1. Suppose we wish to calculate $P(he_1,e_2)$ and we have no conditional independence information. Which of the following sets of numbers are sufficient for the calculation?

1. ${\textbf{P}}(E_1,E_2)$, ${\textbf{P}}(H)$, ${\textbf{P}}(E_1H)$, ${\textbf{P}}(E_2H)$ 2. ${\textbf{P}}(E_1,E_2)$, ${\textbf{P}}(H)$, ${\textbf{P}}(E_1,E_2H)$

3. ${\textbf{P}}(H)$, ${\textbf{P}}(E_1H)$, ${\textbf{P}}(E_2H)$

2. Suppose we know that ${\textbf{P}}(E_1H,E_2)={\textbf{P}}(E_1H)$ for all values of $H$, $E_1$, $E_2$. Now which of the three sets are sufficient?

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 $P(he_1,e_2)$ and we have no
conditional independence information. Which of the following sets of
numbers are sufficient for the calculation?

1. ${\textbf{P}}(E_1,E_2)$, ${\textbf{P}}(H)$,
${\textbf{P}}(E_1H)$,
${\textbf{P}}(E_2H)$
2. ${\textbf{P}}(E_1,E_2)$, ${\textbf{P}}(H)$,
${\textbf{P}}(E_1,E_2H)$

3. ${\textbf{P}}(H)$,
${\textbf{P}}(E_1H)$,
${\textbf{P}}(E_2H)$

2. Suppose we know that
${\textbf{P}}(E_1H,E_2)={\textbf{P}}(E_1H)$
for all values of $H$, $E_1$, $E_2$. Now which of the three sets are
sufficient?