Pairwise comparisons with confidence There is a lot of information available on the subject of Pairwise Comparisons but I haven't found any guidance on how to optimize pair measurements that have confidence values attached to them.

Imagine that an AI black box has been trained on many pairs of objects in order to predict the distance between the objects in novel pairs. All objects are colinear. The black box also provides a confidence level for each distance. Here is a simple example of outputs based on a set of 5 objects ($a$ to $e$):

Pair, Distance, Confidence

a-b,  5,        0.4
a-c,  9,        0.7
a-d,  -3,       0.9
a-e,  2,        0.6
b-c,  6,        0.8
b-d,  -10,      0.7
b-e,  -2,       1.0
c-d,  -6,       0.5
c-e,  -2,       0.6
d-e,  7,        0.9


So, the distance between $a$ and $b$ is predicted to be 5 units ($b$ is to the right of $a$) and the black box's confidence in that measurement is 0.4 (0.0 = no confidence and 1.0 = perfect confidence). The distance between $a$ and $d$ is -3 units ($d$ is to the left of $a$) and the confidence is 0.9.

I would like to understand the process for deriving the solution, which is the optimization of the four distances between these objects given the predicted distances and their associated confidence levels.
There are several possible ways to solve this problem. One way is to assume some measurement model (usually Gaussian noise) and then find the maximum likelihood solution. 

That is, we start with some distribution of the measured distance given the actual distance  and the noise level, for example if the actual distance is $d$ and the noise level is $\sigma$, then perhaps the measured distance is distributed $N(d,\sigma)$.

Given a sequence of measurements, for each potential list of positions (say with $a=0$ to break the symmetry), we can calculate the likelihood that we would get the actual measurements. In your case, we would get
$$ L(a,b,c,d,e) = \frac{1}{\sqrt{2\pi} 0.4} e^{-(5-(b-a))^2/(2\cdot 0.4)} \cdot \frac{1}{\sqrt{2\pi} 0.7} e^{-(9-(c-a))^2/(2\cdot 0.7)} \cdots. $$
The idea is to find the setting of $a,b,c,d,e$ that maximizes $L(a,b,c,d,e)$. To that end, we can calculate
$$ -\log L(a,b,c,d,e) = C + \frac{(5-(b-a))^2}{2\cdot 0.4} + \frac{(9-(c-a))^2}{2\cdot 0.7} + \cdots.$$
We want to minimize the latter. We can substitute $a = 0$, and then differentiate with respect to each of the other variables, and equate to zero. For example, differentiating with respect to $b$, we obtain
$$ 0 = -\frac{\partial \log L}{\partial b} = \frac{b-5}{0.4} + \frac{b-c+6}{0.8} + \frac{b-d-10}{0.7} + \frac{b-e-2}{1.0}. $$
In this way, we get linear equations which we can solve for $b,c,d,e$. (This is the method of least squares.)

For other models of noise, we might have to use numerical optimization instead. You can find more examples in any machine learning textbook.