420 8. GRAPHICAL MODELS
Figure 8.54 Example of a graphical model used to explore the con-
ditional independence properties of the head-to-head
patha–c–bwhen a descendant ofc, namely the node
d, is observed.
c
ab
d
8.10 ( ) Consider the directed graph shown in Figure 8.54 in which none of the variables
is observed. Show thata⊥⊥b|∅. Suppose we now observe the variabled. Show
that in generala⊥ ⊥b|d.
8.11 ( ) Consider the example of the car fuel system shown in Figure 8.21, and suppose
that instead of observing the state of the fuel gaugeGdirectly, the gauge is seen by
the driverDwho reports to us the reading on the gauge. This report is either that the
gauge shows fullD=1or that it shows emptyD=0. Our driver is a bit unreliable,
as expressed through the following probabilities
p(D=1|G=1) = 0. 9 (8.105)
p(D=0|G=0) = 0. 9. (8.106)
Suppose that the driver tells us that the fuel gauge shows empty, in other words
that we observeD=0. Evaluate the probability that the tank is empty given only
this observation. Similarly, evaluate the corresponding probability given also the
observation that the battery is flat, and note that this second probability is lower.
Discuss the intuition behind this result, and relate the result to Figure 8.54.
8.12 ( ) www Show that there are 2 M(M−1)/^2 distinct undirected graphs over a set of
Mdistinct random variables. Draw the 8 possibilities for the case ofM=3.
8.13 ( ) Consider the use of iterated conditional modes (ICM) to minimize the energy
function given by (8.42). Write down an expression for the difference in the values
of the energy associated with the two states of a particular variablexj, with all other
variables held fixed, and show that it depends only on quantities that are local toxj
in the graph.
8.14 ( ) Consider a particular case of the energy function given by (8.42) in which the
coefficientsβ=h=0. Show that the most probable configuration of the latent
variables is given byxi=yifor alli.
8.15 ( ) www Show that the joint distributionp(xn− 1 ,xn)for two neighbouring
nodes in the graph shown in Figure 8.38 is given by an expression of the form (8.58).
