In California, whether it rains or not from each day to the next forms a Markov chain (note: this is a terrible model for real weather). However, sometimes California is in a drought and sometimes it is not. Whether California is in a drought from each day to the next itself forms a Markov chain, and the state of this Markov chain affects the transition probabilities in the rain-or-shine Markov chain. The state diagram for droughts, rain given drought, and rain given no drought appear in Figure S14.8. 

a. Draw a dynamic Bayes net which encodes this behavior. Use variables D_t−1, D_t , D_t+1, R_t−1, R_t , and R_t+1. Assume that on a given day, it is determined whether or not there is a drought before it is determined whether or not it rains that day. 

b. Draw the CPT for D_t in the above DBN. Fill in the actual numerical probabilities. 

c. Draw the CPT for R_t in the above DBN. Fill in the actual numerical probabilities. Suppose we are observing the weather on a day-to-day basis, but we cannot directly observe whether California is in a drought or not. We want to predict whether or not it will rain on day t + 1 given observations of whether or not it rained on days 1 through t. 

d. First, we need to determine whether California will be in a drought on day t+1. Derive a formula for P(D_t+1|r_1:t) in terms of the given probabilities (the transition probabilities on the above state diagrams) and P(D_t |r_1:t) (that is, you can assume we’ve already computed the probability there is a drought today given the weather over time). 

e. Now derive a formula for P(R_t+1|r_1:t) in terms of P(D_t+1|r_1:t) and the given probabilities.

Figure S14.8

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