I feel like HD remake of PS2 titles is kind of done. But if PS4 was BC with PS2 Sony could put basically the whole PS2 library (or at least the good games) on PSN and right there you have a reason to buy a PS4 to replace your PS3 even if the actual PS4 library doesn't have much to offer you.

Sure with PS Now you will be able to stream some PS2 games without having to have built in BC. But it takes time and effor to convert a game to play via internet streaming, I assume, and how long will it be before a substantial number of PS2 games are on PS Now? It takes bugger all effort to plonk a game on PSN for normal old downloading. And then you have the people who still have a PS2 collection who like to play from time to time, and PS2 BC on PS4 would mean only needing 2 consoles in the house to be able to play all 4 generations of Playstation home console games.

PS4 must have sufficient processing power to be able to emulate PS2 without any problem.

Because it takes time and money to write an emulator, if Sony thought there would be an significant increase in hardware sales if they offered free BC for disk based PS2 games they would, but there isnt so they wont.

They will bring you PSNow and rent you the entire back catalogue for a monthly fee, where there is lots of money to be made.

I think they just want to make money from remakes, and save time and money and not develop an emulator that works properly.

Anyway whats the point playing PS2 games on PS4.
If you do that, youre going to end up playing PS4 games on PS6 and PS6 games on PS8.
Just cut the losses and stay with the times.

Because they have a service to sell you.

What? It's a massive improvement over Cell.
Then again, Jaguar is still horrible, but it's still much better than Cell ever was, but that's to be expected, consoles haven't taken CPU performance seriously for a very very long time.

The reason for the lack of backwards compatability though is one of hardware, the Playstation 2 is radically different than the Playstation 4 in many aspects that makes a hardware based solution simply unfeasible.
Sony tried the hardware approach with the early Playstation 3's, the extra processors required drove up costs, it would have been possible if the Playstation 4 was an architectural evolution of the Playstation 2, instead the Playstation 4 is more like a cousin of a Mid-range PC.

However, the Playstation 4 is plenty powerfull enough to offer a fully software based approach and would even be able to employ techniques to improve image quality. (I.E. Upscale to 4k, then downsample to 1080P is one option, then apply some Anti-Aliasing and texture filtering and some post-processing effects.)
The reason why it would work is one of low level hardware understanding by Sony, emulators on the PC work by making the PC "Act" like it's a Playstation 2, some calls/commands require multiple passes to perform the same function that would be done in a single pass on a console, hence why on a PC you need a massively more powerfull system to pull it off.

However, Sony is a for-profit company, they are going to be more inclined to offer a paid approach via the cloud and lock you into the platform than a free one, it's not going to be ideal, but those are the cards that were dealt.

They're not going to spend time and money creating a PS1 emulator for PS4 when they're offering backwards compatibility through Gaikai instead.

The PS1 uses a MIPS CPU which is RISC based ( Reduced instruction set computing )

The PS2 uses a newer version of the PS1 so full BC is easy.

The PS3 used a IBM POWER CELL CPU which is also RISC based as so was capable of dealing with the much older PS1 but not the PS2 BC.

PS4 uses an AMD X86 with X64 extension CISC ( complex instruction set computer ) CPU that in some opertations is a lot slower than a CELL ( the best Intel $1000 CPU's are slower than CELL in some operations ) and as such can not give BC for any previous RISC based console.

For another reason...PS1 games use CD's not DVD's i believe....the reader itself might not even be able to read disc based ps1 games since they took audio CD playback out of the PS4.

I think it's because of the very low latency of the 4MB EDRAM of the PS2. Very hard to emulate that via software using regular XDR ram or GDRR5 ram (even DDR3 ram couldn't do it).

That's why only added hardware on the first PS3 models, or partial hardware on later PS3 models, could only perfectly emulate 99% of PS2 games.

The PS2 games on PSN have been reprogrammed, bypassing the very fast EDRAM vram calls.

It would be possible in theory to use the very low latency GPU caches of the PS4 to emulate the 4MB Vram of the PS2, I dunno.

Prove it, with actuall benchmarks.