Given the symmetric space $D$ of non-compact type for the real semi-simple Lie group $G$, there are $2^r-1$ Satake compactifications $\bar{D}$ for $D$ up to homeomorphism, where $r$ is the real rank of $G$.

These compactifications correspond to the non-empty subsets of a set $S$ of simple roots, and as such they form a semi-lattice: if $S_1 \subset S_2$, then the identity of $D$ extends to a continuous mapping $\bar{D}_{S_2} \to \bar{D}_{S_1}$.

The *Satake-Baily–Borel compactification* is one of the minimal (in the semi-lattice sense) Satake compactifications in the case where $D$ is Hermitian, i.e., has a $G$-invariant complex structure, see this entry in the Encyclopedia of Mathematics.

According to [Faltings-Chai, *Degeneration of Abelian Varieties*, p. 136], the construction of the minimal compactification of $\mathcal{A}_g$ mimics the construction of the the Satake-Baily-Borel compactification of a symmetric space, and this is why the word "minimal" is used.