# Supervised learning with error-range in labels?

I am working in a problem where labels have an error range (we know the range). For instance, a label can be expressed as $$y_i \pm e_i$$ with $$e_i$$ is the error range for the label of the instance $$i^{th}$$.

There are two problems with that:

1. Regression: If I want to infer the real value of $$y$$, how should I integrate the knowledge of error range into a regression model, say linear regression?

2. Classification: If I want to convert the problem to a classification problem by grouping similar $$y$$ values to a class (e.g. group age 11-19, 20-29, etc.) and I have a person with $$age = 20 \pm 2$$ (well, it is stupid, but just assume that we don't know for sure the age of a person), what class I should assume for this person?

I stuck at both. Many thanks for any hints.

## 1 Answer

1. Yes, you probably should and then the likelihood contribution of each observation is a normal (It's some other distribution that corresponds to how these errors are provided) likelihood with the corresponding mean and scale parameter equal to the error. Alternatively, you can (e.g. for a neutral network) use a loss function that penalizes accordingly.
2. For classification you could just calculate to what probability of being in each class each observation corresponds. Then you feed into your classification procedure what you get such as (0.05, 0.25, 0.4, 0.3) - or perhaps (0, 0.01, 0.98, 0.01) for a more precise observation - for the probability that one observation is in each of 4 classes.