# Probability distribution over classes as labels in classification task

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 $\sum_{i=1}^{k}p_i = 1, \forall i : p_i \geq 0$.

• 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. – Sycorax Apr 25 '16 at 13:16