The Mathematics Of Conspiracy Theories. Part 1: Black Eye Club Example

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Elon Muck infamously pitched a fit last week over his frustration politicians were acting like politicians. After he calmed down, or sobered up, he apologized. And we all moved on.

What’s curious about this episode, for us, are two things. 1) He claimed Trump was in the Epstein files, and 2) He, Musk, emerged with a black eye. Not a metaphorical one, though that is also true. A real genuine shiner.

The Epstein files are a well known “conspiracy theory”. A conspiracy theory is a belief that the rich and powerful are up to something and don’t want us to know about whatever it is. Or they do want us to know, so that they know we know they are in charge, but they won’t go so far as to admit they are doing it.

Musk’s eye darkening, some say, is also part of the conspiracy theory called the Black Eye Club. Briefly, one variant of this theory is that those who would be rich and powerful must submit themselves to a humiliation ritual, the black eye, which signals their willingness to be part of the system.

Are either or these, or both, or any conspiracy theory true? What are the chances they are?

Without pretending to come to a full and final answer, let us develop some tools we can use to assess the chance any conspiracy theory is true. Or, today, one tool, to be used when evidence is certain or nearly certain if the conspiracy theory is true. I’ll explain what I mean about this carefully.

We want the chances some conspiracy theory is true. It helps, at least sometimes, to have notation to keep track of complex ideas. Let’s use “C” to stand for our conspiracy theory, whatever it might be. This C is the cause, the machinations behind the scenes, they why and how and all that: the full cause. It pays to be as specific as one can about the full causes of the conspiracy. Shifting definitions cause confusion.

There will always be some evidence that allows you to consider the theory initially. Something you heard in the background, maybe, or on the interwebs. Maybe deductions from various observations you have made yourself. Whatever it is, call all this evidence “E” for short.

All we need is the notation (familiar to regular readers) for the chances, which is “Pr(C|E)”, which stands for the probability the conspiracy is true given or accepting or assuming the evidence E. It is a separate and different question whether the evidence itself is true, likely, unlikely, or false. For now, we’ll assume it’s true, which is entirely non-committal. We’ll deal with the uncertainty of the evidence another day.

Though the Black Eye Club might be new to you, you’ve probably had at least vague notions of the Epstein files. To an extent, the latter is not a conspiracy at all. Epstein did traffic girls to our politicians and elite. The conspiratorial part of this goes deeper. like whether Epstein was an agent for everybody’s (mandatory) favorite foreign country, that he was using blackmail to enrich himself, and to gain secrets to give to his minders. That we commoners do not have direct evidence for.

The Black Eye Club is touted using (many) pictures like this, with elites, celebrities, and rulers, even a Pope!, sporting black eyes.

That there is some explanation (i.e. cause) for these pictures is a truism. That it is the same explanation in most or all of them is the conspiracy.

Incidentally, while searching for the BEC, I came across this peer-reviewed paper (make of it what you will): Adrenochrome deposits in the cornea (“Black cornea”). A clinical-pathological case report.

Let’s take the BEC as our prime example. You’ve just heard of it, seen some of these pictures, read some disturbing theorizing about alien lizards and soul scraping and you wonder. You form a fuzzy idea of the truth of this theory, not willing to believe it, but not entirely dismissing it as impossible, either. Then comes that picture of Musk, which coincidentally showed around the time of his Trump tantrum.

Our new evidence is Musk’s BE pic. Call it “M” for short. We now want the probability the conspiracy theory is true given our old evidence, which might be scant and anyway unconvincing, and we want to add this new evidence to it. That is, we want, in notation, this—but don’t worry if you can’t follow or read this, because we’re going to simplify it greatly.

$$\Pr(C|ME) = \frac{\Pr(M|CE)\Pr(C|E)}{\Pr(M|CE)\Pr(C|E)+\Pr(M|C^FE)(1-\Pr(C|E))}$$,

where $C^F$ is short for “the conspiracy theory is false.” This is, of course, the standard calculation in probability for adding evidence. But we can simplify it, because we notice that if the conspiracy is true, then of course Musk would have a black eye. That makes Pr(M|CE) = 1; which is, the probability Musk’s black eye was caused by the conspiracy, accepting the conspiracy is true is 1. Or close enough to certainty that we don’t care about any differences.

That’s our tool. We’re today taking only cases in which the new probability is true, or close enough, if the conspiracy theory is true.

Now, even though we’re assuming the new evidence is true if the conspiracy is, it doesn’t make the conspiracy itself true. We still have to work out the full math for that. Using Pr(M|CE) = 1, we get:

$$\Pr(C|ME) = \frac{\Pr(C|E)}{\Pr(C|E)+\Pr(M|C^FE)(1-\Pr(C|E))}.$$

The chance the theory is true given we’ve seen Musk’s picture is a function of only two numbers: our initial belief Pr(C|E) and the chance Musk has a black eye pic given the conspiracy is false, i.e. $\Pr(M|C^FE)$.

That number is key to the whole thing, and the only real number we care about. To see that, let’s use a range of numbers for our initial belief. Obviously, if we’re already convinced the conspiracy is true, Musk’s picture adds little to our belief: we still believe, even when we grant it’s possible Musk might have come by his black eye innocently. That is, if Pr(C|E) is large, then (1-Pr(C|E)) is small, and even if believe $\Pr(M|C^FE)$ is high, it’s multiplied by a number close to 0. Then we get

$$\Pr(C|ME) \approx \frac{\Pr(C|E)}{1} = \Pr(C|E).$$

Our level of certainty doesn’t change with the new evidence.

Our tool is thus only useful for those cases where we’re not already convinced, and where we accept as true, or highly likely, the initial information E, even if this information is vague. So, again, let’s only look at situations where we’re doubtful. That is, cases in which the initial belief has less than even odds, i.e. Pr(C|E) < 1/2.

Something like a 10%, or 1 out of 10, chance is doubtful, but still meaty. A 10% belief in the BEC is, many would think, pretty high. One out of 100 is of course less so, and 1 out of 1,000 even less, and so on for increasing number of zeros.

This points the way to another simplification. If we write our initial chance as 1/x, for x = 10, or 100, or 1,000, or even a million, or any “power of 10”, we get:

$$\Pr(C|ME) = \frac{\frac{1}{x}}{\frac{1}{x}+\Pr(M|C^FE)\frac{x-1}{x}}.$$

And that equals

$$\Pr(C|ME) = \frac{1}{1+\Pr(M|C^FE)(x-1)}\approx \frac{1}{1+x\Pr(M|C^FE)}.$$

That is our great simplification, an equation with only two numbers we need plug in: $x$ and $\Pr(M|C^FE)$.

Here’s some pictures of what that looks like, for various x and for varying $\Pr(M|C^FE)$ from 0 to 1.

Take an initial belief of 1/10. If you believe the new evidence is true, or very likely, if the conspiracy is false, your belief in the conspiracy doesn’t change much from 1/10. It’s only as the new evidence becomes more and more unlikely, if the conspiracy is false, that your belief in the conspiracy rises. And the more skeptical you start, the more unlikely the new evidence must be to convince you.

If you start at 1 in a million (not pictured), it becomes very difficult to convince you of the conspiracy. Whereas, if you start closer to believing, then any new evidence can only move you closer to belief. That explains (partially) why people’s beliefs move further apart the more evidence there is. When we another day examine the initial evidence itself, we’ll see why differences persist, even as new evidence which all agree on arises.

Again, the smaller $\Pr(M|C^FE)$ is, the greater the conspiracy theory has of being true. The conspiracy theory is exactly true if it is ruled impossible for M to happen if the theory is false. Why? If the only way for a thing to happen is by a certain cause, then our observation proves the cause is true. But we have to be perfectly strict in that “only”. Absolutely zero deviation or doubt is allowed.

In our case, that means it would be impossible for Musk (or anybody) to get a black eye outside the Black Eye conspiracy. That, I think, is obviously false, because people get black eyes for all kinds of reasons. Like getting popped for mouthing off.

We can simplify things even further. Let’s look for the level of $\Pr(M|C^FE)$ that makes the conspiracy more likely true than false; i.e. when $\Pr(C|ME)> 1/2$. That’s when (simple algebra reveals):

$$\Pr(M|C^FE) \lesssim \frac{1}{x}.$$

What a nice, compact formula! If $x = 10$, i.e. where 1/x = 1/10 is the chance the conspiracy is true initially, then the chance of the new evidence assuming the theory is false must be less than 1/10 to make the conspiracy more likely than not. If $x = 100$, then the new evidence must have a (conditional) chance of less than 1/100. And so on.

That means that the more skeptical of the conspiracy you are at the start, the more surprising, or rather unlikely, the new evidence must be (assuming the conspiracy is false) to bring you anywhere near believing the conspiracy is true. The opposite is also true. The closer you are to believing at the start, the harder it is to budge you from your belief.

So. What, in your mind, is a good starting value for $\Pr(C|ME)$? Then, what is a reasonable value for $\Pr(M|C^FE)$? For the BEC, I’m pretty low on the first, but cannot prove it false. Then I think $\Pr(M|C^FE)$ is not that small. Many get black eyes for all sorts of reasons. I do not believe it is so rare as $1/x$ for my large x. So I start with doubt, and I am left with doubt. So doubtful that I need not bother making any kind of exact calculation.

All right, what about the full blown Epstein conspiracy? You have to be tight about its definition and evidence, of course. I’ll go with he was a bad person trafficking girls, and who knows what else, and surely used some of the information he gathered to enrich himself. He might have been friendly with foreign entities, and passed information on to them. I do not think any of this unlikely, but I don’t have direct proof.

The comes Musk’s revelation, or story, that Trump was in the Epstein files. This is our new evidence. This is where the simplified formula breaks down. Remember that formula started with the approximation Pr(M|CE) = 1 (or close enough). Here I don’t think that’s true. That is, even accepting Epstein was a blackmailer for everybody’s favorite foreign country, it is not true, I think, that Trump was one of the blackmailees.

We have to go all the way back to the initial equation.

M stands for Musk’s accusation that Trump is in the files, and has some secret dirt on him that Epstein had or used. Accepting that the conspiracy is true, again I don’t think it likely that Trump is in the files.

Because why? Because if he were, the obsessed political maniacs who were desperate to nail Trump on anything, even jaywalking, would have found and used the information in those files, even if it meant sacrificing some of their own. Their deranged hatred was (is) so deep, they would have been like Captain Ahab and opened up their chests and shot out their hearts at Trump if they could have.

Whereas, I think that if the conspiracy is false, I think the same. That is, I judge $\Pr(M|CE)=\Pr(M|C^FE)$, or near enough. That makes $\Pr(C|ME)=\Pr(C|E)$, i.e. my beliefs about the conspiracy haven’t really changed with the new information.

What about when we doubt the initial evidence? We’ll save that for another time.

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