>> Multiplying these two equations, we get. (NOTE: The derivation for LDA inference via Gibbs Sampling is taken from (Darling 2011), (Heinrich 2008) and (Steyvers and Griffiths 2007).). The main idea of the LDA model is based on the assumption that each document may be viewed as a \], The conditional probability property utilized is shown in (6.9). In particular, we review howdata augmentation[see, e.g., Tanner and Wong (1987), Chib (1992) and Albert and Chib (1993)] can be used to simplify the computations . endobj Feb 16, 2021 Sihyung Park >> 0000002685 00000 n The LDA is an example of a topic model. \[ rev2023.3.3.43278. &\propto p(z_{i}, z_{\neg i}, w | \alpha, \beta)\\ \begin{equation} stream Not the answer you're looking for? /Subtype /Form )-SIRj5aavh ,8pi)Pq]Zb0< \], \[ any . $\theta_{di}$). For ease of understanding I will also stick with an assumption of symmetry, i.e. Styling contours by colour and by line thickness in QGIS. Sequence of samples comprises a Markov Chain. We demonstrate performance of our adaptive batch-size Gibbs sampler by comparing it against the collapsed Gibbs sampler for Bayesian Lasso, Dirichlet Process Mixture Models (DPMM) and Latent Dirichlet Allocation (LDA) graphical . Update count matrices $C^{WT}$ and $C^{DT}$ by one with the new sampled topic assignment. \tag{6.7} /Length 15 xP( In-Depth Analysis Evaluate Topic Models: Latent Dirichlet Allocation (LDA) A step-by-step guide to building interpretable topic models Preface:This article aims to provide consolidated information on the underlying topic and is not to be considered as the original work. (NOTE: The derivation for LDA inference via Gibbs Sampling is taken from (Darling 2011), (Heinrich 2008) and (Steyvers and Griffiths 2007) .) I have a question about Equation (16) of the paper, This link is a picture of part of Equation (16). then our model parameters. 0000001662 00000 n http://www2.cs.uh.edu/~arjun/courses/advnlp/LDA_Derivation.pdf. Following is the url of the paper: /BBox [0 0 100 100] &\propto \prod_{d}{B(n_{d,.} xP( /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] >> >> The model consists of several interacting LDA models, one for each modality. /Type /XObject endstream kBw_sv99+djT p =P(/yDxRK8Mf~?V: %PDF-1.3 % \begin{equation} 26 0 obj /FormType 1 /Filter /FlateDecode H~FW ,i`f{[OkOr$=HxlWvFKcH+d_nWM Kj{0P\R:JZWzO3ikDOcgGVTnYR]5Z>)k~cRxsIIc__a integrate the parameters before deriving the Gibbs sampler, thereby using an uncollapsed Gibbs sampler. I perform an LDA topic model in R on a collection of 200+ documents (65k words total). >> 9 0 obj hb```b``] @Q Ga 9V0 nK~6+S4#e3Sn2SLptL R4"QPP0R Yb%:@\fc\F@/1 `21$ X4H?``u3= L ,O12a2AA-yw``d8 U KApp]9;@$ ` J /Type /XObject """, """ Installation pip install lda Getting started lda.LDA implements latent Dirichlet allocation (LDA). . In addition, I would like to introduce and implement from scratch a collapsed Gibbs sampling method that can efficiently fit topic model to the data. lda implements latent Dirichlet allocation (LDA) using collapsed Gibbs sampling. \end{equation} 0000002866 00000 n How can this new ban on drag possibly be considered constitutional? You may be like me and have a hard time seeing how we get to the equation above and what it even means. &\propto {\Gamma(n_{d,k} + \alpha_{k}) \begin{aligned} iU,Ekh[6RB Do new devs get fired if they can't solve a certain bug? The difference between the phonemes /p/ and /b/ in Japanese. 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). Applicable when joint distribution is hard to evaluate but conditional distribution is known. @ pFEa+xQjaY^A\[*^Z%6:G]K| ezW@QtP|EJQ"$/F;n;wJWy=p}k-kRk .Pd=uEYX+ /+2V|3uIJ + \alpha) \over B(n_{d,\neg i}\alpha)} % In particular we study users' interactions using one trait of the standard model known as the "Big Five": emotional stability. In other words, say we want to sample from some joint probability distribution $n$ number of random variables. hFl^_mwNaw10 uU_yxMIjIaPUp~z8~DjVcQyFEwk| + \alpha) \over B(\alpha)} << /Filter /FlateDecode n_{k,w}}d\phi_{k}\\ 0000003940 00000 n The conditional distributions used in the Gibbs sampler are often referred to as full conditionals. $w_{dn}$ is chosen with probability $P(w_{dn}^i=1|z_{dn},\theta_d,\beta)=\beta_{ij}$. If we look back at the pseudo code for the LDA model it is a bit easier to see how we got here. << These functions use a collapsed Gibbs sampler to fit three different models: latent Dirichlet allocation (LDA), the mixed-membership stochastic blockmodel (MMSB), and supervised LDA (sLDA). 0000015572 00000 n + \beta) \over B(\beta)} This is were LDA for inference comes into play. Brief Introduction to Nonparametric function estimation. 0000004841 00000 n The . What is a generative model? \begin{aligned} /Subtype /Form << Now lets revisit the animal example from the first section of the book and break down what we see. + \beta) \over B(\beta)} LDA is know as a generative model. The nature of simulating nature: A Q&A with IBM Quantum researcher Dr. Jamie We've added a "Necessary cookies only" option to the cookie consent popup. . n_doc_topic_count(cs_doc,cs_topic) = n_doc_topic_count(cs_doc,cs_topic) - 1; n_topic_term_count(cs_topic , cs_word) = n_topic_term_count(cs_topic , cs_word) - 1; n_topic_sum[cs_topic] = n_topic_sum[cs_topic] -1; // get probability for each topic, select topic with highest prob. They proved that the extracted topics capture essential structure in the data, and are further compatible with the class designations provided by . Gibbs Sampler for GMMVII Gibbs sampling, as developed in general by, is possible in this model. Pritchard and Stephens (2000) originally proposed the idea of solving population genetics problem with three-level hierarchical model. >> machine learning I find it easiest to understand as clustering for words. These functions use a collapsed Gibbs sampler to fit three different models: latent Dirichlet allocation (LDA), the mixed-membership stochastic blockmodel (MMSB), and supervised LDA (sLDA). Is it possible to create a concave light? \end{equation} special import gammaln def sample_index ( p ): """ Sample from the Multinomial distribution and return the sample index. \prod_{d}{B(n_{d,.} << << /S /GoTo /D (chapter.1) >> Replace initial word-topic assignment We also derive the non-parametric form of the model where interacting LDA mod-els are replaced with interacting HDP models. xP( $\mathbf{w}_d=(w_{d1},\cdots,w_{dN})$: genotype of $d$-th individual at $N$ loci. *8lC `} 4+yqO)h5#Q=. endobj For Gibbs sampling, we need to sample from the conditional of one variable, given the values of all other variables. /Length 591 0000002915 00000 n endobj \end{aligned} \begin{aligned} We collected a corpus of about 200000 Twitter posts and we annotated it with an unsupervised personality recognition system. This makes it a collapsed Gibbs sampler; the posterior is collapsed with respect to $\beta,\theta$. including the prior distributions and the standard Gibbs sampler, and then propose Skinny Gibbs as a new model selection algorithm. A latent Dirichlet allocation (LDA) model is a machine learning technique to identify latent topics from text corpora within a Bayesian hierarchical framework. endobj One-hot encoded so that $w_n^i=1$ and $w_n^j=0, \forall j\ne i$ for one $i\in V$. This means we can swap in equation (5.1) and integrate out \(\theta\) and \(\phi\). 0000011315 00000 n xuO0+>ck7lClWXBb4>=C bfn\!R"Bf8LP1Ffpf[wW$L.-j{]}q'k'wD(@i`#Ps)yv_!| +vgT*UgBc3^g3O _He:4KyAFyY'5N|0N7WQWoj-1 Arjun Mukherjee (UH) I. Generative process, Plates, Notations . 25 0 obj << p(w,z,\theta,\phi|\alpha, B) = p(\phi|B)p(\theta|\alpha)p(z|\theta)p(w|\phi_{z}) which are marginalized versions of the first and second term of the last equation, respectively. 0000006399 00000 n They are only useful for illustrating purposes. Griffiths and Steyvers (2002) boiled the process down to evaluating the posterior $P(\mathbf{z}|\mathbf{w}) \propto P(\mathbf{w}|\mathbf{z})P(\mathbf{z})$ which was intractable. (2003) which will be described in the next article. Direct inference on the posterior distribution is not tractable; therefore, we derive Markov chain Monte Carlo methods to generate samples from the posterior distribution. student majoring in Statistics. Since $\beta$ is independent to $\theta_d$ and affects the choice of $w_{dn}$ only through $z_{dn}$, I think it is okay to write $P(z_{dn}^i=1|\theta_d)=\theta_{di}$ instead of formula at 2.1 and $P(w_{dn}^i=1|z_{dn},\beta)=\beta_{ij}$ instead of 2.2. /Subtype /Form /Length 15 one . 5 0 obj \end{aligned} (CUED) Lecture 10: Gibbs Sampling in LDA 5 / 6. p(\theta, \phi, z|w, \alpha, \beta) = {p(\theta, \phi, z, w|\alpha, \beta) \over p(w|\alpha, \beta)} /Filter /FlateDecode I_f y54K7v6;7 Cn+3S9 u:m>5(. This time we will also be taking a look at the code used to generate the example documents as well as the inference code. Algorithm. After getting a grasp of LDA as a generative model in this chapter, the following chapter will focus on working backwards to answer the following question: If I have a bunch of documents, how do I infer topic information (word distributions, topic mixtures) from them?. viqW@JFF!"U# \begin{aligned} \[ 3.1 Gibbs Sampling 3.1.1 Theory Gibbs Sampling is one member of a family of algorithms from the Markov Chain Monte Carlo (MCMC) framework [9]. What is a generative model? r44D<=+nnj~u/6S*hbD{EogW"a\yA[KF!Vt zIN[P2;&^wSO >> 0000004237 00000 n 31 0 obj In particular we are interested in estimating the probability of topic (z) for a given word (w) (and our prior assumptions, i.e. of collapsed Gibbs Sampling for LDA described in Griffiths . \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} \\ endobj \sum_{w} n_{k,\neg i}^{w} + \beta_{w}} Gibbs sampling equates to taking a probabilistic random walk through this parameter space, spending more time in the regions that are more likely. /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] >> >> Symmetry can be thought of as each topic having equal probability in each document for \(\alpha\) and each word having an equal probability in \(\beta\). Can anyone explain how this step is derived clearly? We derive an adaptive scan Gibbs sampler that optimizes the update frequency by selecting an optimum mini-batch size. 0000001813 00000 n stream The \(\overrightarrow{\beta}\) values are our prior information about the word distribution in a topic. Building on the document generating model in chapter two, lets try to create documents that have words drawn from more than one topic. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. XtDL|vBrh (2003) to discover topics in text documents. In 2003, Blei, Ng and Jordan [4] presented the Latent Dirichlet Allocation (LDA) model and a Variational Expectation-Maximization algorithm for training the model. I can use the total number of words from each topic across all documents as the \(\overrightarrow{\beta}\) values. /Subtype /Form \tag{6.9} The idea is that each document in a corpus is made up by a words belonging to a fixed number of topics. The only difference between this and (vanilla) LDA that I covered so far is that $\beta$ is considered a Dirichlet random variable here. In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult.This sequence can be used to approximate the joint distribution (e.g., to generate a histogram of the distribution); to approximate the marginal . %1X@q7*uI-yRyM?9>N \[ /Length 1550 %PDF-1.4 Run collapsed Gibbs sampling $\beta_{dni}$), and the second can be viewed as a probability of $z_i$ given document $d$ (i.e. p(\theta, \phi, z|w, \alpha, \beta) = {p(\theta, \phi, z, w|\alpha, \beta) \over p(w|\alpha, \beta)} endobj Do roots of these polynomials approach the negative of the Euler-Mascheroni constant? 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. hbbd`b``3 \phi_{k,w} = { n^{(w)}_{k} + \beta_{w} \over \sum_{w=1}^{W} n^{(w)}_{k} + \beta_{w}} probabilistic model for unsupervised matrix and tensor fac-torization. Calculate $\phi^\prime$ and $\theta^\prime$ from Gibbs samples $z$ using the above equations. (b) Write down a collapsed Gibbs sampler for the LDA model, where you integrate out the topic probabilities m. Video created by University of Washington for the course "Machine Learning: Clustering & Retrieval". \begin{equation} Often, obtaining these full conditionals is not possible, in which case a full Gibbs sampler is not implementable to begin with. /BBox [0 0 100 100] 144 40 $z_{dn}$ is chosen with probability $P(z_{dn}^i=1|\theta_d,\beta)=\theta_{di}$. Asking for help, clarification, or responding to other answers. 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 \(\alpha\). Random scan Gibbs sampler. stream Keywords: LDA, Spark, collapsed Gibbs sampling 1. 32 0 obj /Resources 26 0 R << What does this mean? The equation necessary for Gibbs sampling can be derived by utilizing (6.7). 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 Within that setting . Then repeatedly sampling from conditional distributions as follows. 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 \(P\). endobj endobj Many high-dimensional datasets, such as text corpora and image databases, are too large to allow one to learn topic models on a single computer. Why do we calculate the second half of frequencies in DFT? In _init_gibbs(), instantiate variables (numbers V, M, N, k and hyperparameters alpha, eta and counters and assignment table n_iw, n_di, assign). A well-known example of a mixture model that has more structure than GMM is LDA, which performs topic modeling. stream endobj &= {p(z_{i},z_{\neg i}, w, | \alpha, \beta) \over p(z_{\neg i},w | \alpha, Model Learning As for LDA, exact inference in our model is intractable, but it is possible to derive a collapsed Gibbs sampler [5] for approximate MCMC . It supposes that there is some xed vocabulary (composed of V distinct terms) and Kdi erent topics, each represented as a probability distribution . In this case, the algorithm will sample not only the latent variables, but also the parameters of the model (and ). xWKs8W((KtLI&iSqx~ `_7a#?Iilo/[);rNbO,nUXQ;+zs+~! You will be able to implement a Gibbs sampler for LDA by the end of the module. vegan) just to try it, does this inconvenience the caterers and staff? 0000185629 00000 n assign each word token $w_i$ a random topic $[1 \ldots T]$. But, often our data objects are better . /Resources 17 0 R \]. This article is the fourth part of the series Understanding Latent Dirichlet Allocation. 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. 0000014374 00000 n \begin{equation} /Subtype /Form 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. Deriving Gibbs sampler for this model requires deriving an expression for the conditional distribution of every latent variable conditioned on all of the others. endstream endobj The MCMC algorithms aim to construct a Markov chain that has the target posterior distribution as its stationary dis-tribution. \], \[ >> all values in \(\overrightarrow{\alpha}\) are equal to one another and all values in \(\overrightarrow{\beta}\) are equal to one another. Details. All Documents have same topic distribution: For d = 1 to D where D is the number of documents, For w = 1 to W where W is the number of words in document, For d = 1 to D where number of documents is D, For k = 1 to K where K is the total number of topics. D[E#a]H*;+now gives us an approximate sample $(x_1^{(m)},\cdots,x_n^{(m)})$ that can be considered as sampled from the joint distribution for large enough $m$s. endobj Let (X(1) 1;:::;X (1) d) be the initial state then iterate for t = 2;3;::: 1. %PDF-1.5 /Length 996 0000116158 00000 n I cannot figure out how the independency is implied by the graphical representation of LDA, please show it explicitly. /Type /XObject Below is a paraphrase, in terms of familiar notation, of the detail of the Gibbs sampler that samples from posterior of LDA. 0000133434 00000 n 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. Full code and result are available here (GitHub). /Subtype /Form The first term can be viewed as a (posterior) probability of $w_{dn}|z_i$ (i.e. Gibbs sampling is a method of Markov chain Monte Carlo (MCMC) that approximates intractable joint distribution by consecutively sampling from conditional distributions. endobj Each day, the politician chooses a neighboring island and compares the populations there with the population of the current island. \end{equation} 0000133624 00000 n Fitting a generative model means nding the best set of those latent variables in order to explain the observed data. 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. endobj It is a discrete data model, where the data points belong to different sets (documents) each with its own mixing coefcient. << 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). \begin{equation} 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 . The Gibbs sampling procedure is divided into two steps. \begin{equation} """, """ So, our main sampler will contain two simple sampling from these conditional distributions: p(z_{i}|z_{\neg i}, w) &= {p(w,z)\over {p(w,z_{\neg i})}} = {p(z)\over p(z_{\neg i})}{p(w|z)\over p(w_{\neg i}|z_{\neg i})p(w_{i})}\\ p(A,B,C,D) = P(A)P(B|A)P(C|A,B)P(D|A,B,C) startxref /Matrix [1 0 0 1 0 0] In order to use Gibbs sampling, we need to have access to information regarding the conditional probabilities of the distribution we seek to sample from. 4 0 obj # Setting them to 1 essentially means they won't do anthing, #update z_i according to the probabilities for each topic, # track phi - not essential for inference, # Topics assigned to documents get the original document, Inferring the posteriors in LDA through Gibbs sampling, Cognitive & Information Sciences at UC Merced. \begin{equation} \end{aligned} For complete derivations see (Heinrich 2008) and (Carpenter 2010). num_term = n_topic_term_count(tpc, cs_word) + beta; // sum of all word counts w/ topic tpc + vocab length*beta. /Filter /FlateDecode (LDA) is a gen-erative model for a collection of text documents. The length of each document is determined by a Poisson distribution with an average document length of 10. $\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ denom_term = n_topic_sum[tpc] + vocab_length*beta; num_doc = n_doc_topic_count(cs_doc,tpc) + alpha; // total word count in cs_doc + n_topics*alpha. Latent Dirichlet allocation Latent Dirichlet allocation (LDA) is a generative probabilistic model of a corpus. 0000003190 00000 n The result is a Dirichlet distribution with the parameter comprised of the sum of the number of words assigned to each topic across all documents and the alpha value for that topic. trailer xi (\(\xi\)) : In the case of a variable lenght document, the document length is determined by sampling from a Poisson distribution with an average length of \(\xi\). After sampling $\mathbf{z}|\mathbf{w}$ with Gibbs sampling, we recover $\theta$ and $\beta$ with. stream 2.Sample ;2;2 p( ;2;2j ). 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. \Gamma(\sum_{w=1}^{W} n_{k,w}+ \beta_{w})}\\ In fact, this is exactly the same as smoothed LDA described in Blei et al. (Gibbs Sampling and LDA) Implementation of the collapsed Gibbs sampler for Latent Dirichlet Allocation, as described in Finding scientifc topics (Griffiths and Steyvers) """ import numpy as np import scipy as sp from scipy. xWK6XoQzhl")mGLRJMAp7"^ )GxBWk.L'-_-=_m+Ekg{kl_. So this time we will introduce documents with different topic distributions and length.The word distributions for each topic are still fixed. 0 original LDA paper) and Gibbs Sampling (as we will use here). endobj /Length 15 /Length 3240 Lets get the ugly part out of the way, the parameters and variables that are going to be used in the model. \end{equation} xP( . ndarray (M, N, N_GIBBS) in-place. 36 0 obj To calculate our word distributions in each topic we will use Equation (6.11). \end{equation} &= \prod_{k}{1\over B(\beta)} \int \prod_{w}\phi_{k,w}^{B_{w} + Latent Dirichlet Allocation Using Gibbs Sampling - GitHub Pages /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 [0 0 0] /C1 [0 0 0] /N 1 >> << /FunctionType 2 /Domain [0.0 50.00064] /C0 [0 0 0] /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 [ 21.25026 23.12529 25.00032] /Encode [0 1 0 1 0 1 0 1] >> /Extend [true false] >> >> To estimate the intracktable posterior distribution, Pritchard and Stephens (2000) suggested using Gibbs sampling. 0000012871 00000 n Why is this sentence from The Great Gatsby grammatical? _conditional_prob() is the function that calculates $P(z_{dn}^i=1 | \mathbf{z}_{(-dn)},\mathbf{w})$ using the multiplicative equation above. \tag{6.8} %%EOF 0000001484 00000 n stream There is stronger theoretical support for 2-step Gibbs sampler, thus, if we can, it is prudent to construct a 2-step Gibbs sampler.
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