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2:unigram;3:bigram;4:bigram_smooth;5:sentences;7:likelihood;8:posterior;9:predictions
📗 (Introduction) In this programming homework, you will build and simulate a simple Markov chain model based on a movie script. You will use it to generate new sentences that hopefully contain sensible words maybe even phrases. In addition, you will build a Naive Bayes classifier to distinguish sentences from the script and sentences from another fake script. Due to the English vocabulary size, you will use characters as features instead of words. In practice, you could replace the 26 letters by (more than 170,000) English words when training these models.
📗 (Part 1) Download the script of one of the following movies: "", "", "" from IMSDb: Link. If you have another movie you really like, you can use that script instead. Go to the website, use "Search IMSDb" on the left to search for this movie, click on the link: "Read ... Script", and copy and paste the script into a text file.
📗 (Part 1) Make everything lower case, then remove all characters except for letters and spaces. Replace consecutive spaces by one single space and make sure that there are no consecutive spaces.
📗 (Part 1) Construct the bigram character (letters + space) transition probability table. Put "space" first then "a", "b", "c", ..., "z". It should be a 27 by 27 matrix.
📗 (Part 1) Construct the trigram transition probability table. It could be a 27 by 27 by 27 array or a 729 by 27 matrix. You do not have to submit this table.
📗 (Part 1) Generate 26 sentences consists of 1000 characters each using the trigram model starting from "a", "b", "c", ..., "z". You should use the bigram model to generate the second character and switch to the bigram model when the current two-character sequence never appeared in the script. For example, when you see "xz", instead of using the trigram model for the probabilities of Pr{? | xz}, switch to use the bigram model for the probabilities of Pr{? | z}. Find and share some interesting sentences.
📗 (Part 2) The following is my randomly generated fake script written according to an non-English language model.
You can either use the button to download a text file, or copy and paste from the text box to a text file. Please do not change the content of the text box.
📗 (Part 2) Train a Naive Bayes classifier based on the characters. You should use the prior that is biased against your script: Pr{Document = your script} = ? and Pr{Document = fake script} = ?, compute the likelihood Pr{Letter | Document} based on your script and the fake script, and compute the posterior probabilities Pr{Document | Letter}, and test your classifier on the 26 random sentences you generated.
📗 [5 points] Input the unigram probabilities (27 numbers, comma-separated, rounded to 4 decimal places, "space" first, then "a", "b", ...). Hint
📗 Please make sure all probabilities are strictly larger than 0 after rounded to 4 decimal places, for example, 0.00002 should be rounded up to 0.0001 instead of 0.0. Also, the numbers should add up to exactly 1, you can do this by adding or subtracting the difference between the sum of the rounded probabilities and 1 to one of the probabilities.
📗 If there are \(n\) characters and \(n_{a}\) "a"s, then the unigram probability of "a" is \(p_{a} = \dfrac{n_{a}}{n}\).
📗 [5 points] Input the bigram transition probabilities without Laplace smoothing (27 lines, each line containing 27 numbers, comma-separated, rounded to 4 decimal places, "space" first, then "a", "b", ...). Hint
📗 If there are \(n_{a}\) "a"s and \(n_{a b}\) "ab"s, then the bigram transition probability from "a" to "b" (row "a" column "b") is \(p_{a b} = \dfrac{n_{a b}}{n_{a}}\).
📗 [5 points] Input the bigram transition probabilities with Laplace smoothing (27 lines, each line containing 27 numbers, comma-separated, rounded to 4 decimal places, "space" first, then "a", "b", ...). Hint
📗 Again, please make sure all probabilities are strictly larger than 0 after rounded to 4 decimal places, for example, 0.00002 should be rounded up to 0.0001 instead of 0.0. Also, the numbers should add up to exactly 1, you can do this by adding or subtracting the difference between the sum of the rounded probabilities and 1 to one of the probabilities.
📗 If there are \(n_{a}\) "a"s and \(n_{a b}\) "ab"s, then the bigram transition probability from "a" to "b" (row "a" column "b") is \(p_{a b} = \dfrac{n_{a b} + 1}{n_{a} + 27}\) with Laplace smoothing.
📗 [10 points] Input the 26 sentences generated by the trigram and bigram models (Laplace smoothed). (26 lines, each line containing 1000 characters, line 1 starts with "a", line 2 starts with "b" ...). Hint
📗 Suppose you start with "a", then you draw a random character according to the distribution specified by row "a" of the bigram transition matrix, i.e. \(p_{a, \text{\;space\;}}, p_{a a}, p_{a, b}, ..., p_{a, z}\).
📗 Suppose you start with "ab" and \(p_{a b} > 0\) without Laplace smoothing, then you draw a random character according to the distribution specified by row "a" column "b" of trigram transition array, i.e. \(p_{a b \text{\;space\;}}, p_{a b a}, p_{a b b}, ..., p_{a b z}\).
📗 Suppose you start with "yz" and \(p_{y z} = 0\) without Laplace smoothing, i.e. "yz" never appeared in your script, then you draw a random character according to the distribution specified by row "z" of the bigram transition matrix again, i.e. \(p_{z \text{\;space\;}}, p_{z a}, p_{z b}, ..., p_{z z}\).
📗 To generate a random character using CDF inversion according to a distribution, for example \(p_{1}, p_{2}, p_{3}\): compute the CDF \(p_{1}, p_{1} + p_{2}, p_{1} + p_{2} + p_{3}\), generate a random number \(u\) between 0 and 1, if \(0 \leq u < p_{1}\), output 1; if \(p_{1} \leq u < p_{1} + p_{2}\), output 2; if \(p_{1} + p_{2} \leq u < p_{1} + p_{2} + p_{3} = 1\), output 3.
📗 [5 points] Enter likelihood probabilities of the Naive Bayes estimator for the fake script. (27 numbers, comma separated, rounded to 4 decimal places, Pr{"space" | D = fake script}, Pr{"a" | D = fake script}, Pr{"b" | D = fake script}, ...). The likelihood probabilities for your script should be the same as your answer to Question 2. Hint
📗 If there are \(n'\) characters and \(n'_{a}\) "a"s, then the likelihood probability of "a" is \(p'_{a} = \dfrac{n'_{a}}{n'}\).
📗 [5 points] Enter posterior probabilities of the Naive Bayes estimator for the fake script. (27 numbers, comma separated, rounded to 4 decimal places, Pr{D = fake script | "space"}, Pr{D = fake script | "a"}, Pr{D = fake script | "b"}, ...). Hint
📗 Suppose your prior probability that the script is fake is \(p\) (you can find this prior value under "Instruction" (Part 2), it is generated randomly based on your ID), then the posterior probability that the script is fake given there is a character "a" is,
\(\dfrac{p \cdot p'_{a}}{p \cdot p'_{a} + \left(1 - p\right) \cdot p_{a}}\).
📗 [5 points] Use the Naive Bayes model to predict which document the 26 sentences your generated in Question 5 came from. Remember to compare the sum of log probabilities instead of the direct product of probabilities. (26 numbers, either 0 or 1, 0 is the label for your script, 1 is the label for the fake script) Hint
📗 Your prediction is supposed to be all 0s, but if there are few 1s, it is okay, but make sure you generated the sentences in Part 1 correctly according to the instruction (especially switching to bigram from trigram when necessary).
📗 The log-likelihood that a sentence "abbccc" came from the fake script is the sum of the log of the posterior probability of each character in the sentence,
\(\log\left(\mathbb{P}\left\{M | a\right\}\right) + 2 \log\left(\mathbb{P}\left\{M | b\right\}\right) + 3 \log\left(\mathbb{P}\left\{M | c\right\}\right)\), \(M\) is the event "D = fake script".
📗 The log-likelihood that a sentence "abbccc" came from your script is similar,
\(\log\left(1 - \mathbb{P}\left\{M | a\right\}\right) + 2 \log\left(1 - \mathbb{P}\left\{M | b\right\}\right) + 3 \log\left(1 - \mathbb{P}\left\{M | c\right\}\right)\), the probability in the log is the probability of the event "D = your script".
📗 [1 points] Please confirm that you are going to submit the code on Canvas under Assignment P3, and make sure you give attribution for all blocks of code you did not write yourself (see bottom of the page for details and examples).
I will submit the code on Canvas.
📗 [1 points] Please enter any comments and suggestions including possible mistakes and bugs with the questions and the auto-grading, and materials relevant to solving the question that you think are not covered well during the lectures. If you have no comments, please enter "None": do not leave it blank.
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% Code attribution: (TA's name)'s P3 example solution.
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% Code attribution: (student name)'s answer on Piazza: (link to Piazza post).
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📗 Solution:
📗 Java: Link
📗 Python: Link
📗 Sample solution from last year: 2022 P3. The homework is slightly different, please use with caution.
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