Imagine that you would like to predict if your favorite table will be free at your favorite restaurant. The only additional piece of information you can collect, however, is if it is sunny or not sunny. Therefore, you would like to predict whether the table will be free or not given the weather. You collect paired samples from visit of the form (is sunny, is table free), where it is either sunny (1) or not sunny (0) and the table is either free (1) or not free (0). Your goal is to learn P (Table is Free|Weather Information), which then also automatically tells you P (Table is Not Free|Weather Information) by taking 1- P (Table is Free|Weather Information). Given these learned probabilities, you can make predictions about whether your table is free or not by checking if P (Table is Free|Weather Information) >0.5. If there is a greater than 50% probability your table is free, you are happy to risk the icy streets to get to your favorite restaurant. Hint: To help you with this question, see Section 5.4, Example 22 about books. (a) [10 MARKS] Formulate this distribution learning problem as a maximum likelihood problem. Start by stating what the random variables are for this problem and what values they can take. Then explain what distributions on these random variables you want to learn and what parameters need to be learned. Finally, write the log likelihood explicitly for those distributions and parameters, p(Dw) where D={{xi,yi)}i=1n is a dataset of the collected paired samples and w are your parameters. The parameters are bold because they might consist of more than one parameter, and so would be a vector rather than a scalar. Note: You do not need to solve this maximum likelihood problem, just formalize it. (b) [10 MARKS] Assume you have collected data for the last 10 days and computed the maximum likelihood solution w to the problem formulated in (a). You do not actually have to compute the solution, just assume that you did and now have estimated the parameters w for your distribution. If it is sunny today, then how would you predict if your table will be free? Be precise and explain how you would use your learned distribution and parameters to do this. (c) [5 MARKs] Imagine you could further gather information about if it is morning, afternoon, or evening. How does this change the maximum likelihood problem? You do not need to write the log likelihood explicitly for this question, just explain if you need any new random variables, what values it can take, what the distribution is, and what parameters you will need