**1. Consider the given joint distribution**

Calculate the value of P(cavity | ¬ toothache) + P(cavity | catch).

Round the answer up to 2 decimal places. (e.g., if the answer is 0.1372, return 0.14)

Answer :- 0.63

**2. There are 1000 coins — 999 are fair, and 1 has heads on both sides. You randomly choose a coin and flip it 10 times. Miraculously, all 10 flips turn up heads. What is the probability that you chose the unfair coin? If the answer is of the form m/n, where m and n are relatively co-prime, return m+n.**

Answer :- 3047.0

**3. Which of the following statements is/are true?**

- X is conditionally independent of B given A, D, and F.
- X is conditionally independent of F given A.
- X is conditionally independent of D given A.
- X is conditionally independent of F given D.

Answer :- a, b

**4. Compute P(A = true | X = true, B = false), given that**

P(A = true) = 0.4, P(X = true | A = true) = 0.3, P(X = true | A = false) = 0.2, P(B = true | A = true) = 0.5, P(B = true | A = false) = 0.4

Return the answer (rounded) up to 2 decimal places.

Answer :- 0.45

**5. What is the minimum number of parameters needed to store the full joint distribution?**

Answer :- 14.0

**6. Which of the following is/are true ?**

- Any boolean formula can be converted to a Bayesian Network, performing inference on which can tell us how many models satisfy the original boolean formula
- Exact Inference for a general Bayes Net is NP-Hard
- Exact Inference for certain kind of Bayes Nets can be done in polynomial time
- Approximate inference can be done for Bayes Nets using sampling in polynomial time

Answer :- a, b, c, d

**7. Which of the following is/are true for a Bayes Network ?**

- Bayesian Network (structures) are Directed Acyclic Graphs
- There exists a unique bayesian network for a given joint probability distribution
- Bayes Networks encode conditional independence relations between variables and can often compactly represent a joint probability distribution.
- For any Bayes Network, a variable is conditionally independent of all other variables given all its parents

Answer :- a, c

**8. We are given the following Bayes Network (as discussed in the videos)**

How many of the following statements are true ?

i. Earthquake and Burglary are independent of each other

ii. Given Alarm, Burglary is independent of Earthquake

iii. Given Alarm, JohnCalls and MaryCalls are independent of each other

iv. JohnCalls and MaryCalls are independent of Each other

v. Given Earthquake and Burglary, Alarm is independent of MaryCalls and JohnCalls

vi. JohnCalls is dependent of Burglary given Earthquake

vii. Given Alarm, JohnCalls is independent of all other variables

viii. Given Alarm, Burglary is independent of both JohnCalls and MaryCalls

Answer :- 5

**9. In the Bayesian Network given below, what is the probability that Mary calls given that an Earthquake has occurred and a burglary has not taken place? Write the solution rounded to 4 decimal places**

Answer :- 0.2101

**10. In the same Bayesian Network given below, what is the probability that Mary calls given that an Earthquake has occurred?**

Write the solution after rounding off to first 4 decimal digits.

Answer :- 0.2106

**11. There is a biased coin with probability of heads as θ. Let the prior on this probability be given by the PDF P(θ) = 6θ (1- θ). The coin is flipped 5 times, and it lands up on heads each time. The MAP estimate of θ is (return answer rounded upto 3 decimal places, eg, if the answer is 0.3456, return 0.346)**

Answer :- 0.857

**12. How many of the following statements are CORRECT**

a. Rejection sampling (before rejection step) samples from the posterior distribution of the Bayes Net given the evidence variables

b. In the limit of infinite sampling, rejection sampling will return consistent posterior estimates

c. In the limit of infinite sampling, likelihood weighting will return consistent posterior estimates

d. Exact Inference in Bayesian networks can be done in Polynomial time in the worst case.

e. MCMC with Gibbs Sampling is guaranteed to converge to the same distribution irrespective of the initial state for all kinds of Bayes Networks

f. Performance of Likelihood weighting degrades in practice when there are many evidence variables that are lower in the Bayes Net

g. Late occurring evidence variables can help guide sample generation in likelihood weighting

Answer :- 3

**13. What is the P(? _{1} = 1) + P(?_{5} = 2), where “?i” indicates “?” in the ith row, after the 1st E-step. (round up to 2 decimal places)**

Answer :- 0.83

**14. What is P(X = 1 | B = 1) – P(B = 0 | A = 1) after the 1st M-step? (round up to 3 decimal places)**

Answer :- 0.397

**15. Consider the bayes-net**

We wish to calculate P(c | ¬w, r). Select the correct statements –

- When using rejection sampling, we will accept the sample ( ¬c, s, ¬w, r).
- When using likelihood weighting, every sample will be weighted 0.5
- When using MCMC with Gibbs sampling, we vary assignments to C and S.
- S is conditionally independent of all other variables, given its parent C.

Answer :- a, b, c

**16. Which of the following is/are INCORRECT?**

- ML (Maximum Likelihood) works better than MAP when the data is sparse.
- In the limit of infinite data, MAP reduces to MLE.
- MAP estimation needs a prior distribution on the parameter values
- MLE assumes a beta distribution prior on the parameter values

Answer :- a, d

**17. How many of the following statements are CORRECT ?**

a. The 15-Puzzle has a single agent setup

b. The game of poker has a partially observable environment

c. The game of poker is deterministic

d. The environment of an autonomous driving car is fully observable

e. Playing the game of chess under time constraints is semi-dynamic

f. The outcomes of the decisions taken by an autonomous driving agent are stochastic in nature

g. The game of poker has a discrete environment

h. Interactive Medical Diagnosis chatbots do episodic decision making

Answer :- 5

**18. Consider the following scenario, where we have a bag of candies. There are only two types of candies: red (R) and green (G) in the bag. We want to estimate the % of candies of each type in the bag. We perform Bayesian learning where the hypothesis space is given by H = {h1, h2, h3, h4, h5}. We have the following 5 hypotheses and their initial probabilities (prior).**

What is the probability of picking a green candy as predicted by our initial estimates:

Answer :- 0.5

**19. Consider the same table as question 8, we now pick a random candy from the bag and it turns out to be green. We then recompute the probabilities of our hypothesis using Bayesian learning (posterior):**

(values in the table percentages)

What is the value of 5a + 4b + 3c + 2d + e

Answer :- 240

**20. Which of the following is/are CORRECT?**

- To learn the structure of Bayes net, we can perform a local search over the space of network structures.
- If the structure of Bayes net is known and we are trying to estimate the parameters of the network from data which has some missing values, we can use an EM algorithm to estimate both the missing values and the parameters.
- Given a training dataset, a fully connected Bayes Net is always guaranteed to fit the dataset the best
- While learning the Bayes net structure given data, If we use a scoring function which is a linear combination of a term which is inversely proportional to model complexity and a term which is proportional to how well the Bayes net fits the data, then the model with the highest score might not be the fully connected Bayes net

Answer :- a, b, c, d

**21. Consider the following Bayesian Network.**

We are given the following samples.

What is the estimated value of P(C=1|B=0) if we use Add-One smoothing? Round off your answer to 2 decimal points. (e.g., if your answer is 0.14159… then write 0.14)

Answer :- 0.71

**22. Choose the CORRECT statements:**

- We will choose decision D if using Maximax criterion.
- Given the current reward table, a rational agent would never prefer decision B over C.
- If using Maximin criteria, we would take decision A.
- If using Equal Likelihood Criterion, we would choose decision B.
- If using Minimax Regret, we would choose decision C.

Answer :- a, b, c

**23. Consider the same reward table. Probability of neutral state happening is 0.6, and that of unfavourable happening is 0.1. Which decision would you take, based on expected reward?**

Answer :- d

**24. For what values of ɑ would you take the action D, according to Hurwicz criterion? The values of ɑ lie in the range (p/q, r], where p and q are co-prime natural numbers. Return p+q+r.**

Answer :- 5.0

**25. Consider the same probability distribution as Q2. What is the amount of money that you should pay for perfect information? (Don’t enter any other symbols such as $, just enter the amount)**

Answer :- 200.0

**26. Choose the INCORRECT statements –**

- A most optimistic agent will choose the Maximax Criterion.
- A realistic agent will choose the Minimax Criterion.
- Equal Likelihood is same as Hurwicz criterion with ɑ = 0.5
- Consider 2 lotteries. Lottery 1 returns $100 always. Lottery 2 returns $10000 with probability 0.01 and $0 with probability 0.99. Both of them have the same expected value.

Answer :- b, c

**27. Consider the Value Iteration algorithm for finding the optimal value function for each state in an MDP specified by < S , A, T, R, S _{0}, γ >. We terminate when the maximum change of the value function across all states is less than ϵ.**Which of the following is/are TRUE?

- The space complexity of the algorithm doesn’t depend on |A|
- The number of iterations taken by the algorithm increases as we decrease ϵ
- The number of iterations taken by the algorithm increases as we decrease γ
- The Time Complexity of a single iteration is O |A|
^{2}|S|

Answer :- a, b

**28. We are given the following cyclic MDP, with 4 states A, B, C, G (goal state). The transition costs and probabilities are mentioned in the figure. We evaluate the policy π according to which we take the action a0 in A, a1 in B and a2 in C and G is a terminal state . What is the value of 29|Vπ (A)-Vπ (C)| ? Here |x| is the absolute value of x .**The value function of a state is the expected discounted cost paid to reach the goal starting from the state following the policy.

Answer :- 5.0

**29. Which of the following is/are TRUE in the context of Markov Decision Processes?**

- In general, Value Iteration converges in fewer iterations compared to Policy Iteration.
- Using a discount factor less than 1 ensures that the value functions do not diverge
- If the MDP has a fixed point solution, then value iteration is guaranteed to converge in a finite number of iterations for any value of epsilon irrespective of the starting point
- In each iteration of policy iteration, value iteration is run as a subroutine, using a fixed policy

Answer :- b, c, d

**30. Consider a deterministic policy 𝜋 such that 𝜋(s0) = a00, 𝜋(s1) = a1, 𝜋(s2) = a20, 𝜋(s3) = a3, 𝜋(s4) = a41. What will be the value of state s0 under this policy?**

Answer :- 9.0