It looks as though you’ll need to re-buy your old PS2 games, as the video displayed the notice “PS2 Classics and PS2-formatted discs are incompatible with the PS4 system.” ‘PS2 Classics’ was the label Sony gave to PS2 games digitally re-released on the PS3, so if you’ve already bought a non-physical version of a game, you could be looking at triple-dipping on the PS4 version.
When Sony stealthily introduced PS2 backwards compatibility, it was initially only for a trio of Star Wars games to coincide with the new Star Wars Battlefront. Soon after though, it asked fans on Twitter which games they wanted to see made available, effectively confirming the emulation process would be expanded to other titles.
Expect more information on backwards compatibility this weekend, as Sony is holding its PlayStation Experience event in San Francisco.