# 1 4 using the intuition in part 33 design a dynamic

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described in part 3.1. 4. Using the intuition in part 3.3, design a dynamic programming algorithm (caching partial results) which computes P ( X i , X j ) for all n choose 2 choices of i and j in time asymptotically much lower than the complexity you described in part 3.2. What is the asymptotic running time of your algorithm? 4 Belief Propagation Two graduate students in University of Tehran have gotten into an argument over the weather. One thinks summer is over and Autumn has already come, while the other thinks it is still summer. In Tehran, there are four seasons – Spring (S), Summer (M), Autumn (A), and Winter (W). Given the season the previous day, the season on a day is conditionally 3

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independent of the season on all previous days. The weather is either Hot (H), Rainy (R) or Freezing (F). Given the season on any given day, the weather that day is independent of all other variables. More formally, if we let C i denote the season on the i–th day (taking values S, M, A, W) and O i denote the observed weather pattern (one of H, R, F). We have j < i - 1 , C i C j | C i - 1 and X, O i X | C i where X is any random variable other than C i , O i . 1. Draw a graphical model over C 1 . . . C N , O 1 . . . O N that satisfies the conditional inde- pendencies listed above. 2. Implement sum-product and max-product algorithms in MATLAB for this graphical model. 3. We has made 20 observation of the weather over the last few months (i.e., O 1 . . . O N ) : { R, F, F, H, F, H, H, H, H, H, H, H, H, R, H, H, H, R, H, H } Some of the values for the conditional probability table (CPT) are as follows.
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