11. Jessica is interested in identifying states in U.S.A. Specifically, she categorizes states into one of three categories based on their longitudes: State = [West, Middle, East]. Jessica further notices three features for each state: (1) Population = [overpopulated, underpopulated] (2) Weather = [cold, warm] (3) Polity = [democratic, republic]. For the sub questions below, we suppose that those 3 features (Population, Weather and Polity) are all conditionally independent given State. Additionally, the following probability values are known to us: P( Population = overpopulated / State = West )=0.5, P( Weather = cold / state = West )=0.5, P( Polity = republic / State = West )=0.9, P( Population = overpopulated / state = Middle )=Q, P( Weather = cold / State = Middle )=0.1, P(Polity= republic State = Middle )=0, P( Population = overpopulated / State = East )=0.1, P( Weather = cold / State = East )=0.4, P( Polity = republic / state = East )=0.5, P( State = West )=0.01, P( State = Middle )=0.01. (1) What is the expression of P(Population, Weather, Polity, State) as combination of P(Population/State), P(Weather/State), P(Polity/State) and P(State) by incorporating conditional independence? (4pts) (2) When you create a joint probability table for P(Population, Weather, Polity, State), what is the number of rows in the table? (4pts) (3) Calculate the probability of P(Population=overpopulated, Polity =republic, Weather =cold, State = West ) and of P(Population=underpopulated, Polity =democratic, Weather =warm, State = Middle ). (4pts) (4) Please express P (State/Population, Weather, Polity) by applying Bayes rule. You may need to involve (one or several) terms where each term is the summation over all possible values for some variables. (4pts) (5) After Jessica selects a state, she identifies the following features: Population=overpopulated, Weather=cold, Polity=republic. What is the probability that the state is one of west states? (4pts)