This
assignment will give you practice in determining conditional
independence and performing exact inference.
Consider
the Bayes net with 5 binary random variables. (Thank you
to Pedro Domingos of U. Washington for the example.)
1. What
is the Markov blankot of Sneeze?
2. What is the Markov blanket of Take Medicine?
3. Write the expression for the joint probability, P(L, C, RN, S, TM),
in terms of the conditional probability distributions.
4. Moralize the above graph to obtain an equivalent Markov net.
5. Write the joint probability function for the Markov net, with one
term per maximal clique. Show the correspondence between the potential
functions in this equation with the conditional probability
distributions in question 3.
6. Is this graph a polytree? If not, what links might you add
or
remove to make it into a polytree? (A polytree is a directed
graph with no undirected
cycles; it is a generalization of a tree that allows for a node to have
multiple parents.)
7. Which of the following are true:
(a) C ? TM | RN, S
(b) TM ? C | S
(c) C ? L
(d) C ? L | TM
(e) RN ? L | TM
(f) RN ? L
(g) RN ? L | S
(h) RN ? L | C, S
8. Compute P(C=1 | TM=1, RN=0, L=0). Note that this conditional can be
expressed as the ratio of P(C=1, TM=1, RN=0, L=0) and P(TM=1,
RN=0, L=0).
1. Write the joint distribution over A, D, U, H, and P in this
graphical model (from Barber, Fig 3.14). Ignore the shading of H and A.
2. Write out an expression for P(H) as a summation over nuisance
variables in a manner that would be appropriate for efficient variable
elimination. (Don't do any numerical
computation; simply write the expression
in terms of the conditional probabilities and summations over the
nuisance variables. Arrange the summations as we did in the
variable elimination examples to be efficient in computing partial
results.)
3. Write out a simplified expression for P(U=u | D=d) in a manner
that would be
appropriate for variable elimination. By 'simplified expression', I
mean to reduce the expression to the simplest possible form,
eliminating constants and unnecessary terms.