1. Which of these Bayesian networks are correct (but not necessarily efficient) representations of the preceding information?

2. Which is the best network? Explain.

3. Write out a conditional distribution for ${\textbf{P}}(M_1N)$, for the case where $N\{1,2,3\}$ and $M_1\{0,1,2,3,4\}$. Each entry in the conditional distribution should be expressed as a function of the parameters $e$ and/or $f$.

4. Suppose $M_11$ and $M_23$. What are the

*possible*numbers of stars if you assume no prior constraint on the values of $N$?

5. What is the

*most likely*number of stars, given these observations? Explain how to compute this, or if it is not possible to compute, explain what additional information is needed and how it would affect the result.

Two astronomers in different parts of the world
make measurements $M_1$ and $M_2$ of the number of stars $N$ in some
small region of the sky, using their telescopes. Normally, there is a
small possibility $e$ of error by up to one star in each direction. Each
telescope can also (with a much smaller probability $f$) be badly out of
focus (events $F_1$ and $F_2$), in which case the scientist will
undercount by three or more stars (or if $N$ is less than 3, fail to
detect any stars at all). Consider the three networks shown in
FigureĀ telescope-nets-figure.

1. Which of these Bayesian networks are correct (but not
necessarily efficient) representations of the preceding information?

2. Which is the best network? Explain.

3. Write out a conditional distribution for
${\textbf{P}}(M_1N)$, for the case where
$N\{1,2,3\}$ and $M_1\{0,1,2,3,4\}$. Each
entry in the conditional distribution should be expressed as a
function of the parameters $e$ and/or $f$.

4. Suppose $M_11$ and $M_23$. What are the
*possible* numbers of stars if you assume no prior
constraint on the values of $N$?

5. What is the *most likely* number of stars, given these
observations? Explain how to compute this, or if it is not possible
to compute, explain what additional information is needed and how it
would affect the result.