Class 60: Does This Guy Have ESP? 1

A guy correctly “sees” 4 cards out of 52. A wee P hypothesis test says he has psychic powers. Does he? We only begin this difficult question today.

Video

Links: YouTube * Twitter – X * Rumble * Bitchute * Class Page * Jaynes Book * Uncertainty

HOMEWORK: Given below; see end of lecture.

Lecture

This lecture, when I make it prettier, will appear in the updated version of So, Your Think You’re Psychic?

What we want, in all science, indeed in all life:
Pr(Hypothesis true | Evidence).

This hypothesis can be any and every proposition you want to know, in and out of science. Rain tomorrow, my glasses were left in the bathroom, Ed is going to call and ask for help moving his couch, the toast has one minute to go, this new drug works, this fellow has ESP. Anything. We do this well, too, in ordinary life, without making much about it, and rarely making it formal with math. We do it differently in science.

Here is what we do in science:
Pr(Data we did not see | Hypothesis false).

Worse, when the second probability is less than the magic number—when it is wee—we believe the first probability is high. This is bad because that move is absolutely strictly verboten in the theory that gives us the second probability, but every soul who makes the calculation violates the law more forcefully than a sixteen-year-old visiting Epstein Island.

Why don’t people calculate what they want? Why do they instead calculate that weird, wrong, strange second thing? I’ll tell you: because it’s easy.

The second way removes the burden of thought and the angst of making a decision. It does it all for you in one trivial calculation (trivial because the software will do it all for you). True, the calculation too often gets it wrong, and always provides far too much certainty, but these are small prices to pay to automate science. Where results are demanded, and needed! Without results, no scientist can have a career. The second calculation produces them aplenty.

We’ll discuss that second calculation another day. For today, and next week, we instead do it the right way. I picked a subject which is easy to understand, and which sees great skepticism.

The first calculation, when formal, can be incredibly hard. The math might be easy enough, but knowing which math to use can be a brutal battle in thought, as we see in the video.

Let’s go through a deck of 52 cards, where I will try to transmit the image of each card to you, and you will try to pick it up and guess it. We want:

Pr(You have ESP | Evidence).

The total evidence will be the rules of our experiment, and what happened during it. But it will also contain what “You have ESP” means. Last week we saw that this is a far from easy question to answer.

But! It’s simplicity itself to think about its contrary, which is “You do not have ESP”. That’s easy! It means just what it says: you have no telepathic powers. Couldn’t be simpler. We’re done.

That’s why the second calculation is sought out. “You do not have ESP” is “Hypothesis false”.

It’s true “You have ESP” is the logical contrary of “You do not have ESP”, but it doesn’t mean much. What precisely do you mean by “You have ESP”? We have to be precise, too, if we want to do a calculation.

Being nauseously precise is what we did last week. We saw there are many, very many, ways to be precise about “She has the ability” (in tasting tea or milk poured into a cup first).

Maybe you can’t read my mind, directly, but I have the power to put images in your mind. If you got a lot of cards right, this could be me demonstrating my powers, not you demonstrating yours. We can’t tell using only our experiment.

Maybe you can read minds, but I can’t send. Maybe you just can’t read me. Maybe you can, but like lifting weights you can only do so much then become fatigued.

There are lots of possibilities! And this is one of the simplest problems we can do. Imagine how tough it is in other areas, like asking whether a drug works. How much easier to work with “You do not have ESP.”

It seems there are any numbers to be precise. So let’s pick a couple of them. These will not exhaust the possibilities. It is important crucial absolute that you get this, because it will be the same all the time.

Remember! All probability is conditional on the evidence we assume. Change the evidence, change the probability.

This feature, which is not a bug, is another reason people run to the second calculation. It seems to only give one answer. They pursue the false certainty it seems to offer.

There are endless ways an ESP experiment can go wrong. I can signal to you, even unknowingly, whether your previous guesses were right or wrong. You might be able to see parts of some cards by accident. This lets you seem to have powers when you don’t. This and many similar things are called “sensory leakage”. The cards can be badly shuffled, you could cheat, I could cheat, et cetera, et cetera.

Suppose we program a computer to generate a guess of 52 cards, then I go through my own deck, and pretend the computer was trying to read my mind. Obviously here the hypothesis “You have ESP” (where you = computer) is false. Any results we see will not be caused by the ESP power of the computer. We therefore deduce the probability of guessing any card correctly when “You have ESP” is false is 1/52. This is, of course, conditional on evidence of the experiment.

Call this hypothesis $\Psi$_1, i.e. “You do not have ESP”. This will cover all situations, including sensory leakage, cheating, and the like. Of course, if you cheat in the experiment you will get many “hits”, but this is not evidence you have ESP. Because you are cheating. That probability is still correct conditional (only) on the hypothesis “You do not have ESP”.

All right, that’s the easy part.

What does “You have ESP” mean! That’s harder. Think about Ty Cobb, who some say was the greatest baseball player. He did not get a hit at every at bat. So it is with true ESP, maybe. That is, supposing you have ESP you are not able to get every card signaled. Only some.

If we insist “You have ESP” means you get every card, then our test is easy. The first mistake proves you do not have ESP. That is, Pr(H|E & one mistake) = 0, where E includes the definition of H = “You have ESP” to mean “no mistakes”.

Except in silly fiction, nobody claims ESP powers like this for man. Something like $\Psi$_2 = 2/52 or $\Psi$_3 = 3/52 is closer to what most think. That is, the chance of guessing a card correctly assuming you do have some ESP is 2/52 conditional on hypothesis $\Psi$_2, and 3/52 on hypothesis $\Psi$_3.

Are there other possibilities? Sure. And that is the problem. I have to constrain the universe of possibilities, where there an infinite number of hypotheses, down to something I can work with.

Maybe you don’t like these ($\Psi$_2 and $\Psi$_3; we deduced $\Psi$_1) and have others in mind, given whatever experience or other evidence you have, but I don’t. Then you work with those.

Again, this is a feature and not a bug. Change the evidence, change the probability. Repeat that until you never forget. That is the way evidence works in real life. Not all agree on all premises. You might not like that, but tough cookies. That’s the Way Things Are, and, even more importantly, The Way Things Will Always Be.

“Briggs, I see what you’re doing, but suppose the real hypothesis, for me, is $\Psi$_4 = 4/52? You’ll never get that with your method. You’ll miss the truth.”

Yep.

I’ll also miss $\Psi$_5 = 4.001/52 and $\Psi$_6 = 34.01/52. And hey, it might even be true your psychic power levels fluctuate with the moon, or by the amount of health-giving tobacco you consumed. I’ll miss all that. I’ll miss any possibility I haven’t considered. That’s the way life is.

Consider the benefits I gain, though. I have left out an infinity of other hypotheses! I have cut down my labor to something possible to accomplish. I give almost every hypothesis a “prior” probability of 0 by not including or considering them. So, incidentally, do those who only make the calculation. Only they never realize that’s what they have done.

But what do we do if we really do fail to consider the true hypothesis? Same thing we do in all of life. We keep plugging away, consider other hypotheses, move on, adapt, overcome.

Write this in all caps and bolded: THERE IS NO MAGIC FORMULA THAT WILL TELL YOU WHAT HAPPENED.

NUMBERS

With all that in mind, let’s do some numbers. Here is a table which gives the probability (in percentage form) of the number of hits assuming each of our three hypotheses, from a deck of N = 52 cards. I only show hits up to 5, as the other outcomes are unlikely. So there’s a 36% you’ll get no hits if you have no psychic powers, and so on.

X (hits)	X|$\Psi$_1	X|$\Psi$_2	X|$\Psi$_3
0	36	13	5
1	37	27	14
2	19	28	23
3	6	18	23
4	1	9	17
5	0	3	10

Let’s suppose we got X = 4 hits. With that, we can calculate

Pr(Data we did not see | $\Psi$_1 E) = 0.003,

where the E is all the other evidence we’re using, such as the experimental setup. The “Data we did not see” is X = 5, X = 6, X = 7, X = 8 and so on (sometimes it’s only larger numbers we didn’t see, and sometimes it all numbers we didn’t see, depending on choices we will cover another day). This is less than the magic number (of 0.05), so we are entitled to pass peer review, get a grant, and proclaim you have ESP.

And we’d be done.

But that’s the wrong number. Let’s keep going and calculate the right one, which is the chance you have ESP.

To do that, we use Bayes’s formula, as we learned earlier. It’s very simple here:

$$\Pr(\Psi_i|X=4 E) = \frac{\Pr(X=4|\Psi_i E)\Pr(\Psi_i|E) }{\sum_i \Pr(X=4|\Psi_i E)\Pr(\Psi_i|E)}.$$

That table gives us Pr(X=4|$\Psi$_i E) for each i. All we need is Pr($\Psi$_i|E) for each i. That is, we need the “priors”, as the lingo has it.

Which are the proper “priors” to use here? There aren’t any. They do not exist. There is no right answer. There is no single answer. There is no “science” which says “Pick these and none other!”

“Briggs, what about maximum entropy? That’s non-informative and minimum information.”

What about it? Is that what the evidence you have tells you? If so, have a ball, use it. If not, don’t. It’s anyway not “non-informative”. Information by definition is informative. It’s not even minimum. If it’s what you have, then it’s what you have. If it’s not, it’s not. We’ll go over all that again another day.

The priors you pick depend on the evidence E you—you, dear reader—bring to the problem. Which might not be the same evidence I have. This, too, is a feature. Expecting agreement is likely unwise.

Yet I do want to communicate our results. We need to pick some Pr($\Psi$_i|E). The solution is simple: I pick what I believe, according to my E, and, interested in you, I pick another E I think you might hold, and give the results according to it, too.

What if I guess wrong? Well then, I guess wrong. But the way I set it up, you can plug your own numbers in. This is a huge plus.

My E is this: Pr($\Psi$_1|E_1) = 94, Pr($\Psi$_2|E_1) = 5, Pr($\Psi$_3|E_1) = 1. I base this on my long reading and experience with this field. And, I note, I consider this on the generous side for the sake of argument. My real E_1 is even more lopsided.

For your E, I’ll let you be utterly agnostic, and a strict maximum entropyist. You get Pr($\Psi$_1|E_2) = 33, Pr($\Psi$_2|E_2) = 33, Pr($\Psi$_3|E_2) = 33. (Rounded here, but in calculations, you get the real thing.)

Here’s the probabilities of each hypothesis given X = 4 and my E_1:

X (hits)	$\Psi$_1|XE_1	$\Psi$_2|XE_1	$\Psi$_3|XE_1
0	98	2	0
1	96	4	0
2	92	7	1
3	83	13	3
4	69	23	9
5	48	32	19

Again, I’ve stopped at 5, but you get the idea. With X = 4, and my E_1, I give the chance $\Psi$_1 is true at 69%, that $\Psi$_2 is true at 23%, and that $\Psi$_3 is true at 9%.

Here’s the same table for E_2 (maximum entropy):

X (hits)	$\Psi$_1|XE_2	$\Psi$_2|XE_2	$\Psi$_3|XE_2
0	67	24	8
1	47	34	18
2	27	40	33
3	13	39	49
4	5	32	62
5	2	25	73

At X = 4, you give only a 5% chance for $\Psi$_1, a 32% chance for $\Psi$_2, and a 62% for $\Psi$_3.

I cannot stress too highly that these are the deduced probabilities on these assumptions. Change the assumptions, change the probability!

And that’s what we’ll do next time, because it’s clear few would accept either E_1 or E_2. And we also have to think about adding additional evidence. For instance, what if you X = 4 was only one result out of many more, and this is the only one I published?

This is easy to think about now, because we have our hypotheses and definitions in hand. The math gets a bit harder, but we can do it. We see next time.

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