Okay, I believe you. I was honestly confused, so once again, I’m sorry about that detour. You really do have a legitimate PhD, studying under one of the leading lights in the ID movement. What you are putting forward here deserves serious attention. I’m glad we have clarified this.
The EricMH Information Argument
I pressed you to define your terms and produce a simulation because this statement is not remotely specified enough to understand and analyze.
Nonetheless, how this is formulated, your entire argument depends on this equation: I(X:Y) \geq I(E(X):Y). If this is true, we still have work to do to see if it is relevant to evolution or not. However, if it is false, we are done. There is no further reason to engage in this argument. It has been falsified. Whether or not its true depends on exaclty what you mean by these poorly defined terms.
Perhaps you were abbreviating a well considered argument, and would stand and deliver. Or perhaps not. I cannot tell from that paragraph. I interpolated some of your meaning, guessing what you meant, and antipicpated…
The reason I needed a simulation is to just get a basic idea of what you were even talking about. It is not clear, from how you write, what it is you mean. That is why I asked you a battery of questions.
The Simulation
You did not really answer my questions. What you did was far more surprising. Your simulation actually falsified your own equation. This at least makes clear what you were thinking. It also shows your equation is wrong in that you have not articulated or understood its limits. If you had, you would never have written the simulation this way.
So took your code and modified it to run 1000 times, and report how often the main experiment, and the control did or did not violate your equation. This is a test of (1) the equation itself and (2) your understanding of the equation. https://repl.it/repls/DarkorangeSomberCopycat
Here are the results from one run:
main experiment equation holds 512 out of 1000 trials
control experiment equation holds 700 out of 1000 trials
First off, you have the main experiment, which is supposed follow your equation. In this, you pick an X and Y, using a particular E. Then you compute I(E(X):Y). You test to see if this is greater or less than I(X:Y). Here is the thing, ~50% of the time it is greater, and ~50% of the time it is less than. That is exactly what we expect for your choice of E, X, and Y. That is the definition of falsifying your primary equation.
What ever you think it means, you are wrong. What is remarkable, however, is that instead of collecting this data and understanding it, you write it off as “stochasticity.” I’m not going to do the work for you, but you’ll find that the deviations from equal are of equal magnitude in each direction. The key issue is that you do not know when this equation applies and when it does not.
I can tell you exactly what you did wrong here. In fact, looking at your code, I knew exactly what would happen before I ran the experiment. I know what I am doing here. That is all beside the point though. Your own simulation demonstrates there is an error in your understanding. That is all we need to know for this conversation.
The Control Experiment
Second, you have a positive control, where you set Y = X. At no point do you describe what you expect, so we have to infer. This is a degenerate case, and in this case the two sides of the question should be equal.
However, what we find is that about 70% of the time your equation is violated. Why?
The reason why is because you are only computing an approximation of the mutual information that has a tiny positive bias (in this case). It turns out that it is impossible to exactly compute mutual information content. Once again this is exactly what i would have expected in this scenario.
Why Uncomputability Matters
Remember our exchange on this?
Now you are seeing why it matters. If you can’t keep this straight you will end up totally misunderstanding what your proofs mean. This one of the most important results in information theory, and must be thoroughly grasped to apply it correctly. This is why your control failed worse than your main experiment.
Other Controls?
This is just one control experiment. Where are the rest? Typically you want to have a large battery of controls to test every edge case, and ensure you can precisely demarcate the domain of applicability of a proof like this. Any surprise indicates lack of understanding. Every surprise sends us back to the drawing board to understand what we missed from first principles. Mathematical proofs in information theory can be very difficult to get right.
They are very subtle ways they can go wrong. The same is true in population genetics, and in many areas of computational biology. That is why simulation is such a fundamental part of the field. We do not trust our intuitions. We do not trust our proofs. We verify them. We understand them. We attempt to falsify them as many ways as possible, before we advance them with any confidence.
What all This Means
It means you are back to the drawing board. I was not bluffing when I said your argument was not clearly specified. I couldn’t tell what you were saying. Based on some definitions you might be right on some claim or another. On others, not.
I’m concerned, however, that even on having the simulation in front of you that you did not even realize you had made an error.
At this point, it seems we are done @EricMH. You have a lot of work to do if you here. Next time, please take my questions seriously. Make sure you’ve done your home work too. You are obviously a good computer programmer. Test your claims before you make them publicly. This this post really important read (including the linked philosophy entry): The Role of Simulation in Science. Always verify your proofs with simulation, so you can carefully understand what you are really learning.
Good luck, and send my kind regards to Marks and @Winston_Ewert.