Suppose that we have
- k classes, $C_1$,...$C_k$.
- A classifier with precision $p_i$ and recall $c_i$, for $i=1 \dots k$.
- A sample S, such that the classifier assigns $s_i$ elements of S to class $C_i$ fo $i=1 \dots k$.
I'm not interested in which particular elements correspond to each class, but in the true number of elements that correspond to each class $r_i$.
That is, I'm interested in obtaining $r_i$ (the real number of elements in class $C_i$) from $s_i$ (the number of elements classified as $C_i$), the precision $p_i$ and the recall $r_i$.
An obvious answer would be
$$ (1) \qquad r_i = s_ip_i + (1-r_i)s_ip_i $$ that is, of the $s_i$ elements classified as $C_i$, only $s_ip_i$ are indeed in $C_i$. This number must be incremented taking into account the recall, or more precisely $(1-r_i)$, which gives the second value $(1-r_i)s_ip_i$.
I would like to know if $(1)$ is ok. It sounds strange to me, because apparentely it allows obtaining the value $r_i$ even if the precision $p_i$ and recall $r_i$ are very low, that is, we can obtain the real number of elements even if the classification method is really poor.
What is the answer to this paradox?