a. Assuming that A is independent of B given C and that A and C are absolutely independent, what is the most factored representation of P(A, B, C)? 

b. Assuming that A is independent of B given C, what is the most factored representation of P(A, B | C)? 

c. Given no independence assumptions, what is the most factored representation of P(A | B, C)? 

d. Assuming that A is independent of B given C, what is the most factored representation of P(A | B, C)? 

e. Assuming that A is absolutely independent of B, what is the most factor representation of P(A | B, C)? 

f. Assuming that A is independent of B given C, write an expression equivalent to P(A | B) using A, B, and C. 

g. Which of the following expressions are equal to 1, given no independence assumptions? 

(i) Σa P(A = a | B). 

(ii)  Σb P(A | B = b). 

(iii) Σa Σb P(A = a, B = b). 

(iv) Σa Σb P(A = a | B = b). 

(v) Σa Σb P(A = a) P(B = b). 

h. Which of the following expressions hold for any distribution over four random variables A, B, C and D? 

(i) P(A, B | C, D) = P(A | C, D)P(B | A, C, D). 

(ii) P(A, B) = P(A, B | C, D)P(C, D). 

(iii) P(A, B | C, D) = P(A, B)P(C, D)P(C, D | A, B). 

(iv) P(A, B | C, D) = P(A, B)P(D)P(C, D | A, B).