Homework 7: Sequence Models

1. Homework 7: Sequence Models

2. 1. What model is most appropriate? For each prediction problem below, what model (e.g., linear regression, linear classifier, HMM, particle filter) would you use?

3. 2. Predicting states in a Markov Model 0.6 0.2 0.4 • Suppose the weather starts out in a Sunny state. What is the probability that it is Sunny • One time step later • Two time steps later • Three time steps later • Suppose the weather starts out with a 0.7 probability of Sunny, and 0.3 probability of Rainy. Now, what is the probability that it is Sunny • One time step later • Two time steps later • Three time steps later Sunny Rainy 0.8

4. 3. Estimating parameters of a MM ? ? Initial state is Sunny: ? is Rainy: ? ? • For each sequence of (S)unny and (R)ainy below, determine the corresponding parameters of the MM using maximum likelihood. • RRRSRRSRRRRR • SRSSRSRSSRSRS • For the sequence in (a) above, estimate the parameters for the MM again, but this time use Laplace smoothing with k=1. Sunny Rainy ?

5. 4. Finding the stationary distribution Determine the stationary distribution for the two MMs on the right. 0.6 0.2 0.4 Sunny Rainy 0.8 0.8 0.8 0.2 Sunny Rainy 0.2

6. 5. HMM Computations Notice that the parameters of this HMM do not match the slides from lecture. • Compute P(H1=Sleep | O1=Ans) • Compute P(H2=Sleep | O2=-Ans) H1 H2 O1 O2