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Classical classification problem has next formulation. Given a set of $n$ attributes, a set of $k$ classes and a set of labelled training instances: $(i_i, l_j),...,(i_j, l_j)$, where $ i = (v_1, v_2, ..., v_n)$ and $ l \in \{c_1, ..., c_k \}$

We want to determine a classification rule, that predicts the class of any instance from the values of its attributes.

Ok, in this case we must have training set, where every instance is labelled with precision class. We could annotate a set of examples in a "cheap way": not 100% correct every time, but with low cost.

So, I'd like to consider next task: we have a set of $n$ attributes, a set of $k$ classes and a set of labelled training instances: $(i_i, l_j),...,(i_j, l_j)$, where $i = (v_1, v_2, ..., v_n) , l = (p_1, p_2, ..., p_k)$

$p_i (i \in \{1,..,k\}) $ - probability that instance belongs to class $c_i$ and $\sum_{i=1}^{k}p_i = 1, \forall i : p_i \geq 0$.

Unfortunately, i can't found anything about this task. Maybe anybody heard anything?

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    $\begingroup$ you probably additionally mean that $p_i\ge 0\forall i$! But note that classical formulae like log-likelihood may be naturally extended in this way, so augmenting the objective functions of these methods immediately yields the desired result. $\endgroup$ – Sycorax Apr 25 '16 at 13:16
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    $\begingroup$ Logistic regression outputs the predicted probability that an instance is a member of class A, if that's what you're asking. $\endgroup$ – gung Apr 25 '16 at 13:25
  • $\begingroup$ I'd like to increase classification precision, if I'll use probability distribution over classes in training set (instead just class labels). $\endgroup$ – Simplex Apr 25 '16 at 13:28
  • $\begingroup$ @Simplex I'm not convinced that this will increase class precision. But it will average over your uncertainty, so that will be "baked in" to the estimation procedure and the weak information of uncertain classes will be reflected in the variance of the model estimates. $\endgroup$ – Sycorax Apr 25 '16 at 13:34

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