- Consider the Bayes net shown in Figure 1. Write answers with a scale of 4, i.e., 0
*.*

(You don’t need to show the calculation steps.)

- (2 points) Calculate the value of
*P*(*b,i,*¬*m,g,j*). - (3 points) Calculate the value of (c) (4 points) Calculate the value of

Figure 1: A simple Bayes net with Boolean variables.

- (6 points) In Figure 2, suppose we observe an unending sequence of days on which the umbrella appears. As the days go by, the probability of rain on the current day increases toward a fixed point, we expect that . Find
*ρ*with a scale of 4, i.e., 0*.*(You don’t need to show the calculation steps.)

Figure 2: Bayesian network structure and conditional distributions describing the umbrella world. The transition model is) and the sensor model is

- A professor wants to know if students are getting enough sleep. Each day, the professorobserves whether they have red eyes. The professor has the following domain theory:
- The prior probability of getting enough sleep, with no observations, is 0.7.
- The probability of getting enough sleep on night
*t*is 0.8 given that the student got enough sleep the previous night, and 0.3 if not. - The probability of having red eyes is 0.2 if the student got enough sleep, and 0.7 if not.

- (6 points) Formulate this information as a hidden Markov model. Give a Bayesiannetwork and conditional distributions.
- (8 points) Consider the following evidences, and compute) with a scale of 4, i.e., 0
*.*(You don’t need to show the calculation steps.)*~e*_{1 }= red eyes*~e*_{2 }= not red eyes

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