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CS 475/675 Graphical Models and Inference

创建日期：2024-06-27 15:36:34

所属分类：计算机computer science

标签： CS675

Graphical Models and Inference

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CS 475/675 Machine Learning: Homework 6
Graphical Models and Inference

1 Introduction
In this assignment you will be exploring a technique for natural language understanding
known as Topic Modeling. You will be implementing a Variational Inference method that
uses the Mean Field approximation for learning the parameters of the probabilistic LDA
graphical model.
2 Data
We will be using State of the Union (SotU) speeches given by US presidents1
. SotU addresses have been delivered annually by the president since 1790. The purpose of these
speeches is to summarize the affairs of the US federal government. Typically, the SotU
addresses given by a newly inaugurated president contain a different tone. Your task in
this assignment is to uncover topics important to US federal affairs over the years.
The corpus contains D = 232 documents, comprising years 1790-2020 (there were two
SotU addresses in 1961 by Eisenhower and Kennedy). Each document d has a length
Nd on the order of 103 words. The overall corpus vocabulary contains approximately 104
words. Like in homework 4, we have pre-processed the data for you using standard techniques from NLP, including tokenization, converting all words to lowercase, and removing
stopwords2
, punctuation, and numbers.

3 Topic Modeling
Natural language is complex; modeling and extracting information from it requires breaking down language layer by layer. In linguistics, semantics is the study of meaning of words
and sentences4
. At the document level, one of the most useful techniques for understanding text is analyzing its topics. In this model, a topic is just a collection of words and an
observed document is a mixture of topics. The key idea of the topic model is that the
semantic meaning of a document is influenced by these latent topics.
3.1 Latent Dirichlet Allocation
Latent Dirichlet Allocation (LDA) is a popular topic modeling technique for uncovering latent topics. LDA is a generative probabilistic model of a corpus (collection of
text)56. In the LDA model, documents are represented as random mixtures over latent
topics, whereas topics are characterized as distributions over words.
The Dirichlet distribution7
is a member of the exponential family of distributions, whose
conjugate prior is the multinomial distribution. This means that when computing the
MLE or MAP, we can simplify our expressions to the addition of sufficient statistics. Additionally, we have two sparsity requirements for the topic model: (1) document sparsity
and (2) word sparsity. Each document d should contain only a small number of topics
and each topic k should be determined by a small subset of words in the total vocabulary.
Dirichlet distributions encode these sparsity requirements. The Dirichlet distribution is
paramaterized by a vector of concentration parameters α. As α → 0, the multinomials
drawn from the Dirichlet tend to be more sparse.8
3.2 Inference
In order to use LDA, we must learn the latent parameters z, θ, and φ using inference. To
do this, we must compute the posterior distribution of the latent variables Z, Θ, and Π
given the the model parameters α and β, and observed data W:
P(Z, Θ, Φ|W, α, β) = P(W, Z, Θ, Φ|α, β)
P(W|α, β)
(2)
Unfortunately, this posterior is intractable to compute in general. To normalize the
distribution, we must marginalize over the latent variables. Specifically, examining the
form of P(W|α, β) we have:
P(W|α, β) = Z
Φ
Z
Θ
X
Z
P(W, Z, Θ, Φ|α, β)dΘdΦ (3)
=
Z
Φ
P(Φ|β)
Z
Θ
p(Θ|α)
X
Z
P(Z|Θ)P(W|Z, Φ)dΘdΦ (4)
Computing the posterior exactly is intractable because of the coupling between Θ and Φ in
the summation over latent topic assignments. In other words, we cannot make statements
about maximizing a particular term since all the latent variables are coupled together.
Since we cannot compute this posterior exactly, we turn to approximate inference methods. Specifically, in this assignment you will be using variational inference to approximate
the posterior distribution.
3.3 Mean Field Variational Inference
In Variational Inference, we design a family of tractable distributions Q, which have
parameters that can be tuned to approximate the true posterior. To obtain Q, we will
use the Mean Field approximation, which approximates the complex p(Z, Θ, Φ|W, α, β)
with a fully factorized approximation.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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