MMiDS 6.3: Self-Assessment Quiz

Which of the following statements is not true about conditional probability?

• a) \(P[A|B] = \frac{P[A \cap B]}{P[B]}\) for events \(A\) and \(B\) with \(P[B] > 0\).
• b) If \(A\) and \(B\) are independent, then \(P[A|B] = P[A]\).
• c) Conditional probabilities can be used to express the multiplication rule and the law of total probability.
• d) \(P[A|B] = P[B|A]\) for any events \(A\) and \(B\).

Let \(X\) and \(Y\) be discrete random variables. Which expression defines the conditional expectation of \(X\) given \(Y = y\)?

• a) \(\mathbb{E}[X|Y = y] = \sum_{x \in S_X} x p_{X,Y}(x,y)\)
• b) \(\mathbb{E}[X|Y = y] = \sum_{x \in S_X} x p_X(x)\)
• c) \(\mathbb{E}[X|Y = y] = \sum_{x \in S_X} x p_{X|Y}(x|y)\)
• d) \(\mathbb{E}[X|Y = y] = \sum_{y \in S_Y} y p_{Y|X}(y|x)\)

Which of the following is a valid expression for the joint probability \(P[X=x, Y=y, Z=z]\) using the chain rule of probability?

• a) \(P[X=x]P[Y=y]P[Z=z]\)
• b) \(P[X=x]P[Y=y|X=x]P[Z=z|X=x,Y=y]\)
• c) \(P[X=x|Y=y]P[Y=y|Z=z]P[Z=z]\)
• d) \(P[X=x|Z=z]P[Y=y|Z=z]P[Z=z]\)

Which of the following is the correct mathematical expression for the conditional independence of events \(A\) and \(B\) given event \(C\), denoted as \(A \perp\!\!\!\perp B \mid C\)?

• a) \(P[A \cap B \mid C] = P[A \mid C] + P[B \mid C]\)
• b) \(P[A \cup B \mid C] = P[A \mid C] P[B \mid C]\)
• c) \(P[A \cap B \mid C] = P[A \mid C] P[B \mid C]\)
• d) \(P[A \mid B \cap C] = P[A \mid C]\)

In the fork configuration \(Y \leftarrow X \rightarrow Z\), which of the following conditional independence relations always holds?

• a) \(X \perp\!\!\!\perp Y \mid Z\)
• b) \(Y \perp\!\!\!\perp Z \mid X\)
• c) \(X \perp\!\!\!\perp Z \mid Y\)
• d) \(Y \perp\!\!\!\perp Z\)

In the chain configuration \(X \rightarrow Y \rightarrow Z\), which of the following conditional independence relations always holds?

• a) \(X \perp\!\!\!\perp Y \mid Z\)
• b) \(Y \perp\!\!\!\perp Z \mid X\)
• c) \(X \perp\!\!\!\perp Z \mid Y\)
• d) \(X \perp\!\!\!\perp Z\)

In the collider configuration \(X \rightarrow Z \leftarrow Y\), which of the following conditional independence relations always holds?

• a) \(X \perp\!\!\!\perp Y \mid Z\)
• b) \(Y \perp\!\!\!\perp Z \mid X\)
• c) \(X \perp\!\!\!\perp Z \mid Y\)
• d) \(X \perp\!\!\!\perp Y\)

What is the key conditional independence assumption in the Naive Bayes model for document classification?

• a) The words in a document are conditionally independent given the document's length.
• b) The words in a document are conditionally independent given the document's topic.
• c) The topics of documents are conditionally independent given the words in the documents.
• d) The topics of documents are conditionally independent given the document lengths.

In the Naive Bayes model for document classification, what does the variable \(X_m\) represent?

• a) The presence or absence of the \(m\)-th word in the vocabulary in a document.
• b) The frequency of the \(m\)-th word in the vocabulary in a document.
• c) The topic of the \(m\)-th document in the dataset.
• d) The length of the \(m\)-th document in the dataset.

Which of the following best describes the graphical representation of the Naive Bayes model for document classification?

• a) A chain with the topic variable at the center and word variables as the links.
• b) A collider with the topic variable at the center and word variables as the parents.
• c) A fork with the topic variable at the center and word variables as the prongs.
• d) A complete graph with edges between all pairs of variables.

In the sentiment analysis example, what does the variable \(X\) represent after the following code is executed?

X = (count > 0).astype(int)

• a) The document-term matrix with word frequencies.
• b) The document-term matrix with binary indicators for word presence.
• c) The term-document matrix with word frequencies.
• d) The term-document matrix with binary indicators for word presence.

In the sentiment analysis example, what does the variable \(N_{km}\) represent?

• a) The number of documents in the training set with topic \(k\) and word \(m\) present.
• b) The number of words in the vocabulary that are present in at least one document with topic \(k\).
• c) The probability of word \(m\) being present in a document given that the document has topic \(k\).
• d) The probability of a document having topic \(k\) given that word \(m\) is present in the document.