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📗 [2 points] Consider the following directed graphical model over binary variables: \(A \to B \leftarrow C\) with the following training set.
A
B
C
0
0
0
0
0
1
0
1
1
0
1
0
1
1
1
1
What is the MLE (Maximum Likelihood Estimate) with Laplace smoothing of the conditional probability that \(\mathbb{P}\){ \(B\) = | \(A\) = , \(C\) = }?
📗 Answer: .
📗 [3 points] Consider the following directed graphical model over binary variables: \(A \leftarrow B \to C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{B = 1\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 1\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 0\right\}\)
\(\mathbb{P}\left\{A = 1 | B = 1\right\}\)
\(\mathbb{P}\left\{A = 1 | B = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = , \(B\) = , \(C\) = }?
📗 Answer: .
📗 [5 points] Consider the following directed graphical model over binary variables: \(A \leftarrow B \to C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{B = 1\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 1\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 0\right\}\)
\(\mathbb{P}\left\{A = 1 | B = 1\right\}\)
\(\mathbb{P}\left\{A = 1 | B = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = \(|\) \(C\) = }?
📗 Answer: .
📗 [3 points] Consider the following directed graphical model over binary variables: \(A \to B \to C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{A = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 0\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 1\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = , \(B\) = , \(C\) = }?
📗 Answer: .
📗 [5 points] Consider the following directed graphical model over binary variables: \(A \to B \to C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{A = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 0\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 1\right\}\)
\(\mathbb{P}\left\{C = 1 | B = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = \(|\) \(C\) = }?
📗 Answer: .
📗 [3 points] Consider the following directed graphical model over binary variables: \(A \to B \leftarrow C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{A = 1\right\}\)
\(\mathbb{P}\left\{C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 0, C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 1, C = 0\right\}\)
\(\mathbb{P}\left\{B = 1 | A = C = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = , \(B\) = , \(C\) = }?
📗 Answer: .
📗 [5 points] Consider the following directed graphical model over binary variables: \(A \to B \leftarrow C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{A = 1\right\}\)
\(\mathbb{P}\left\{C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 0, C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 1, C = 0\right\}\)
\(\mathbb{P}\left\{B = 1 | A = C = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = \(|\) \(C\) = }?
📗 Answer: .
📗 [3 points] Consider the following directed graphical model over binary variables: \(A \to B \leftarrow C\). Given the CPTs (Conditional Probability Table):
Variable
Probability
Variable
Probability
\(\mathbb{P}\left\{A = 1\right\}\)
\(\mathbb{P}\left\{C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 0, C = 1\right\}\)
\(\mathbb{P}\left\{B = 1 | A = 1, C = 0\right\}\)
\(\mathbb{P}\left\{B = 1 | A = C = 0\right\}\)
What is the probability that \(\mathbb{P}\){ \(A\) = , \(B\) = , \(C\) = }?
📗 Answer: .
📗 [4 points] Consider the following Bayesian Network containing 5 Boolean random variables. How many numbers must be stored in total in all CPTs (Conditional Probability Table) associated with this network (excluding the numbers that can be calculated from other numbers)?
📗 Answer: .
📗 [2 points] Given the following network \(A \to B \to C\) where A can take on values, B can take on values, C can take on values. Write down the minimum number of conditional probabilities that define the CPTs (Conditional Probability Table).
📗 Answer: .
📗 [3 points] Given the following Bayesian network and the estimated CPTs (conditional probability table), and a new data point with = , = , what is estimated probability of = ?
\(\hat{\mathbb{P}}\left\{A\right\}\)
\(\hat{\mathbb{P}}\left\{B\right\}\)
\(\hat{\mathbb{P}}\left\{C | A, B\right\}\)
\(\hat{\mathbb{P}}\left\{C | A, \neg B\right\}\)
\(\hat{\mathbb{P}}\left\{C | \neg A, B\right\}\)
\(\hat{\mathbb{P}}\left\{C | \neg A, \neg B\right\}\)
\(\hat{\mathbb{P}}\left\{D | C\right\}\)
\(\hat{\mathbb{P}}\left\{D | \neg C\right\}\)
\(\hat{\mathbb{P}}\left\{E | C\right\}\)
\(\hat{\mathbb{P}}\left\{E | \neg C\right\}\)
📗 Answer: .
📗 [5 points] Andy is a three-month old baby. He can be happy (state 0), hungry (state 1), or having a wet diaper (state 2). Initially when he wakes up from his nap at 1pm, he is happy. If he is happy, there is a chance that he will remain happy one hour later, a chance to be hungry by then, and a chance to have a wet diaper. Similarly, if he is hungry, one hour later he will be happy with chance, hungry with chance, and wet diaper with chance. If he has a wet diaper, one hour later he will be happy with chance, hungry with chance, and wet diaper with chance. He can smile (observation 0) or cry (observation 1). When he is happy, he smiles of the time and cries of the time; when he is hungry, he smiles of the time and cries of the time; when he has a wet diaper, he smiles of the time and cries of the time.
What is the probability that the particular observed sequence (or \(Y_{1}, Y_{2}\) = ) happens (in the first two periods)?
Note: if the weights are not shown clearly, you could move the nodes around with mouse or touch.
📗 Answer: .
📗 [3 points] You have a joint probability table over \(k\) = random variables \(X_{1}, X_{2}, ..., X_{k}\), where each variable takes \(m\) = possible values: \(1, 2, ..., m\). To compute the probability that \(X_{1}\) = , how many cells in the table do you need to access (at most)?
📗 Answer: .
📗 [3 points] Given a Bayesian network \(A \to B \to C \to D\) of 4 binary event variables with the following conditional probability table (CPT), what is the probability that none of the events happen, \(\mathbb{P}\left\{\neg A, \neg B, \neg C, \neg D\right\}\)?
\(\mathbb{P}\left\{A\right\}\) =
\(\mathbb{P}\left\{B | A\right\}\) =
\(\mathbb{P}\left\{C | B\right\}\) =
\(\mathbb{P}\left\{D | C\right\}\) =
\(\mathbb{P}\left\{\neg A\right\}\) =
\(\mathbb{P}\left\{B | \neg A\right\}\) =
\(\mathbb{P}\left\{C | \neg B\right\}\) =
\(\mathbb{P}\left\{D | \neg C\right\}\) =
📗 Answer: .
📗 [3 points] Given a Bayesian network \(A \to B \to C \to D \to E\) of 5 binary event variables with the following conditional probability table (CPT), what is the probability that none of the events happen, \(\mathbb{P}\left\{\neg A, \neg B, \neg C, \neg D, \neg E\right\}\)?
\(\mathbb{P}\left\{A\right\}\) =
\(\mathbb{P}\left\{B | A\right\}\) =
\(\mathbb{P}\left\{C | B\right\}\) =
\(\mathbb{P}\left\{D | C\right\}\) =
\(\mathbb{P}\left\{E | D\right\}\) =
\(\mathbb{P}\left\{\neg A\right\}\) =
\(\mathbb{P}\left\{B | \neg A\right\}\) =
\(\mathbb{P}\left\{C | \neg B\right\}\) =
\(\mathbb{P}\left\{D | \neg C\right\}\) =
\(\mathbb{P}\left\{E | \neg D\right\}\) =
📗 Answer: .
📗 [3 points] If the joint probabilities of the Bayesian network \(X_{1} \to X_{2} \to X_{3} \to ... \to X_{n}\) with \(n\) = binary variables are stored in a table (instead of the conditional probability tables (CPT)), what is the size of the table?
📗 For example, if the network is \(X_{1} \to X_{2}\), then the size of the joint probability table is 3, containing entries \(\mathbb{P}\left\{X_{1}, X_{2}\right\}, \mathbb{P}\left\{X_{1}, \neg X_{2}\right\}, \mathbb{P}\left\{\neg X_{1}, X_{2}\right\}\), because the joint probability \(\mathbb{P}\left\{\neg X_{1}, \neg X_{2}\right\} = 1 - \mathbb{P}\left\{X_{1}, X_{2}\right\} - \mathbb{P}\left\{X_{1}, \neg X_{2}\right\} - \mathbb{P}\left\{\neg X_{1}, X_{2}\right\}\) can be computed based on the other entries in the table.
📗 Answer: .
📗 [3 points] Given the following Bayesian network and the training set, what is the MLE estimate of \(\mathbb{P}\){|} without smoothing?
\(A\)
\(B\)
\(C\)
\(D\)
\(E\)
📗 Answer: .
📗 [3 points] Suppose ( + + ) entries are stored in conditional probability tables of three binary variables \(X_{1}, X_{2}, X_{3}\)? What is the configuration of Bayesian network? Enter 1 for causal chain (e.g. \(X_{1} \to X_{2} \to X_{3}\)), enter 2 for common cause (e.g. \(X_{1} \leftarrow X_{2} \to X_{3}\)) and enter 3 for common effect (e.g. \(X_{1} \to X_{2} \leftarrow X_{3}\)), and enter -1 if more information is needed or more than one of the previous configurations are possible.
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