# Summary

M6,
P3, P6,
Q9, Q10, Q11, Q12, Q13, Q14,
D5, D6,

# Lectures

Blank Slides: Part 1: PDF,


Image by xkcd via towards data science

# Other Materials

Lecture 10 Part 1 (Generative Models): Link
Lecture 10 Part 2 (Natural Language): Link
Lecture 10 Part 3 (Sampling): Link
Lecture 11 Part 1 (Probability Distribution): Link
Lecture 11 Part 2 (Bayesian Network): Link
Lecture 11 Part 3 (Network Structure): Link
Lecture 11 Part 4 (Naive Bayes): Link

Lecture 12 Part 1 (Hidden Markov Model): Link
Lecture 12 Part 2 (HMM Evaluation): Link
Lecture 12 Part 3 (HMM Training): Link
Lecture 12 Part 4 (Recurrent Neural Network): Link
Lecture 12 Part 5 (Backprop Through Time): Link
Lecture 12 Part 6 (RNN Variants): Link


Zipf's Law: Link
Markov Chain: Link
Google N-Gram: Link

Simple Bayes Net: Link, Link 2
ABNMS: Link, pathfinder: Link

RNN Visualization: Link
LTSM and GRU: Link


How to compute the probability of A given B knowing the probability of A given not B? Link (Part 4)
How to compute the marginal probabilities given the ratio between the conditionals? Link (Part 1)
How to compute the conditional probabilities given the same variable? Link (Part 1)
What is the probability of switch between elements in a cycle? Link (Part 2)
Which marginal probabilities are valid given the joint probabilities? Link (Part 3)
How to use the Bayes rule to find which biased coin leads to a sequence of coin flips? Link
Please do NOT forget to submit your homework on Canvas! Link
How to use Bayes rule to find the probability of truth telling? Link (Part 6)
How to estimate fraction given randomized survey data? Link (Part 12)
How to write down the joint probability table given the variables are independent? Link (Part 13)
Given the ratio between two conditional probabilities, how to compute the marginal probabilities? Link (Part 1)
What is the Boy or Girl paradox? Link (Part 3)
How to compute the maximum likelihood estimate of a conditional probability given a count table? Link (Part 1)
How to compare the probabilities in the Boy or Girl Paradox? Link (Part 3)

How to compute the MLE probability of a sentence given a training document? Link (Part 1)
How to find maximum likelihood estimates for Bernoulli distribution? Link
How to generate realizations of discrete random variables using CDF inversion? Link
How to find the sentence generated given the random number from CDF inversion? Link (Part 3)
How to find the probability of observing a sentence given the first and last word using the transition matrix? Link (Part 14)
How many conditional probabilities need to be stored for a n-gram model? Link (Part 2)

How to compute the joint probability given the conditional probability table? Link
How to compute conditional probability table given training data? Link
How to do inference (find joint and conditional probability) given conditional probability table? Link
How to find the conditional probabilities for a common cause configuration? Link
What is the size of the conditional probability table? Link
How to compute a condition probability given a Bayesian network with three variables? Link
What is the size of a conditional probability table of two discrete variables? Link (Part 2)
How many joint probabilities are needed to compute a marginal probability? Link (Part 3)
How to compute the MLE conditional probability with Laplace smoothing given a data set? Link (Part 2)
What is the number of conditional probabilities stored in a CPT given a Bayesian network? Link (Part 3)
How to compute the number of probabilities in a CPT for variables with more than two possible values? Link (Part 14)
How to find the MLE of the conditional probability given the sum of two variables? Link (Part 5)
How many joint probabilities are used in the computation of a marginal probability? Link (Part 4)
How to find the size of an arbitrary Bayesian network with binary variables? Link (Part 3)

How to use naive Bayes classifier to do multi-class classification? Link
How to find the size of the conditional probability table for a Naive Bayes model? Link
How to compute the probability of observing a sequence under an HMM? Link
What is the number of conditional probabilities stored in a CPT given a Naive Bayes model? Link (Part 4)
How to find th obervation probabilities given an HMM? Link (Part 2)
What is the size of the CPT for a Naive Bayes network? Link (Part 1)
How to detect virus in email messages using Naive Bayes? Link (Part 2)
What is the relationship between Naive Bayes and Logistic Regression? Link

# Keywords and Notations

Distance: (Euclidean) \(\rho\left(x, x'\right) = \left\|x - x'\right\|_{2} = \sqrt{\displaystyle\sum_{j=1}^{m} \left(x_{j} - x'_{j}\right)^{2}}\), (Manhattan) \(\rho\left(x, x'\right) = \left\|x - x'\right\|_{1} = \displaystyle\sum_{j=1}^{m} \left| x_{j} - x'_{j} \right|\), where \(x, x'\) are two instances.
K-Nearest Neighbor classifier: \(\hat{y}_{i}\) = mode \(\left\{y_{\left(1\right)}, y_{\left(2\right)}, ..., y_{\left(k\right)}\right\}\), where mode is the majority label and \(y_{\left(t\right)}\) is the label of the \(t\)-th closest instance to instance \(i\) from the training set.

Unigram model: \(\mathbb{P}\left\{z_{1}, z_{2}, ..., z_{d}\right\} = \displaystyle\prod_{t=1}^{d} \mathbb{P}\left\{z_{t}\right\}\) where \(z_{t}\) is the \(t\)-th token in a training item, and \(d\) is the total number of tokens in the item.
Maximum likelihood estimator (unigram): \(\hat{\mathbb{P}}\left\{z_{t}\right\} = \dfrac{c_{z_{t}}}{\displaystyle\sum_{z=1}^{m} c_{z}}\), where \(c_{z}\) is the number of time the token \(z\) appears in the training set and \(m\) is the vocabulary size (number of unique tokens).
Maximum likelihood estimator (unigram, with Laplace smoothing): \(\hat{\mathbb{P}}\left\{z_{t}\right\} = \dfrac{c_{z_{t}} + 1}{\left(\displaystyle\sum_{z=1}^{m} c_{z}\right) + m}\).
Bigram model: \(\mathbb{P}\left\{z_{1}, z_{2}, ..., z_{d}\right\} = \mathbb{P}\left\{z_{1}\right\} \displaystyle\prod_{t=2}^{d} \mathbb{P}\left\{z_{t} | z_{t-1}\right\}\).
Maximum likelihood estimator (bigram): \(\hat{\mathbb{P}}\left\{z_{t} | z_{t-1}\right\} = \dfrac{c_{z_{t-1}, z_{t}}}{c_{z_{t-1}}}\).
Maximum likelihood estimator (bigram, with Laplace smoothing): \(\hat{\mathbb{P}}\left\{z_{t} | z_{t-1}\right\} = \dfrac{c_{z_{t-1}, z_{t}} + 1}{c_{z_{t-1}} + m}\).

Conditional probability: \(\mathbb{P}\left\{Y = y | X = x\right\} = \dfrac{\mathbb{P}\left\{Y = y, X = x\right\}}{\mathbb{P}\left\{X = x\right\}}\).
Joint probability: \(\mathbb{P}\left\{X = x\right\} = \displaystyle\sum_{y \in Y} \mathbb{P}\left\{X = x, Y = y\right\}\).
Bayes rule: \(\mathbb{P}\left\{Y = y | X = x\right\} = \dfrac{\mathbb{P}\left\{X = x | Y = y\right\} \mathbb{P}\left\{Y = y\right\}}{\displaystyle\sum_{y' \in Y} \mathbb{P}\left\{X = x | Y = y'\right\} \mathbb{P}\left\{Y = y'\right\}}\).
Law of total probability: \(\mathbb{P}\left\{X = x\right\} = \displaystyle\sum_{y' \in Y} \mathbb{P}\left\{X = x | Y = y'\right\} \mathbb{P}\left\{Y = y'\right\}\).
Independence: \(X, Y\) are independent if \(\mathbb{P}\left\{X = x, Y = y\right\} = \mathbb{P}\left\{X = x\right\} \mathbb{P}\left\{Y = y\right\}\) for every \(x, y\).
Conditional independence: \(X, Y\) are conditionally independent conditioned on \(Z\) if \(\mathbb{P}\left\{X = x, Y = y | Z = z\right\} = \mathbb{P}\left\{X = x | Z = z\right\} \mathbb{P}\left\{Y = y | Z = z\right\}\) for every \(x, y, z\).

Conditional Probability Table estimation: \(\hat{\mathbb{P}}\left\{x_{j} | p\left(X_{j}\right)\right\} = \dfrac{c_{x_{j}, p\left(X_{j}\right)}}{c_{p\left(X_{j}\right)}}\), where \(p\left(X_{j}\right)\) is the list of parents of \(X_{j}\) in the network.
Conditional Probability Table estimation (with Laplace smoothing): \(\hat{\mathbb{P}}\left\{x_{j} | p\left(X_{j}\right)\right\} = \dfrac{c_{x_{j}, p\left(X_{j}\right)} + 1}{c_{p\left(X_{j}\right)} + \left| X_{j} \right|}\), where \(\left| X_{j} \right|\) is the number of possible values of \(X_{j}\).
Bayesian network inference: \(\mathbb{P}\left\{x_{1}, x_{2}, ..., x_{m}\right\} = \displaystyle\prod_{j=1}^{m} \mathbb{P}\left\{x_{j} | p\left(X_{j}\right)\right\}\).
Naive Bayes estimation: .
Naive Bayes classifier: \(\hat{y}_{i} = \mathop{\mathrm{argmax}}_{y} \mathbb{P}\left\{Y = y | X = X_{i}\right\}\).