This will be an OVERLY SIMPLISTIC comparison. I am in no way trying to be scientific here, just trying to give a ROUGH IDEA about the UNOPTIMIZED power of PS4, which is quite relevant with the FIRST GEN THIRD PARTY TITLES.

Now, theoretically PS4 is 10x PS3 in terms of theoretical horse power. Let's take an average 3rd party PS3 game running at 640p & 30fps.

Bump up the resolution to 1080p : 3x theoretical performance cost

Bump up the frame rate to 60fps : 2x theoretical performance cost

Enable Anti-aliasing : about 1.3X - 1.5X theoretical performance cost

That totals to 8X theoretical performance cost

You are still left with 20-30% performance to add additional detail and effects. In short, the EASIEST WAY to maximize the PS4 over a PS3 port is

PS3 original  : 640p, 30fps, No AA, medium quality settings

PS4 port : 1080p, 60 fps, 4x AA, high quality setting  (10x performance cost)

That is of course not the OPTIMUM performance, there probably would be much better trade offf choices as well as platform specific enhancement, which are mostly irrelevant for 1st gen 3rd party titles. You can roughly get twice the graphics detail at 30 fps for example, this alone will wow you, but at the cost of higher development spending (likely to be materialized in 1st party exclusives).

wasn't it like 4-5 times more powerful?

It's been several times quoted to be 10X more powerful by Mark Cerny, which sounds very reasonable and reliable. XB1, on the other hand is quoted to be about 8 times the power of X360, but it's more like 6-7 times due to the bottlenecks and memory related issues. Especially running 2 Virtual Machines in XB1 will cost them dearly!

As an anecdotal evidence from Forza 5, which runs in 1080p & 60 fps;

XB1 has at least 5-6X the power of X360 hence the following:

720p to 1080p : 2.25X
30 fps to 60 fps : 2X
Totals to : 5.5X

We don't quite see much of an improvement in graphic fidelity otherwise. This, being a first party title, 6X unoptimized first gen theoretical performance increase seems adequate.

I suspect the increase might be higher, around 8X but it's the system bottlenecks, the virtual machines and the reserved CPU/GPU cycles that are keeping the theoretical performance down. So 6x performance increase may be more realistic.

Ehmm... no. Really, I don't know where to start but first of all: Not everything in computation scales linearly. An appropriate answer would take so many lines of text I don't think it's worth it.
Let's keep it at no.

I am aware that not everything scales linearly. Just like pushing 2.25x times many pixel does not necessarily require 2.25x horsepower. But also it's true that 10x theoretical performance does not necessarily give this kind of performance in real life. Things are oversimpliefied as I stated initially. So they are not exactly accurate in "absolute" terms but "relatively" they seem to make sense. Just like the performance differential between XB1 (6x) and PS4 (10x) seems to make perfect sense even if it is rough on the edges.

Sometimes, it's best to simplify things, a phenomenon that I run into, each and everyday. When you try to address every little bit of detail,

a) It costs you to too much time and effort

b) Things tend to cancel each other out, and the end results don't drastically differ from the simplified ones

c) Unnecessary detail leads to unbalancing issues; where you represent one side precisely and accurately while other unaddressed points are left uncovered or oversimplified, which turns out to yield worse results...

I am an economics professor, and I can tell you wholeheartedly that the best economic models are usually the simplest yet intiutive ones. The more complicated ones tend to produce either flat-out worse results or the benefits associated with them are so tiny that it's not worth the extra cost and effort. This usually applies in COMPARATIVE ANALYSIS though, rather than ABSOLUTE PRECISION.

But your simplication is just wrong. This doesn't translate nearly to actual performance because of so many other facts (10x is a strange number and we are not told in which way it was measured) like programming, dev-tools, other chips involved, improvement of algorithms, unknown bottlenecks and so on and this is without just arguing the scaling not being linear.

But as I see you are coming from economics: please go on with this No offense but it made me laugh

I understand your arguments, and you may be right. We'll just see how it unveils in the next coming years. I believe my simplifications are quite right and logical, based on my past experience with the game console, PC hardware, and the history of those companies BUT I DON'T HAVE the conclusive evidence to have the opposite claim against what you said.

Feel free to share with me evidence on EITHER side; I am open to counter-arguments as well, based on some illustration etc, not just "opinions".

youre right in saying its over simplistic....even I could understand it....which must mean something has gone wrong.