question.md

This exercise investigates the way in which conditional independence relationships affect the amount of information needed for probabilistic calculations.

1. Suppose we wish to calculate $P(h{{,|,}}e_1,e_2)$ and we have no conditional independence information. Which of the following sets of numbers are sufficient for the calculation?
   

   1. ${\textbf{P}}(E_1,E_2)$, ${\textbf{P}}(H)$, ${\textbf{P}}(E_1{{,|,}}H)$, ${\textbf{P}}(E_2{{,|,}}H)$

   2. ${\textbf{P}}(E_1,E_2)$, ${\textbf{P}}(H)$, ${\textbf{P}}(E_1,E_2{{,|,}}H)$
      

   3. ${\textbf{P}}(H)$, ${\textbf{P}}(E_1{{,|,}}H)$, ${\textbf{P}}(E_2{{,|,}}H)$
      

2. Suppose we know that ${\textbf{P}}(E_1{{,|,}}H,E_2)={\textbf{P}}(E_1{{,|,}}H)$ for all values of $H$, $E_1$, $E_2$. Now which of the three sets are sufficient?