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Let us say I have labelled training data. P predictors, C classes, N exemplars. I also have one additional dimension (not calling it a predictor for now) which is the confidence of the label in the Class column.

Now what if I have some additional information regarding the confidence of the class labels. Perhaps I had more agreement by curators on the class label of a particular exemplar than another.

Is there a way to take advantage of this information?

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Sure, but it depends on what kind of classifier you're using.

For most classifiers, it's relatively easy to add an instance weight to each example. For example, you might minimize $$\sum_{i=1}^n w_i \ell(f(x_i), y_i),$$ where $\ell$ is the loss function for each data point, and $w_i$ is some relative weight you've come up with to put more emphasis on the points whose labels you're more confident of.

You might also treat it probabilistically: assuming that a label is $1$ with probability $p_i$ and $0$ with probability $1-p_i$, and that these labels are independent of one another, you could minimize the expected loss as $$\sum_{i=1}^n p_i \ell(f(x_i), 1) + (1-p_i) \ell(f(x_i), 0).$$

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