

![0_image_0.png](0_image_0.png)

Figure 8: Belief network for Example 22.

-x, t2(B,C, S) (39)
Suppose that we have the following confactors for T:

$\langle\overline{x},t_6(C)\rangle$. 
-x, t3(A, B,C, T) (40)
-x, t4(C, T) (41)
Suppose first that T is observed. Then the confactors for T are replaced by:
-x, t5(A, B,C) (42)
-x, t6(C) (43)
Let's now eliminate A. In both CVE and TVE, confactors (38) and (42) and the prior on A are multiplied together, and A is summed out resulting in:
-x, t7(B,C, S) (44)
In TVE, confactors (39) and (43) are also multiplied together resulting in:
-x, t8(B,C, S) (45)
Next we eliminate B. In both CVE and TVE, confactor (44) is multiplied by the factors representing P(C|B) and P(B|W). We assume that the factor representing P(B|W) is multiplied last as this minimises the number of multiplications. This involves 8008 multiplications (as the product has a table size of 8000 and the intermediate table is of size 8). Then B is summed out resulting in the factor:
-x, t9(C, S, W) (46)
In CVE, confactor (39) is multiplied by the factors representing P(C|B) and P(B|W). This also involves 8008 multiplications. Then B is summed out from the product resulting in:
-x, t10(C, S, W) (47)
In TVE, confactor (45) is multiplied by the factors representing P(C|B) and P(B|W). This also involves 8008 multiplications. Then B is summed out from the product resulting in:
-x, t11(C, S, W) (48)