I have read some materials on Hodge structures and all of them state the equivalence of definitions from the following perspectives: (suppose that we are considering a Hodge structure of weight $n\in \mathbb{Z})$
- Hodge decomposition: $$V_\mathbb{C}=\bigoplus_{p+q=n}V^{p,q}\quad \text{with}\quad V^{p,q}=\overline{V^{q,p}},$$
- Hodge filtration: $$V_\mathbb{C}\supset\cdots\supset F^{p-1}\supset F^p\supset\cdots\quad\text{such that}\quad V_\mathbb{C}=F^p\oplus\overline{F^{q+1}}\quad\text{and}\quad F^p\cap \overline{F^{q+1}}=0 \text{ for }p+q=n.$$
In order to show the equivalence we need $F^p=\bigoplus_{i\geq p}V^{i,n-i}.$ This is clear if we know the Hodge decomposition(i.e. by definition). But if we know the Hodge filtration how can we see this equality?
Hint: Show that $\mathrm{Gr}^p_FV_\mathbb{C}:=F^p/F^{p+1}\cong V_\mathbb{C}^{p,q}:=F^p\cap\overline{F^q}.$
We may consider the map $\varphi: F^p/F^{p+1}\rightarrow F^p\cap\overline{F^q}$ which is a well-defined linear transformation. Being an isomorphism is directly from the condition $F^p\cap \overline{F^{q+1}}=0$.