### An Introduction to Artificial Intelligence Assignment 9 Answers

**Q1.** Which of the following is true about the MAP (Maximum a posteriori estimate) estimation learning framework?

a. It is equivalent to Maximum Likelihood learning with infinite data

b. It is equivalent to Maximum Likelihood learning if P(θ) is independent of θ

c. it can be used without having any prior knowledge about the parameters

d. The performance of MAP is better with dense data compared to sparse data

**Answer:- a, d**

**Q2.** What facts are true about smoothing?

- Smoothed estimates of probabilities fit the evidence better than un-smoothed estimates.
- The process of smoothing can be viewed as imposing a prior distribution over the set of parameters.
- Smoothing allows us to account for data which wasn’t seen in the evidence.
- Smoothing is a form of regularization which prevents overfitting in Bayesian networks.

**Answer: a, c**

**Q3.** Consider three boolean variables X, Y, and Z. Consider the following data:

There can be multiple Bayesian networks that can be used to model such a universe. Assume that we assume a Bayesian Network as shown below:

If the value of the parameter P(¬z|x,¬y) is m/n such that m and n have no common factors. Then, what is the value of m+n? Assume add-one smoothing.

**Answer: 343.6**

**Q4.** Consider the following Bayesian Network from which we wish to compute P(x|z) using rejection sampling:

**Answer: 86.9**

**Q5.** Assume that we toss a biased coin with heads probability p, 100 times. We get heads 66 times out of 100. If the Maximum Likelihood estimate of the parameter p is m/n where m and n don’t have common factors,

then the value of m+n is?

**Answer: 77**

**Q6.** Now, assume that we had a prior distribution over p as shown below:

**Answer:- 6.5**

**Q7.** Which of the following task(s) are not suited for a goal based agent?

**Answer: b, c**

**Q8.** Which of the following are true ?

- Rejection sampling is very wasteful when the probability of getting the evidence in the samples is very low.
- We perform conditional probability weighting on the samples while doing Gibbs Sampling in MCMC algorithm since we have already fixed the evidence variables.
- We perform random walk while sampling variables in Likelihood Weighting, MCMC with Gibbs sampling, but not in Rejection sampling.
- Likelihood Weighting functions well if we have many evidence wars with some samples having nearly all the total weight

**Answer: a**

**Q9.** Consider the following Bayesian Network:

- P(C|A,B,D,F,E) = α. P(C|A). P(C|B)
- P(C|A,B,D,F,E) = α. P(C|A,B)
- P(C|A,B,D,F,E) = α. P(C|A,B). P(D|C,E)
- P(C|A,B,D,F,E) = α. P(C|A,B,D,E)

**Answer: b, c**

**Q10.** Which of the following options are correct about the environment of Tic Tac Toe?

- Fully observable
- Stochastic
- Continuous
- Static

**Answer: a, c**