain [0.0 100.00128] /Functions [ << /FunctionType 2 /Domain [0.0 100.00128] /C0 [0 0 0] /C1 [0 0 0] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [0 0 0] /C1 [1 1 1] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [1 1 1] /C1 [1 1 1] /N 1 >> ] /Bounds [ 25.00032 75.00096] /Encode [0 1 0 1 0 1] >> /Extend [false false] >> >> /Filter /FlateDecode 0000006399 00000 n Perhaps the most prominent application example is the Latent Dirichlet Allocation (LDA . p(\theta, \phi, z|w, \alpha, \beta) = {p(\theta, \phi, z, w|\alpha, \beta) \over p(w|\alpha, \beta)} 0000005869 00000 n The perplexity for a document is given by . >> Griffiths and Steyvers (2004), used a derivation of the Gibbs sampling algorithm for learning LDA models to analyze abstracts from PNAS by using Bayesian model selection to set the number of topics. /BBox [0 0 100 100] In the last article, I explained LDA parameter inference using variational EM algorithm and implemented it from scratch. 0000014374 00000 n Gibbs sampling is a standard model learning method in Bayesian Statistics, and in particular in the field of Graphical Models, [Gelman et al., 2014]In the Machine Learning community, it is commonly applied in situations where non sample based algorithms, such as gradient descent and EM are not feasible. stream p(A, B | C) = {p(A,B,C) \over p(C)} In particular we study users' interactions using one trait of the standard model known as the "Big Five": emotional stability. Xf7!0#1byK!]^gEt?UJyaX~O9y#?9y>1o3Gt-_6I H=q2 t`O3??>]=l5Il4PW: YDg&z?Si~;^-tmGw59 j;(N?7C' 4om&76JmP/.S-p~tSPk t endstream Relation between transaction data and transaction id. >> An M.S. PDF Collapsed Gibbs Sampling for Latent Dirichlet Allocation on Spark &\propto (n_{d,\neg i}^{k} + \alpha_{k}) {n_{k,\neg i}^{w} + \beta_{w} \over Gibbs sampling from 10,000 feet 5:28. endobj Gibbs sampler, as introduced to the statistics literature by Gelfand and Smith (1990), is one of the most popular implementations within this class of Monte Carlo methods. A feature that makes Gibbs sampling unique is its restrictive context. The probability of the document topic distribution, the word distribution of each topic, and the topic labels given all words (in all documents) and the hyperparameters \(\alpha\) and \(\beta\). 17 0 obj Gibbs sampling equates to taking a probabilistic random walk through this parameter space, spending more time in the regions that are more likely. xWK6XoQzhl")mGLRJMAp7"^ )GxBWk.L'-_-=_m+Ekg{kl_. Introduction The latent Dirichlet allocation (LDA) model is a general probabilistic framework that was rst proposed byBlei et al. We introduce a novel approach for estimating Latent Dirichlet Allocation (LDA) parameters from collapsed Gibbs samples (CGS), by leveraging the full conditional distributions over the latent variable assignments to e ciently average over multiple samples, for little more computational cost than drawing a single additional collapsed Gibbs sample. /Type /XObject The topic, z, of the next word is drawn from a multinomial distribuiton with the parameter \(\theta\). Topic modeling is a branch of unsupervised natural language processing which is used to represent a text document with the help of several topics, that can best explain the underlying information. /BBox [0 0 100 100] In 2003, Blei, Ng and Jordan [4] presented the Latent Dirichlet Allocation (LDA) model and a Variational Expectation-Maximization algorithm for training the model. Generative models for documents such as Latent Dirichlet Allocation (LDA) (Blei et al., 2003) are based upon the idea that latent variables exist which determine how words in documents might be gener-ated. p(w,z,\theta,\phi|\alpha, B) = p(\phi|B)p(\theta|\alpha)p(z|\theta)p(w|\phi_{z}) /Filter /FlateDecode The Gibbs sampling procedure is divided into two steps. We also derive the non-parametric form of the model where interacting LDA mod-els are replaced with interacting HDP models. p(w,z|\alpha, \beta) &= . 3. Details. alpha (\(\overrightarrow{\alpha}\)) : In order to determine the value of \(\theta\), the topic distirbution of the document, we sample from a dirichlet distribution using \(\overrightarrow{\alpha}\) as the input parameter. iU,Ekh[6RB Henderson, Nevada, United States. \begin{aligned} The documents have been preprocessed and are stored in the document-term matrix dtm. /Subtype /Form + \alpha) \over B(\alpha)} 0000002237 00000 n /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 100.00128] /Coords [0 0.0 0 100.00128] /Function << /FunctionType 3 /Domain [0.0 100.00128] /Functions [ << /FunctionType 2 /Domain [0.0 100.00128] /C0 [1 1 1] /C1 [1 1 1] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [1 1 1] /C1 [0 0 0] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [0 0 0] /C1 [0 0 0] /N 1 >> ] /Bounds [ 25.00032 75.00096] /Encode [0 1 0 1 0 1] >> /Extend [false false] >> >> >> \]. >> endstream << Moreover, a growing number of applications require that . To start note that ~can be analytically marginalised out P(Cj ) = Z d~ YN i=1 P(c ij . hyperparameters) for all words and topics. >> p(w,z|\alpha, \beta) &= \int \int p(z, w, \theta, \phi|\alpha, \beta)d\theta d\phi\\ stream The les you need to edit are stdgibbs logjoint, stdgibbs update, colgibbs logjoint,colgibbs update. the probability of each word in the vocabulary being generated if a given topic, z (z ranges from 1 to k), is selected. 7 0 obj 0000004237 00000 n one . The value of each cell in this matrix denotes the frequency of word W_j in document D_i.The LDA algorithm trains a topic model by converting this document-word matrix into two lower dimensional matrices, M1 and M2, which represent document-topic and topic . 22 0 obj Key capability: estimate distribution of . /Shading << /Sh << /ShadingType 2 /ColorSpace /DeviceRGB /Domain [0.0 100.00128] /Coords [0.0 0 100.00128 0] /Function << /FunctionType 3 /Domain [0.0 100.00128] /Functions [ << /FunctionType 2 /Domain [0.0 100.00128] /C0 [1 1 1] /C1 [1 1 1] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [1 1 1] /C1 [0 0 0] /N 1 >> << /FunctionType 2 /Domain [0.0 100.00128] /C0 [0 0 0] /C1 [0 0 0] /N 1 >> ] /Bounds [ 25.00032 75.00096] /Encode [0 1 0 1 0 1] >> /Extend [false false] >> >> This estimation procedure enables the model to estimate the number of topics automatically. 0000015572 00000 n 1 Gibbs Sampling and LDA - Applied & Computational Mathematics Emphasis endobj }=/Yy[ Z+ Why is this sentence from The Great Gatsby grammatical? xP( x]D_;.Ouw\ (*AElHr(~uO>=Z{=f{{/|#?B1bacL.U]]_*5&?_'YSd1E_[7M-e5T>`(z]~g=p%Lv:yo6OG?-a|?n2~@7\ XO:2}9~QUY H.TUZ5Qjo6 Latent Dirichlet allocation - Wikipedia B/p,HM1Dj+u40j,tv2DvR0@CxDp1P%l1K4W~KDH:Lzt~I{+\$*'f"O=@!z` s>,Un7Me+AQVyvyN]/8m=t3[y{RsgP9?~KH\$%:'Gae4VDS 57 0 obj << Skinny Gibbs: A Consistent and Scalable Gibbs Sampler for Model Selection /BBox [0 0 100 100] In this paper a method for distributed marginal Gibbs sampling for widely used latent Dirichlet allocation (LDA) model is implemented on PySpark along with a Metropolis Hastings Random Walker. /BBox [0 0 100 100] 94 0 obj << << Latent Dirichlet Allocation with Gibbs sampler GitHub \tag{6.10} lda.collapsed.gibbs.sampler : Functions to Fit LDA-type models PPTX Boosting - Carnegie Mellon University Apply this to . 'List gibbsLda( NumericVector topic, NumericVector doc_id, NumericVector word. The word distributions for each topic vary based on a dirichlet distribtion, as do the topic distribution for each document, and the document length is drawn from a Poisson distribution. /Resources 20 0 R \end{aligned} stream \begin{equation} Implementing Gibbs Sampling in Python - GitHub Pages Gibbs Sampling in the Generative Model of Latent Dirichlet Allocation %PDF-1.4 xMS@ p(z_{i}|z_{\neg i}, \alpha, \beta, w) By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The authors rearranged the denominator using the chain rule, which allows you to express the joint probability using the conditional probabilities (you can derive them by looking at the graphical representation of LDA). You may notice \(p(z,w|\alpha, \beta)\) looks very similar to the definition of the generative process of LDA from the previous chapter (equation (5.1)). \tag{6.7} Share Follow answered Jul 5, 2021 at 12:16 Silvia 176 6 Question about "Gibbs Sampler Derivation for Latent Dirichlet Allocation", http://www2.cs.uh.edu/~arjun/courses/advnlp/LDA_Derivation.pdf, How Intuit democratizes AI development across teams through reusability. 0000013318 00000 n 0000002685 00000 n \tag{6.2} ewLb>we/rcHxvqDJ+CG!w2lDx\De5Lar},-CKv%:}3m. Draw a new value $\theta_{2}^{(i)}$ conditioned on values $\theta_{1}^{(i)}$ and $\theta_{3}^{(i-1)}$. *8lC `} 4+yqO)h5#Q=. >> Latent Dirichlet Allocation (LDA), first published in Blei et al. Update $\alpha^{(t+1)}$ by the following process: The update rule in step 4 is called Metropolis-Hastings algorithm. /FormType 1 /FormType 1 Keywords: LDA, Spark, collapsed Gibbs sampling 1. << We collected a corpus of about 200000 Twitter posts and we annotated it with an unsupervised personality recognition system. /Length 351 0000370439 00000 n CRq|ebU7=z0`!Yv}AvD<8au:z*Dy$ (]DD)7+(]{,6nw# N@*8N"1J/LT%`F#^uf)xU5J=Jf/@FB(8)uerx@Pr+uz&>cMc?c],pm# XcfiGYGekXMH/5-)Vnx9vD I?](Lp"b>m+#nO&} 0000004841 00000 n

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