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3. Gibbs Sampling Graphical models are often useful for modeling phenomena involving multiple variables. In this problem, you'll formulate a graphical model, then demonstrate how to sample from the posterior using Gibbs sampling. (a) Consider the following scenario: suppose the probability that a burglar breaks into your car is 111,, and the probability that an innocent passerby accidentally touches your car is m. Let Z, be a binary random variable that is 1 if there is a burglar, and 0 otherwise. Likewise, let Zz- be a binary random variable that is 1 if there is an innocent passerby, and 0 otherwise. Suppose Z1, and Zi are independent of each other. Let X be a binary random variable that is 1 if your car alarm goes off. The probability your car alarm goes off depends on Zb and Zi, and is known to be: Draw the graphical model depicting the direct relationships between 7a,, 75-, Zb, Zi, and X. (b) Suppose you know the parameters 7n, and 7\(i) Suppose we are running Gibbs sampling, and on each iteration we sample Zo first and then sample Zi. We observed X = 0, and the values of Zo and Zi from iteration t are z(t) = 0 and Z,") = 1. Derive the distribution used for the Gibbs sampling update of Z. . Your solution should be in terms of Tb, Ti, and constants. (ii) Now, suppose we draw Z. = 1 from the distribution derived in Part (b.i). Derive the distribution used for the Gibbs sampling update of Z.". Your solution should be in terms of Tb, Ti, and constants