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? Prediction ProblemModel Each example is a set of websites that a user has visited (with timestamps for the time when the user opened the page). The task is to select the best advertisement to display to the user. Each example is a set of temperature recordings collected every hour in several locations around the city. The task is to predict the temperature 24 hours after the last recorded temperature in the example. Each example is a video stream from a sports broadcast. The video consists of a set of frames, where each frame is a 2-dimensional array of pixel brightness values. The task is to predict the location (subset of the pixels) of the ball in each frame. (This is known as object-tracking.)

3. 2. Predicting states in a Markov Model 1.Suppose the weather starts out in a Sunny state. What is the probability that it is Sunny a.One time step later b.Two time steps later c.Three time steps later 2.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 a.One time step later b.Two time steps later c.Three time steps later SunnyRainy 0.2 0.6 0.4 0.8

4. 3. Estimating parameters of a MM 1.For each sequence of (S)unny and (R)ainy below, determine the corresponding parameters of the MM using maximum likelihood. a.RRRSRRSRRRRR b.SRSSRSRSSRSRS 2.For the sequence in (a) above, estimate the parameters for the MM again, but this time use Laplace smoothing with k=1. SunnyRainy ? ? ? ? Initial state is Sunny: ? is Rainy: ?

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

6. 5. HMM Computations Notice that the parameters of this HMM do not match the slides from lecture. 1.Compute P(H1=Sleep | O1=Ans) 2.Compute P(H2=Sleep | O2=-Ans) H1H1 O1O1 H1H1 H2H2 P(H 2 |H 1 ) Sleep 0.7 StudySleep0.5 H1H1 O1O1 P(O 1 |H 1 ) SleepAns0.2 StudyAns0.7 H1H1 P(H 1 ) Sleep0.7 Study0.3 H2H2 O2O2