Description
1.1 Markov Chain Monte Carlo Methods
1. In class, we studied two Monte Carlo estimation methods: rejection sampling and importance sampling.
Given a proposal distribution Q(x), answer the following questions:
(a) (1 point) If sampling from Q(x) is computationally expensive, which of the following methods is
likely to be more efficient?
Rejection Sampling
Importance Sampling
Both are equally inefficient
(b) (1 point) If Q(x) is high-dimensional, which of the following methods is more efficient?
Rejection Sampling
Importance Sampling
Both are equally inefficient
(c) (1 point) For high-dimensional distributions, MCMC methods such as Metropolis Hastings are
more efficient than rejection sampling and importance sampling.
True
False
(d) (1 point) For low-dimensional distributions, MCMC methods such as Metropolis Hastings produce better sammples than rejection sampling and importance sampling.
True
False
2. (2 points) Suppose you are using MCMC methods to sample from a distribution with multiple modes.
Briefly explain what complications may arise while using MCMC.
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1.2 Metropolis Hastings
1. (4 points) Recall that conjugate priors lead to the posterior belonging to the same probability distribution family as the prior. Show that the beta distribution is the conjugate prior for the binomial distribution.
2. Show that
(a) (5 points) The Metropolis Hastings algorithm satisfies detailed balance.
(b) (5 points) A variant of the Metropolis Hastings algorithm with the acceptance probability defined
without the minimum, ie acceptance probability a = A(x ←− xi) = p(x)q(xi|x)
p(xi)q(x|xi)
, wouldn’t satisfy
the detailed balance.
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(c) (5 points) A sampler which resamples each variable conditioned only on the parents of a Bayesian
Network wouldn’t satisfy the detailed balance.
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1.3 Gibbs Sampling and Topic Modeling
1. (3 points) Consider the following graphical model:
A
B C
E
D
F
Figure 1.1: Bayesian Network Structure
Assume that we run Gibbs sampling on this graph, write the conditional probabilities which will be
computed by the sampler for each variable. The variables are sampled in the order A−B−C−D−F−E
2. Now let’s run a collapsed Gibbs sampler for a topic model on a toy dataset. We will run sampling for a
regular LDA (Latent Dirichlet Allocation) model. Suppose our dataset consisting of 4 documents is as
follows:
X = {fizz fizz buzz fizz fizz buzz,
fizz buzz fizz buzz,
foo bar foo bar foo bar,
fizz buzz foo bar}
where xmn refers to the mth word in document n. The vocabulary of this dataset consists of 4 unique
words. Assume that we run our LDA model with 2 topics, prior parameters set to α = 0.1, β = 0.1 and
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start with the following sample of topic assignments:
Z = {0 0 1 0 1 1,
1 0 0 1,
0 1 0 1 1 0,
0 1 0 1}
where zmn refers to the topic assigned to the mth word in document n. Note that α and β are the prior
parameters for the document-topic and word-topic distributions respectively.
(a) (1 point) Compute the word-topic counts table with this assignment
0 1
fizz
buzz
foo
bar
(b) (1 point) Compute the document-topic counts table with this assignment
0 1
0
1
2
3
(c) (2 points) Suppose we want to sample a new topic assignment z00 for word x00. Compute the
full conditional probability distribution p(z00|z
−0
0
, X, α, β) from which this assignment will be
sampled under the collapsed Gibbs sampler.
0 1
p(z00|z
−0
0
, X, α, β)
(d) (2 points) Assume that the new topic assignment is z00 = 1. What is the updated word-topic
distribution?
0 1
fizz
buzz
foo
bar
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1.4 Empirical Questions
The following questions should be completed after you work through the programming portion of this
assignment (Section 2).
1. (10 points) After training your model, report the top 10 most probable words for 5 of your favorite
discovered topics. Note that here most probable is defined as the word whose embedding maximizes
the t distribution probability for a given class.
Rank Topic 0 Topic 1 Topic 2 Topic 3 Topic 4
1
2
3
4
5
6
7
8
9
10
2. (10 points) Plot the log-likelihood of the data defined by equation 2.2 over training for 50 iterations.
Record this probability every 10 iterations. Note: Your plot must be machine generated.
1.5 Wrap-up Questions
1. (1 point) Multiple Choice: Did you correctly submit your code to Autolab?
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Yes
No
2. (1 point) Numerical answer: How many hours did you spend on this assignment?.
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1.6 Collaboration Policy
After you have completed all other components of this assignment, report your answers to the collaboration
policy questions detailed in the Academic Integrity Policies for this course.
1. Did you receive any help whatsoever from anyone in solving this assignment? If so, include full
details including names of people who helped you and the exact nature of help you received.
2. Did you give any help whatsoever to anyone in solving this assignment? If so, include full details
including names of people you helped and the exact nature of help you offered.
3. Did you find or come across code that implements any part of this assignment? If so, include full
details including the source of the code and how you used it in the assignment.
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2 Programming [ pts]
2.1 Task Background
Topic modeling is the task of finding latent semantic categories that group words together called topics. The
typical approach to this task is to build a model by telling a generative story: For each document we assign
a probability of picking a set of topics and for each topic I have some probability of picking any given word.
The decision of what these probabilities are and how our generative story plays out determines not just how
successful our model is, but also how computationally feasible inference is.
Because we’ve seen the power of vector representations of words in previous assignments, let’s assume that
we want to model words as vectors in RM. A reasonable assumption is that words that belong to similar
topics are close to each other in embedding space. One way to encode this intuition is to say that given a
topic k, a word embedding v is drawn from the distribution N (µk
, I). This implies that if we can discover
these µk
s we can discover a set of topics that explain our observed words. In order to make this practical,
we’ll need to flesh out our generative story.
Suppose we’re given a corpus of D documents and a fixed number of topics K. Our model will look like
this:
1. For topic k = 1 to K:
(a) Draw topic mean µk ∼ N (µ,
1
κ
I)
2. For each document d in corpus D:
(a) Draw topic distribution θd ∼ Dir(α)
(b) For each word index n from 1 to Nd
i. Draw a topic zn ∼ Categorical(θd)
ii. Draw vd,n ∼ N (µzn
, I)
In our case, we will fix K = 5, |D| = 1208, and our embeddings live in R
50
.
We will see in section 2.3 how this can be translated into a practical sampling algorithm.
2.2 Data
For this task we’ve provided you with a selection of news articles in the numpy file news_corpus.npy.
This array is |D| × |V | where V is our vocabulary. Each i, j entry corresponds to the number of instances
of word j in document i.
In order to save you the trouble of training word embeddings before you can even start the real task, we
have also provided 50 dimensional GloVe embeddings in the numpy file named news_embeddings.npy
along with a dictionary to map between words to indices in the pickle file named mews_word2index.pkl.
The embedding array is a |V | × 50 array.
2.3 Gibbs Sampler
Given the generative model described earlier we’d like to know the posterior distribution over our parameters. As is the case for almost all interesting models, we don’t have an analytical form for this. We’ll make
do by approximating the posterior with a collapsed Gibbs sampler. It’ll be a Gibbs sampler because we will
iteratively sample word-topic assignments conditioned on all over assignments and parameters. It’s said to
be collapsed because we will have integrated over nuisance parameters to provide a simpler distribution to
sample from.
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p(zd,i|z−(d,i)
, Vd, ξ, α) ∝ (nk,d + αk) × tvk−M+1
vd,i|µk
, I
(2.1)
Equation 2.1 shows the full conditional on which we can build a Gibbs sampler. This equation has a lot to
unpack.
Prior Parameters
In the above equation, our prior parameters were summarized as the tuple ξ = (µ, κ, Σ, ν) along with the
vector of parameters α. These are the same parameters as those in the introductory section. We will treat
the hyperparameter α as αi = 10 for all classes i. We will also set κ = 0.01, Σ = IM, and ν = M.
Data Statistics
Nk = #{words assigned to topic k across all documents}
v¯k =
P
d
P
i:zd,i=k
(vd,i)
Nk
Updated Parameters
κk = κ + Nk
νk = ν + Nk
µk =
κµ + Nkv¯k
κk
Important Distributions
The most important distribution for you to be familiar with for this assignment is the multivariate t distribution given parameters (µk
, κk, ν0
). The final parameter is the degrees of freedom for the distribution and is
denoted in the subscript of the distribution tν
0(·). We have provided a function in the file utils.py called
multivariate_t_distribution that will return log-probabilities according to this model.
2.4 Implementation Details
Now that we have our notation sorted out, we can ensure that we understand the steps to implement our
sampler.
Implementation Outline
1. Load in the provided embeddings and the text of our corpus.
2. Randomly initialize a dictionary that places each word in each document into a given topic.
3. Calculate statistics and update our posterior parameters and calculate the full conditional in equation
2.1 for each word and each possible topic assignment. Note: This will require adjusting the statistics
to not include the current word.
4. Normalize the above distributions and sample from a multinomial over the k topics with the posterior
distribution.
5. Reassign words to topics based on the above samples.
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6. Go back to step 4 and repeat until convergence.
Numerical Issues
For numerical stability, you must calculate log-probabilities and only convert to regular probability distribution before the normalization step.
One way to increase stability is to keep track of the maximum log-probability encountered while iterating
over topics in step 5. Then before normalization, subtract of this maximum log-probability from each logprobability.
2.5 Evaluation
We will track the joint probability of zk, vd,i, µk
, θd, for each word i, document d, and currently assigned
category k. This will be calculate using the following equation.
p(zk, vd,i, µk
, θd) = p(vd,i|zk, µk
)p(zk|θk)p(µk
)p(θd) (2.2)
Please refer to the generative model for Gaussian-LDA to define each of these probabilities. Also note that
because we’re using a collapsed Gibbs sampler, we will not have estimate one of these parameters before.
2.6 Autolab Submission [35 pts]
You must submit a .tar file named gaussianlda.tar containing gaussianlda.py, which contains
all of your code.
You can create that file by running:
tar -cvf gaussianlda.tar gaussianlda.py
from the directory containing your code.
Some additional tips: DO NOT compress your files; you are just creating a tarball. Do not use tar -czvf.
DO NOT put the above files in a folder and then tar the folder. Autolab is case sensitive, so observe that all
your files should be named in lowercase. You must submit this file to the corresponding homework link on
Autolab.
Your code will not be autograded on Autolab. Instead, we will grade your code by hand; that is, we will
read through your code in order to grade it. As such, please carefully identify major sections of the code via
comments.
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