[SOLVED] Si 650 / eecs 549: homework 1 — probabilities, text, and ranking
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managed to subscribe to thousands of accounts. Their feed is now a useless mess of spam
accounts, weird dances, and outrage videos. Sam doesn’t have time to deal with all of this
manually so they decide to find a programmatic way of unfollowing accounts, starting with
getting rid of spam accounts.Luckily, Sam attended SI 650 / EECS 549 and decided to filter the accounts in a Bayesian
way. They went through 12 accounts manually and noted down 4 observations for each
account in Table 1 to figure out which features are more likely used in spam accounts.
1. S: whether the account posts mostly spam (1 for yes, 0 for no);
2. B: whether the account’s bio has the word “based” (1 for yes, 0 for no);
3. U: whether the account’s bio has an URL link (1 for yes, 0 for no);
4. E: whether the account’s bio has an emoji (1 for yes, 0 otherwise).Sam now wants to build a filter with these observations. Now in terms of probabilistic reasoning, we can formulate the question as evaluating the conditional probability P(S|B, U, E),
we say that the account is a spam if P(S = 1|B, U, E) > P(S = 0|B, U, E). We make a
further conditional independence assumption that P(B, U, E|S) = P(B|S)P(U|S)P(E|S).In other words, we assume that if the status whether a account is a spam is known (i.e.,
value of S is known), the values of B, U, and E would be independent to each other.
a) (5 points) Fill in the Table 2 with conditional probabilities using only the information
present in the 12 samples.
b) (5 points) With the independence assumption, use the Bayes formula and the calculated
conditional probabilities to compute the probabilities that account a with B = 0,
U = 1, E = 0 is a spam. That is, compute P(S = 1|B = 0, U = 1, E = 0) and
P(S = 0|B = 0, U = 1, E = 0). Would you conclude that account a is a spam? Show
your computation.c) (5 points) Now, compute P(S = 1|B = 0, U = 1, E = 0) and P(S = 0|B = 0, U =
1, E = 0) directly from the 12 examples in Table 1, just like what you did in problem
A. Do you get the same value as in problem B? Why?
1
S B U E
0 1 1 1
1 0 1 0
1 1 1 1
0 1 1 0
1 1 0 1
0 0 0 0
1 1 1 0
0 0 1 1
0 0 0 0
1 0 1 1
1 1 1 0
1 1 0 0
Table 1: Sample observations of the accountsd) (5 points) Now, ignore Table 1, and consider any possibilities you can fill in Table 2.
Are there any constraints on these values that we must respect when assigning these
values? In other words, can we fill in Table 2 with 8 arbitrary values between 0 and 1?
If not, are there any constraints on some values that we must follow? Describe your
answer.e) (5 points) Can you change your conclusion of problem a (i.e., whether account a is
a spam) by only changing the value E (i.e., if the account bio has an emoji) in one
example of Table 1? Describe your answer.
f) (5 points) Explain why the independence assumption P(B, U, E|S) = P(B|S)P(U|S)P(E|S)
does not necessarily hold in reality.In this exercise, we are going to get our hands dirty and play with some data in the wild.
Download two collections from Canvas, reddit-questions.10k.txt and wiki-bios.10k.txt. The
first collection are 100,000 questions randomly sampled from r/AskReddit. The second collection is 100,000 biographies of people taken from Wikipedia. You can also find a stopword
list in stoplist.txt.For text processing, we’ll use the SpaCy library, which is a modern NLP library that
is well documented and highly performant. You want to use nlp function from SpaCy for
tokenizing and part-of-speech (POS) tagging.
S P(B = 1|S) P(U = 1|S) P(E = 1|S) prior P(S)
1 0.71428 ? ? ?
0 ? ? ? 0.41666
Table 2: Conditional Probabilities and Prior1. (5 points) Tokenize the text using SpaCy and compute the frequency of words. Then,
plot the frequency distribution of words in each collection after the removal of the
stopwords: x-axis – each point is a word, sorted overall by frequency (number of times
a word appears in the collection)1
; y-axis – how many times the word occurred. Plot
this using a log scale on each axis. Does each plot look like a power-law distribution?
Are the two distributions similar or different?2. (10 points) Now compare the two collections more rigorously. Report the following
properties of each collection, using SpaCy to POS tag. Can you explain these differences based on the nature of the two collections?
a) frequency of stopwords (percentage of the word occurrences that are stopwords.);
b) percentage of capital letters;
c) average number of characters per word;
d) percentage of nouns, adjectives, verbs, adverbs, and pronouns;
e) the top 10 nouns, top 10 verbs, and top 10 adjectives.3. (10 points) We would like to summarize each document with a few words. However,
picking the most frequently used words in each document would be a bad idea, since
they are more likely to appear in other document as well. Instead, we pick the words
with the highest TF-IDF weights in each document.In this problem, term frequency (TF) and inverse document frequency (IDF) are defined as:
T F(t, d) = log(c(t, d) + 1)
IDF(t) = 1 + log(N/k).
c(t, d) is the frequency count of term t in doc d, N is the total number of documents
in the collection, and k is the document frequency of term t in the collection.
For each of the first 10 documents in the Wikipedia biographies collection, print out
the 5 words that have the highest TF-IDF weights. Write whether you think these
could be a good summary of the documents.4. (5 points) As discussed in the class, TF-IDF is a common way to weight the terms in
each document. It can also be easily calculated from the inverted index (covered in
Week 3), since TF can be obtained from the postings and IDF can be summarized as
a dictionary. Could you think of another weighting that cannot be calculated directly
from inverted index? What is the advantage of such a weighting?
• Hint 1: You can find a tutorial for SpaCy at https://spacy.io/usage/spacy-101
which covers all of the functionality you’ll need here as well as many more advanced
preprocessing steps. )
1This also gets called the “rank” of the word in the collection• Hint 2: You may find a lot of decision to make: Should I lower-case the words?
Should I use stemmer or a lemmatizer? What to do with the punctuation? How
should I handle html, markdown, or emoji? There is not always right and wrong,
different answers are accepted. But you should write down clearly how you process the
data in each parts and explain your decisions.Suppose we have a query with a total of 20 relevant documents in a collection of 100 documents. A system has retrieved 20 documents whose relevance status is [++, -, +, ++, -, +,
-, ++, +, -, -, +, ++, -, -, +, +, ++, +, -] in the order of ranking. A + or ++ indicates that
the corresponding document is relevant, while a – indicates that the corresponding document
is non-relevant.• (10 points) Compute the precision, recall, F1 score, and the mean average precision
(MAP).
• (10 points) Consider ++ as the corresponding document being highly relevant (ri = 2),
while + indicates somewhat relevant (ri = 1), – being non-relevant (ri = 0). For the
nine rest relevant documents, treat them as somewhat relevant (ri = 1) Calculate the
Cumulative Gain (CG) at rank 10, Discounted Cumulative Gain (DCG) at rank 10,
and Normalized Cumulative Gain (NDCG), at rank 10. Use log2 for the discounting
function.Note You may find the definition of DCG in Wikipedia is different from the definition
in our lecture. Please use the one in our lecture to calculate DCG and NDCG. (i.e.
DCGp = rel1 +
Pp
i=2
reli
log2iLet’s build a simple search engine using some of the techniques we learned in class and
evaluate it in practice both for its ability to retrieve relevant documents and its speed of
retrieval. Here, we’ll use our Reddit questions and play the role of an auto-suggest: If
someone is about to ask a question on Reddit, they can quickly see if someone else has
already asked that question by searching for a few keywords. Specfically, you’ll implement a
very simple search that (1) measures the cosine similarity of a bag-of-words representation
of the query and a bag-of-words representation for a document and (2) returns the k most
similar documents to the query.To support the basic search functionality, we’ll use the scikit-learn package to convert
our Reddit questions and queries to bag of words vectors.
• (5 points) Write a function that uses CountVectorizer to convert the Reddit questions
corpus to vectors. Write a function that given a new query, will convert its text to a
vector (using the same vectorizer), estimate the cosine similarity between the query
and each document (i.e., each Reddit question) and return the 10 most similar.• (10 points) Using your method,
– run the following queries and show the questions they return: (1) programming,
(2) pets, (3) college, (4) love, and (5) food.
– Score each retrieved question for relevance using a three point scale as in Problem
3 (very relevant, somewhat relevant, not relevant)
– compute NDCG for each query and report it.• (5 points) In a 2-3 sentences, describe how well you think your IR system is doing.
What kids of queries do you think it would work well on? What kinds of queries do
you think it will perform poorly on? (Feel free to describe example queries if you want
to test things!)Finally, let’s get a sense of how scalable our system is. Because we’re not imposing a
minimum frequency for including a word in our CountVectorizer, we’re effectively indexing
every wold. For this exercise, we have 100K questions, but what if we had 1M or 1B—how
many terms would we be indexing? Let’s estimate this by looking at how our index size (i.e.,
the number of words recognized by the vectorizer) grows relative to our corpus size.• (5 points) Re-run your CountVectorizer code with 1K, 5K, 10K, 50K, and 100K
questions and for each plot how many terms appear in the vocabulary field for the
vectorizer. In general, we recommend using Seaborn for all plotting.2 Write 2-3 on
why you think this approach will or won’t scale as we get more documents and justify
you answer.If you’re feeling particularly curious, try repeating this process with random samples of each number of
questions (e.g., a random 1K questions) and show the mean and standard error.