Weirded Vampire

Madness works independently of why you're discarding the card. You could discard it to pay a cost, because a spell or ability tells you to, or because you have too many cards in your hand during your cleanup step. You can't discard a card with madness just because you want to, though.

A spell cast for its madness cost is put onto the stack like any other spell. It can be countered, copied, and so on. As it resolves, it's put onto the battlefield if it's a permanent card or into its owner's graveyard if it's an instant or sorcery card.

Cards are discarded in a Magic game only from a player's hand. Effects that put cards into a player's graveyard from anywhere else do not cause those cards to be discarded.