Summary: Hierarchical Classification Problem Hierarchical classification labels involves hierarchical class labels. The hierarchical class labels maybe predefined or inferred. 1
Class Taxonomy A hierarchical classification problem comes with a class taxonomy.
“IS-A” operator: $\prec$, “IS-NOT-A” operator: $\nprec$ A IS-A relationship of the labels $c_a$ class set $C$ is
one root $R$ in the tree, asymmetric, i.e., $c_i \prec c_j$ and $c_j\prec c_i$ can not be both true, anti-reflexive, i.e., $c_i \nprec c_i$, transitive, i.e., $c_i \prec c_j$ and $c_j\prec c_k$ $\Rightarrow$ $c_i \prec c_k$. There are different representations of the hierarchical taxonomies.
Figure 2 in Silla2011, showing the difference between tree taxonomy and DAG taxonomy.

Summary: Ask for valid confidence:
“Valid”: validate for test data, train data, or the generating process? “Confidence”: $P(Y \notin C(X)) \le \alpha$ To avoid too much attention on data based validation, a framework called conformal inference was proposed by Vovk et al. in 2005,
$n$ observations, desired confidence level $1-\alpha$, construct confidence sets $C(x)$ using conform methods so that the sets capture the underlying the distribution a new pair $(X_{n+1}, Y_{n+1})$ from the same distribution, $P(Y_{n+1}\in C(X_{n+1})) \le 1-\alpha$