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📗 [3 points] Suppose the likelihood probabilities of observing "a", "o", "c" in a real movie script is , and the likelihood probabilities of observing "a", "o", "c" in a fake movie script is . Given the prior probabilities, of the scripts are real. How would a Naive Bayes classifier classify a script ""? Enter \(1\) if it is classified as real, enter \(-1\) if it is classified as fake, and enter \(0\) if it's a tie (equally likely to be real and fake).
📗 Answer: .
📗 [4 points] Consider a classification problem with \(n\) = classes \(y \in \left\{1, 2, ..., n\right\}\), and two binary features \(x_{1}, x_{2} \in \left\{0, 1\right\}\). Suppose \(\mathbb{P}\left\{Y = y\right\}\) = , \(\mathbb{P}\left\{X_{1} = 1 | Y = y\right\}\) = , \(\mathbb{P}\left\{X_{2} = 1 | Y = y\right\}\) = . Which class will naive Bayes classifier produce on a test item with \(X_{1}\) = and \(X_{2}\) = .
📗 Answer: .
📗 [4 points] Say we use Naive Bayes in an application where there are features represented by variables, each having possible values, and there are classes. How many probabilities must be stored in the CPTs (Conditional Probability Table) in the Bayesian network for this problem? Do not include probabilities that can be computed from other probabilities.
📗 Answer: .
📗 [4 points] Consider the problem of detecting if an email message is a spam. Say we use four random variables to model this problem: a binary class variable \(S\) indicates if the message is a spam, and three binary feature variables: \(C, F, N\) indicating whether the message contains "Cash", "Free", "Now". We use a Naive Bayes classifier with associated CPTs (Conditional Probability Table):
📗 [4 points] Consider the problem of detecting if an email message contains a virus. Say we use four random variables to model this problem: Boolean (binary) class variable \(V\) indicates if the message contains a virus or not, and three Boolean feature variables: \(A, B, C\). We decide to use a Naive Bayes Classifier to solve this problem so we create a Bayesian network with arcs from \(V\) to each of \(A, B, C\). Their associated CPTs (Conditional Probability Table) are created from the following data: \(\mathbb{P}\left\{V = 1\right\}\) = , \(\mathbb{P}\left\{A = 1 | V = 1\right\}\) = , \(\mathbb{P}\left\{A = 1 | V = 0\right\}\) = , \(\mathbb{P}\left\{B = 1 | V = 1\right\}\) = , \(\mathbb{P}\left\{B = 1 | V = 0\right\}\) = , \(\mathbb{P}\left\{C = 1 | V = 1\right\}\) = , \(\mathbb{P}\left\{C = 1 | V = 0\right\}\) = . Compute \(\mathbb{P}\){ \(A\) = , \(B\) = , \(C\) = }.
📗 Answer: .
📗 [2 points] Let \(X \in\) and \(Y \in\) . What is the least number of probabilities needed to fully specify the CPT (Conditional Probability Table) of the \(Y\) given \(X\) (i.e. \(\mathbb{P}\left\{Y | X\right\}\))? Note that this is not a part of the CPTs in the Naive Bayes model. Do not include probabilities that can be computed from other probabilities.
📗 Answer: .
📗 [2 points] Let \(A \in\) and \(B \in\) . What is the least number of probabilities needed to fully specify the conditional probability table of B given A (\(\mathbb{P}\left\{B | A\right\}\))?
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