Distance between Vectors with Confidence Intervals I have a machine learning application where I extract numerical features $a_{i1}, a_{i2}, \dots, a_{ik}$ for each object $a_i$ to study. Objects are then compared using standard euclidean distance. 
The problem is that the features entail uncertainty. The good message is that I have confidence intervals, meaning that I can tell with probability $\alpha$ that $a_{ij}-c_{ij} \le a_{ij} \le a_{ij}+c_{ij}$. Since these confidence intervals are independent of each other, I end up with k-dimensional boxes instead of k-dimensional points (see below).
My question is whether there's a standard approach to extend euclidean distance to account for uncertainty. The right way to go is probably using standard euclidean distance, and deriving new confidence intervals for that distance. Maybe it wouldn't even be too hard to derive them, but I'm also interested in a paper that I could cite.

 A: Bayesian machine learning relies on probability distribution to represent uncertainty. In the present case, using multivariate normal distributions instead of finite boxes may lead to simpler calculations. 
Currently, you assme that the interval $a_{ij}−c_{ij} \leq a_{ij} \leq a_{ij}+c_{ij}$ 
contains an amount $\alpha$ of the uncertainty mass. This can me converted to a normal distribution by using the z-table. Basically, it consists in finding the standard deviation $\sigma$ of a normal distribution such that the following equality holds:$$\alpha = \int_{-c_{ij}}^{c_{ij}}\mathcal{N}(0, \sigma^2)$$
This change in uncertainty representation turns the problem into computing the Euclidiean distance between 2 multivariates normal distributions. Kettani and Ostrouchov (2005) computed the distribution over the distance resulting from this operation under various hypothesis regarding 2 multivariate normal random variables. 
Kettani, H., & Ostrouchov, G. (2005). On the distribution of the distance between two multivariate normally distributed points. Department of Computer Science and Information Systems Engineering, Fort Hays State University, Fort Hays (KS).
A: A similar problem has been framed in M. Kumar, N. R. Patel, and J. Woo, “Clustering seasonality patterns in the presence of errors,” KDD '02: Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining. 
From the Abstract:

[...] We present a new distance function that is based on the distribution of errors in data. Using a Gaussian model for errors, the distance function follows a Chi-Square distribution and is easy to compute. [...] The distance function is scale-invariant so that clustering results are independent of units of measuring data. In the special case when the error distribution is the same for each attribute of data points, the rank order of pairwise distances is the same for our distance function and the Euclidean distance function.

