"No, that is false. It would be like flipping a coin and seeing heads, then claiming that this outcome increases the probability that heads will be the preferred outcome in future tosses, or tosses in general. Think about what you're saying."

There's nothing wrong with that. If you flip enough straight heads, at some point you decide there's something going on with the coin - even if you're not sure what.

"Without an explanation, it's a coincidence."

If a coincidence is sufficiently unlikely, the chance that it's not simply a coincidence becomes more likely. Just because I don't know what's stripping entropy from the system doesn't mean I can't conclude that something almost certainly is. Let's say that you and I set up an experiment where we flip coins together - each 1000 flips - and we compare our flips. We find that we've both flipped a lot of heads and a lot of tails, but with no particular observable pattern - except that we both got the same sequence. 1000 flips, and the two coins matched every time.

You don't immediately have an explanation (do you?), but you'd set about looking for one right? Why would it be crazy to start looking for some interaction between the coins before observing those 1000 flips, and make sense after? Because the existence of some causal relationship just became 1000 bits more likely.

"Try to remember that science is not simply drawing conclusions on the basis of observations -- there's more to it than that. It's not science until an explanation (a theory) is proposed, is tested, and has resisted falsification."

You keep going back to this, but my goal isn't "TO DO SCIENCE", it's to draw correct conclusions about the world so that I can make my way around it. If the particular boundaries you've drawn around Science(TM) help with that, then great. If not, then I don't have use for your "Science" - or at least I have a lot of use for things that are not Science and your condemnation of them for simply "not being science" is misleading and harmful and silly.

"What if I told you that someone correctly predicted the direction of the stock market for 20 months straight, all in advance? Does that prove he's a stock market genius? Here's a better explanation, one that a scientist would want to test first -- if 1,048,576 (i.e. 2^20) people try to predict the stock market's direction for 20 months straight by flipping coins, there is a better than even chance that one of them will succeed by chance alone."

No it doesn't "prove that he is a stock market genius" - but it is more likely that he is a stock market genius when you've observed that then when you have not, because P(20 consecutive predictions | he is a genius) > P(20 consecutive predictions | he is not a genius), and that's true regardless of how many people were picking. Given the pool you're drawing from, it remains spectacularly unlikely. You keep insisting that my "we end up thinking X is more likely than we started thinking it was" means "we think X is more likely than alternatives" in order to draw absurd conclusions. Of course that will give you absurd conclusions, it's an absurd position - but it's never been what I've said.

"Next question -- what's the chance that the one random winner will assume that his success resulted from chance, or will he start selling a worthless book called "Secrets of the Winners" on late-night TV?"

Yes, the high probability of that and the low probability of actually being a "stock market genius" means those books are likely to be worthless. That's true whether or not what I say is true.