You claimed that Popper's theory conflicted with Bayesian probability. That was an error, on your part.
First of all, I have not claimed this. Instead, I have claimed that I personally follow, in large parts, Popperian ideas, but also prefer the Bayesian probability interpretation. Which I would not do if I would see a conflict between those parts of Popper I accept and Bayesian probability.
Then, Popper himself has explicitly rejected Bayesian probability, and proposed his own propensity interpretation of probability. This is wiki level information, see
https://en.wikipedia.org/wiki/Propensity_probability I have seen nothing from your side, no argument about this. So, as usual, only an unjustified claim.
In this situation, I would ask you, if you make a claim that I have made an error, to quote this error.
Well, that's what your broker will tell you - in the fine print, you will find a disclaimer along the lines of "you can lose all your money, and it's not our fault".
Who cares about disclaimers people are obliged to make if they don't want to be faced by the absurd American law system to pay huge penalties for the stupidity of their customers?
But the point is: This investment is not hidden, and it's not taken as income by the rich guy. Allow me to repeat: Not hidden. Not income. Rich guy can't buy a house, or pay for his children's college, with that money. It appears on the corporation's books, in excruciating detail available on demand for government audit, and summaries for the stockholder's report are produced every year by professional accounting firms.
First, I would not be so sure about what the rich guys can't do. To pay for the college of the children is, I would guess, an easy exercise. Simply because education the workers of the firm has always been part of what firms do. And who cares if the house is officially owned by the guy himself or by the family corporation (which makes it easier to minimize inheritance taxes, btw)?
It is quite funny to observe how you defend the rich here. A contradiction with your left position? Hm. But, no, it is not. What you defend here is the existing tax system, as being able to provide justice by progressive income taxation, if only the correct good guy is elected. So you have to present it as very powerful, knowing everything ("not hidden") and being just ("not income") at least in some parts I have criticized.
And you honestly don't know, do you. You have no idea how someone would do that.
Maybe, who knows? You would be able to show that I'm wrong and stupid by explaining here, with quotes, how Piketty does the job. For some unknown reason, you don't use this possibility. I guess, because it does not exist in reality.
But, no problem, let's admit that I have no idea how Piketty could find out, with sufficient certainty, the wealth distribution in society. What I have seen from Piketty is a nice try in plausible reasoning, essentially a quite good one, given the large problems with all the available data.
He has better and more comprehensive methods - two or three chapters worth - but the obvious and easy way is to read the financial documents filed by the corporations involved, or if you're feeling lazy and just need a quick answer, the market capitalization reported in the newspaper as the stock price, and add them up. Is this mysterious to you?
No, this is not a problem at all. The problem is how to find out how much of the stock is owned the poor, the middle class, the rich, and the superrich. Ok, the poor usually do not own anything - except, if they had some times a job, some pension entitlements, which are, afaik, backed in the US by pension fonds, which also own stocks. So, how to distinguish the stocks owned by the superrich from those implicitly owned by a bum who has, because of some work in the past, some pension entitlement?
Without doubt, one can make some plausibility arguments that the last part is negligible. Which is certainly not implausible. There are also some data about all these pension fonds and so on to estimate this. One can, I would guess, also find statistics about pensions, to estimate how much of the pension fonds is owned by which group of people. But for the middle class vs. upper class this is already much more problematic, because these groups own stocks themselves. And the rich and especially superrich use firms to manage (and tax-optimize) their shares. So even if, say, Trump really owns a share of, say, the Deutsche Bank, this will not be easily visible even if you know everything the Deutsche Bank has given as information to the authorities.
Your answer, by the way, has another weak point, which I have to repeat again, once you continue to ignore it: If you would have all the documents known to the IRS, you could probably find out a lot of interesting information. And there would be no problem at all about Trump not opening his income tax declaration. (Which is, of course, a very natural decision - if he is really clever as a businessman, the sum will be quite small, because he knows how to minimize it, and, therefore, in some conflict with his own claims that he is very rich.) But I would guess (feel free to correct me) that all this information is not open to the public, and, therefore, also not open to Piketty.
So, all he has is secondary information, a few statistics published by the government. Worth two or three chapters, not more.