Assignment 3: Probability and Statistical Learning


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B551 Assignment 3: Probability and Statistical Learning
This assignment will give you a chance to practice probabilistic inference for some real-world problems in
natural language processing and computer vision.
Guidelines for this assignment
Coding requirements. For fairness and efficiency, we use a semi-automatic program to grade your submissions. As usual, we require that: 1. You must code this assignment in Python 3; 2. You should test your
code on; 3. Your code must obey the input and output specifications given below. 4.
You may import standard Python modules for routines not related to AI, such as basic sorting algorithms
and data structures, as long as they are already installed on; and 5. Make sure to use
the program file name we specify.
Groups. You’ll work in a group of 1-3 people for this assignment; we’ve already assigned you to a group (see
details below) according to your preferences. You should only submit one copy of the assignment for your
team, through GitHub. All the people on the team will receive the same grade on the assignment, except in
unusual circumstances; we will collect feedback about how well your team functioned in order to detect these
circumstances. The requirements for the assignment are the same regardless of team size, but we expect
that teams with more people will submit answers that are more “polished” — e.g., better documented code,
faster running times, more thorough answers to questions, etc.
Coding style and documentation. We will not explicitly grade coding style, but it’s important that you
write your code in a way that we can easily understand it. Please use descriptive variable and function names,
and use comments when needed to help us understand code that is not obvious. For each of these problems,
you will face some design decisions along the way. Your primary goal is to write clear code that finds the
correct solution in a reasonable amount of time. To encourage innovation, we will conduct a competition
among programs to see which can solve the hardest problems in the shortest amount of time.
Report. Please put a report describing your assignment in the file in your Github repository. For
each problem, please include: (1) a description of how you formulated each problem; (2) a brief description
of how your program works; (3) and discussion of any problems you faced, any assumptions, simplifications,
and/or design decisions you made. These comments are especially important if your code does not work
perfectly, since it is a chance to document the energy and thought you put into your solution.
Academic integrity. We take academic integrity very seriously. To maintain fairness to all students in
the class and integrity of our grading system, we will prosecute any academic integrity violations that we
discover. Before beginning this assignment, make sure you are familiar with the Academic Integrity policy
of the course, as stated in the Syllabus, and ask us about any doubts or questions you may have. To briefly
summarize, you may discuss the assignment with other people at a high level, e.g. discussing general strategies
to solve the problem, talking about Python syntax and features, etc. You may also consult printed and/or
online references, including books, tutorials, etc., but you must cite these materials (e.g. in source code
comments). We expect that you’ll write your own code and not copy anything from anyone else, including
online resources. However, if you do copy something (e.g., a small bit of code that you think is particularly
clever), you have to make it explicitly clear which parts were copied and which parts were your own. You can
do this by putting a very detailed comment in your code, marking the line above which the copying began,
and the line below which the copying ended, and a reference to the source. Any code that is not marked in
this way must be your own, which you personally designed and wrote. You may not share written answers
or code with any other students, nor may you possess code written by another student, either in whole or in
part, regardless of format.
Part 0: Getting started
We’ve assigned you to a team; find it as usual by logging into IU Github, look for a repo called userid1 -a3,
userid1-userid2 -a3, or userid1-userid2-userid3 -a3, where the other user ID(s) correspond to your teammate(s). Now that you know their userid(s), you can write them an email at To get started,
clone the github repository using one of the two commands:
git clone -a3
git clone
Part 1: Part-of-speech tagging
A basic problems in Natural Language Processing is part-of-speech tagging, in which the goal is to mark
every word in a sentence with its part of speech (noun, verb, adjective, etc.). Sometimes this is easy: a
sentence like “Blueberries are blue” clearly consists of a noun, verb, and adjective, since each of these words
has only one possible part of speech (e.g., “blueberries” is a noun but can’t be a verb).
But in general, one has to look at all the words in a sentence to figure out the part of speech of any individual
word. For example, consider the — grammatically correct! — sentence: “Buffalo buffalo Buffalo buffalo
buffalo buffalo Buffalo buffalo.” To figure out what it means, we can parse its parts of speech:
Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
Adjective Noun Adjective Noun Verb Verb Adjective Noun
(In other words: the buffalo living in Buffalo, NY that are buffaloed (intimidated) by buffalo living in Buffalo,
NY buffalo (intimidate) buffalo living in Buffalo, NY.)
That’s an extreme example, obviously. Here’s a more mundane sentence:
Her position covers a number of daily tasks common to any social director.
where DET stands for a determiner, ADP is an adposition, ADJ is an adjective, and ADV is an adverb.1
Many of these words can be different parts of speech: “position” and “covers” can both be nouns or verbs,
for example, and the only way to resolve the ambiguity is to look at the surrounding words. Labeling parts
of speech thus involves an understanding of the intended meaning of the words in the sentence, as well as
the relationships between the words.
Fortunately, statistical models work amazingly well for NLP problems. Consider the Bayes net shown in
Figure 1(a). This Bayes net has random variables S = {S1, . . . , SN } and W = {W1, . . . , WN }. The W’s
represent observed words in a sentence. The S’s represent part of speech tags, so Si ∈ {VERB, NOUN, . . .}.
The arrows between W and S nodes model the relationship between a given observed word and the possible
parts of speech it can take on, P(Wi
|Si). (For example, these distributions can model the fact that the word
“dog” is a fairly common noun but a very rare verb.) The arrows between S nodes model the probability
that a word of one part of speech follows a word of another part of speech, P(Si+1|Si). (For example, these
arrows can model the fact that verbs are very likely to follow nouns, but are unlikely to follow adjectives.)
Data. To help you with this assignment, we’ve prepared a large corpus of labeled training and testing data.
Each line consists of a sentence, and each word is followed by one of 12 part-of-speech tags: ADJ (adjective),
ADV (adverb), ADP (adposition), CONJ (conjunction), DET (determiner), NOUN, NUM (number), PRON
(pronoun), PRT (particle), VERB, X (foreign word), and . (punctuation mark).2
If you didn’t know the term “adposition”, neither did I. The adpositions in English are prepositions; in many languages,
there are postpositions too. But you won’t need to understand the linguistic theory between these parts of speech to complete
the assignment; if you’re curious, check out the “Part of Speech” Wikipedia article for some background.
2This dataset is based on the Brown corpus. Modern part-of-speech taggers often use a much larger set of tags – often over
S1 S2 S3 S4 SN
W1 W2 W3 W4 WN
… S1 S2 S3 S4 SN
W1 W2 W3 W4 WN

(a) (b) (c)
Figure 1: Bayes Nets for part of speech tagging: (a) HMM, and (b) simplified model, and (c) complicated
What to do. Your goal in this part is to implement part-of-speech tagging in Python, using Bayes networks.
1. To get started, consider the simplified Bayes net in Figure 1(b). To perform part-of-speech tagging,
we’ll want to estimate the most-probable tag s

for each word Wi

i = arg max
P(Si = si
Implement part-of-speech tagging using this simple model.
2. Now consider Figure 1(a), a richer Bayes net that incorporates dependencies between words. Implement
Viterbi to find the maximum a posteriori (MAP) labeling for the sentence,

, . . . , s∗
N ) = arg max
P(Si = si
3. Consider the Bayes Net of Figure 1c, which could be a better model because it incorporates richer
dependencies between words. But it’s not an HMM, so we can’t use Viterbi. Implement Gibbs Sampling
to sample from the posterior distribution of Fig 1c, P(S|W). Then estimate the best labeling for each
word (by picking the maximum marginal for each word, s

i = arg maxsi P(Si = si
|W). (To do this,
just generate many (thousands?) of samples and, for each individual word, check which part of speech
occurred most often.)
Your program should take as input a training filename and a testing filename. The program should use the
training corpus to estimate parameters, and then display the output of Steps 1-3 on each sentence in the
testing file. For the result generated by each of the three approaches (Simple, HMM, Complex), as well as
for the ground truth result, your program should output the logarithm of the joint probability P(S, W) for
each solution it finds under each of the three models in Figure 1. It should also display a running evaluation
showing the percentage of words and whole sentences that have been labeled correctly so far. For example:
[djcran@raichu djc-sol]$ python3 ./ training_file testing_file
Learning model…
Loading test data…
Testing classifiers…
Simple HMM Complex Magnus ab integro seclorum nascitur ordo .
0. Ground truth -48.52 -64.33 -73.43 noun verb adv conj noun noun .
1. Simplified -47.29 -66.74 -75.29 noun noun noun adv verb noun .
2. HMM -47.48 -63.83 -74.12 noun verb adj conj noun verb .
3. Complex -47.50 -64.21 -72.02 noun verb adv conj noun noun .
==> So far scored 1 sentences with 17 words.
100 tags, depending on the language of interest – that carry finer-grained information like the tense and mood of verbs, whether
nouns are singular or plural, etc. In this assignment we’ve simplified the set of tags to the 12 described here; the simple tag set
is due to Petrov, Das and McDonald, and is discussed in detail in their 2012 LREC paper if you’re interested.
Words correct: Sentences correct:
0. Ground truth 100.00% 100.00%
1. Simplified 42.85% 0.00%
2. HMM 71.43% 0.00%
3. Complex 100.00% 100.00%
We’ve already implemented some skeleton code to get you started, in three files:, which is the
main program, pos, which has the scoring code, and pos, which will contain the actual
part-of-speech estimation code. You should only modify the latter of these files; the current version of
pos we’ve supplied is very simple, as you’ll see. In your report, please make sure to include your
results (accuracies) for each technique on the test file we’ve supplied, bc.test. Your code should finish
within about 10 minutes.
Part 2: Ice tracking
In this problem, we’ll help solve global warming. 🙂
To understand how rising global temperatures affect ice at the Earth’s north and south poles, glaciologists
need information about the structure of the ice sheets. The traditional way of doing this is to drill into the
ice and remove an ice core. But a single ice core can take many months to drill, and only gives information
about the ice at a single latitude-longitude point. To expedite this process, scientists have developed radar
systems that allow an airplane to collect an approximate “cross section” of the ice below the airplane’s flight
path (Fig 2a). This produces a radar echogram like the one shown in Fig 2b. The horizontal axis is the
distance along the flight path, while the vertical axis is the depth below the plane. The echogram shows
two prominent features. One is the very dark line near the top, which is the boundary between the air
and the ice. There’s also a deeper line which shows the boundary between the ice and the bedrock. Fig
2c shows the same echogram but with the air-ice (red) and ice-rock (green) boundaries manually labeled.
These echograms reveal the complex structure of the ice — note the ridges and valleys in the bedrock in Fig
2c, for example — and contain rich information for glaciologists to calculate volumes of ice and to estimate
how it will change with warming temperatures. But as you can see from the figure, these echograms are also
extremely noisy, so finding the layer boundaries is quite challenging. Even human experts, when presented
with the same echogram, often disagree on where the boundaries are.
In this part, we’ll create code to try to find these two boundaries (air-ice and ice-rock). We’ll make some
assumptions to make this possible. First, you can assume that the air-ice boundary is always above the
ice-rock boundary by a significant margin (say, 10 pixels). Second, you can assume that the two boundaries
span the entire width of the image. Taken together these, two assumptions mean that in each column of the
image, there is exactly one air-ice boundary and exactly one ice-rock boundary, and the ice-rock boundary is
always below. Third, we assume that each boundary is relatively “smooth” — that is, a boundary’s row in
one column will be similar in the next column. Finally, you can assume that the pixels along the boundaries
are generally dark and along a strong image edge (sharp change in pixel values)..
We’ve given you code that reads an image file into a two-dimensional array. For an m × n echogram, the
code creates a 2-d array I(x, y) that stores the radar return at each pixel (x, y) ∈ [1, m]×[1, n] where smaller
values indicate greater probability of a boundary. Your goal is to estimate, for each column x ∈ [1, m], the
row ax corresponding to the air-ice boundary, and the row rx corresponding to the ice-rock boundary. (In
image processing, the coordinate system convention is for the upper-left of the image to be the point (1,1),
and for column indices to increase as you go right and row indices to increase as you go down.) Each of
these boundaries can be solved for using a Bayes net. For example, for the air-ice problem, the Bayes net
would have hidden variables a1, a2, …am, and observed variables (specifically w1, w2, …wm, where w1 is a
vector corresponding to column 1 of the image).
Since the air-ice boundary is generally stronger than ice-rock and easier to estimate, it makes sense to
(a) (b) (c)
Figure 2: Figure 2: (a) An airplane with a ground-penetrating radar system flies along the ice, generating
an echogram such as (b). The annotations in (c) show key features of the echogram.
first estimate the air-ice boundary, and then estimate the ice-rock boundary but require that it be at least
(for example) 10 pixels below the air-ice boundary in each column (i.e., require that rx ≥ ax + 10 for all
x ∈ [1, m]).
1. Perhaps the simplest technique would be to use the Bayes’ Net in Figure 1b to estimate the two
boundaries. Implement this technique, showing the boundaries in yellow.
(Hint: What should your emission probabilities look like? This is for you to decide, but intuitively a
stronger edge and/or darker (lower) pixel value should correspond to higher probability. Our skeleton
code includes a function to compute edge strength, in case this is helpful.)
2. A better approach would use the Bayes Net in Figure 1a. Use Viterbi, showing the boundaries in blue.
(Hint: What should your transition probabilities look like? It’s up to you, but you probably want to
use a distribution that encourages “smoothness,” i.e. that P(si+1|si) is high if si+1 = si and is low if
they are very different.)
3. A simple way of improving the results further is to incorporate some human feedback. Assume that
a human gives you two (x, y) points, one that is known to lie somewhere on the air-ice boundary,
and one that is known to lie somewhere on the ice-rock boundary. Modify your answer to step 2 to
incorporate this additional information. (Hint: you can do this by just tweaking the HMM’s probability
distributions – no need to change the algorithm itself!) Show the resulting boundaries in red and the
human-provided point with an asterisk (our skeleton code already does this for you).
Your program should be run like this:
python3 ./ input_file.jpg airice_row_coord airice_col_coord icerock_row_coord icerock_col_coord
where airice row coord and airice col coord are the image row and column coordinates of the human-labeled
air-ice pixel, and icerock row coord and icerock col coord are the coordinates of the human-labeled ice-rock
pixel. The program should generate two files: air ice output.jpg, which should show the original image
with the yellow, blue, and red lines superimposed on the echogram for the estimated air-ice layers, and
ice rock output.jpg, which should show the same for the ice-rock layers.
Run your code on our sample images and please show a few sample outputs in your report.
Part 3: Reading text
To show the versatility of HMMs, let’s try applying them to another problem; if you’re careful and you plan
ahead, you can probably re-use much of your code from Parts 1 and 2 to solve this problem. Our goal is to
Figure 3: Our goal is to extract text from a noisy scanned image of a document.
recognize text in an image – e.g., to recognize that Figure 2 says “It is so ordered.” But the images are noisy,
so any particular letter may be difficult to recognize. However, if we make the assumption that these images
have English words and sentences, we can use statistical properties of the language to resolve ambiguities,
just like in Part 2.
We’ll assume that all the text in our images has the same fixed-width font of the same size. In particular,
each letter fits in a box that’s 16 pixels wide and 25 pixels tall. We’ll also assume that our documents only
have the 26 uppercase latin characters, the 26 lowercase characters, the 10 digits, spaces, and 7 punctuation
symbols, (),.-!?’”. Suppose we’re trying to recognize a text string with n characters, so we have n observed
variables (the subimage corresponding to each letter) O1, …, On and n hidden variables, l1…, ln, which are
the letters we want to recognize. We’re thus interested in P(l1, …, ln|O1, …, On). As in part 1, we can rewrite
this using Bayes’ Law, estimate P(Oi
|li) and P(li
|li−1) from training data, then use probabilistic inference
to estimate the posterior, in order to recognize letters.
What to do. Write a program called that is called like this:
python3 ./ train-image-file.png train-text.txt test-image-file.png
The program should load in the train-image-file, which contains images of letters to use for training (we’ve
supplied one for you). It should also load in the text training file, which is simply some text document
that is representative of the language (English, in this case) that will be recognized. (The training file from
Part 1 could be a good choice). Then, it should use the classifier it has learned to detect the text in testimage-file.png, using (1) the simple Bayes net of Figure 1b and (2) the HMM of Fig 1a with MAP inference
(Viterbi). The last two lines of output from your program should be these two results, as follows:
[djcran@tank]$ python3 ./ train-image-file.png train-text.txt test-image-file.png
Simple: 1t 1s so orcerec.
HMM: It is so ordered.
Hints. We’ve supplied you with skeleton code that takes care of all the image I/O for you, so you don’t have
to worry about any image processing routines. The skeleton code converts the images into simple Python
list-of-lists data structures that represents the characters as a 2-d grid of black and white dots. You’ll
need to define an HMM and estimate its parameters from training data. The transition and initial state
probabilities should be easy to estimate from the text training data. For the emission probability, we suggest
using a simple naive Bayes classifier. The train-image-file.png file contains a perfect (noise-free) version of
each letter. The text strings your program will encounter will have nearly these same letters, but they may
be corrupted with noise. If we assume that m% of the pixels are noisy, then a naive Bayes classifier could
assume that each pixel of a given noisy image of a letter will match the corresponding pixel in the reference
letter with probability (100 − m)%.



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