Another Non-Anthropic Paradox: The Unsurprising Rareness of Rare Events

post by Ape in the coat · 2024-01-21T15:58:14.236Z · LW · GW · 15 comments

Contents

  Intoduction
  The Impossibility of Random Number Generators
  Intuition vs Math
  Even More Extreme Version of the Problem
  Solving the Paradox
  Conclusion
None
15 comments

This is the fifth post in my series on Anthropics. The previous one is Anthropical Paradoxes are Paradoxes of Probability Theory [LW · GW]. The next one is Why Two Valid Answers Approach is not Enough for Sleeping Beauty [LW · GW].

Intoduction

As all anthropic problems are just probability theory problems, let's look deeper at probability theory itself. In my previous post I claimed that there are a lot of paradoxes^[1] there, and people are just not paying attention to them, unless they become relevant to anthropical reasoning, and then these paradoxes are mistakenly attributed to anthropics. So let's highlight one more of such paradoxes, solve it fully in the realm of probability theory, and then in a future post, this result will turn out to be helpful for deconfusing an anthropical problem.  

The Impossibility of Random Number Generators

It's well known that rare events are rare. If an event has 1/n probability then we should expect it to happen about 1 time per n tries. Thus, if you observe some event that you believe to be rare, it should be surprising, proportionate to the rarity of the event. If you keep observing that events you believe to be low probability keep happening, then it's a clear signal that something is wrong with your model of the world.

So far so good. But then, how are we not constantly mind blown by the existence of random number generators? 

After all, it's a device that can produce events of arbitrary rarity at will. Toss a coin ten times and you get a sequence of Heads and Tails which has only 1/2^10 probability. Run python function randint(0, 1000) ten times and you've just witnessed an event that has only 1/1000^10 probability. And it's not because the function was called so many times that some of the outcomes happen to be low probability. Every call of a function leads to an improbable outcome! What is going on? 

Intuition vs Math

Let's notice that some outcomes of random number generators intuitively surprise us much more than others. If you throw a coin ten times and you get a sequence of ten Heads - that would be more surprising than a sequence of $HHTHTTHTTH$. And yet, according to probability theory $P\left(HHTHTTHTTH\right)=P\left(HHHHHHHHHH\right)$. So we shouldn't really be surprised more. Is it just a bug of our human psyche that some sequences feel more random to us than others? Surely it has to be! We are biased but math can't be^[2].

On the other hand, there is something right about our naive human intuition. As we've already noticed, if we consider every outcome of ten coin tosses to be an improbable event we have to be constantly surprised and soon to start doubting our reality. But if we are surprised only by the outcome of 10 Heads in a row, the situation adds up to normality! After all, 10 Heads in a row will happen only in a rare subset of all outcomes. In this sense our biased human intuition satisfies the Law of Conservation of Expected Evidence.

Doesn't this contradict the fact that $P\left(HHTHTTHTTH\right)=P\left(HHHHHHHHHH\right)$? Not at all! There are creatures in the possible mind space^[3] whose intuition works in the opposite way. They are surprised specifically by the sequence of $HHTHTTHTTH$ and do not mind the sequence of $HHHHHHHHHH$. As a result, they would also satisfy the Law of Conservation of Expected Evidence, as they are surprised in a similarly rare, though different, subset of all possible outcomes.

We may say that the mathematical model here is not describing one specific intuition but a general principle. And that there is actually another coherent mathematical model describing specifically our intuition. These two models do not contradict each other, they are just applicable in different circumstances.

Okay, so our intuition is pointing to something true. That's all fine and good. Still, how can it all works out together? Both statements can't be true:

1. We have observed an event, which probability is 1/2^10
2. We are not supposed to be surprised, as if we observed an event which probability is 1/2^10

Even More Extreme Version of the Problem

Before I reveal the solution to the initial problem, let's look into a more radical version of it. What about random number generators that produce real numbers? There is an uncountable amount of them, so the probability of seeing a particular real number produced by a random generator should be zero! We can witness literally impossible events at any moment we want and yet, we are not surprised by that at all.

Let's notice our confusion [LW · GW] and apply the standard technique for resolving it. 

Our strength as rationalists is our ability to be more confused by fiction than by reality. If we are confused it means that something we believe in is false. We couldn't witness an impossible event. So, it means that we didn't. What event did we observe, then?

Random number generators do not actually produce real numbers. They produce float numbers with a specified accuracy. So, their probability isn't zero.

The same principle applies every time we deal with continuous distributions. When we check a thermometer, we do not actually observe an event "a specific real value of temperature was shown", which would've been impossible according to our mathematical model. Thermometers can only show us an interval: value +/- measurement error, each value of which has a non-zero probability. 

Now we are back to square one. The probability of a specific float number to be generated is not zero, but it's still very small. But now I think we have a pretty good hint of what's going on. Neither math nor our intuition is wrong. We just do not observe the event we thought we observed.

Solving the Paradox

Let's apply the confusion resolving procedure one more time. It's extremely unlikely that we observed an extremely unlikely event. Then, with all likelihood, we didn't.  What event did we observe, though?

In case of ten coin tosses in a row, what we actually observed is an event which can be called "any non-specific combination of Heads and Tails with length 10 was produced". This happens most of the time when we throw a coin ten times. But also, there are specific combinations which particularly capture our attention. This event encompasses such elementary outcomes as "10 Heads" and "10 Tails". And, very rarely, such combinations are produced. And then we are lawfully surprised, as we should be.

What I've just described seems as a usual default setting that human psyches has. But we can easily change it.

All it takes is just committing to track a specific set of combinations in your mind. And when the generator produces it and not any other combination - you will be rightfully surprised. The more elementary outcomes are being tracked, the more likely that one of them will be produced by a random generator, and thus the less surprising it is.

Does it mean that we can manipulate probabilities with our minds [? · GW]? Isn't it an example of weird anthropic psychic power, against existence of which I've been arguing the whole time? 

Well, there is clearly nothing anthropical about this power. Not only we are not talking about self-location, it's not at all available only for minds.

Yes, your mind does have some settings allowing you to regulate which event can be observed. But so does an electric thermometer with a customizable measurement error. Or a line of code, changing the number of digits that are shown for float numbers. 

You can also just shut your eyes, or disable the random number generator - does it also feel counterintuitive that these actions affect which events you can observe?

The opposite would be weird. If you could observe events regardless of the sharpness of your senses or the state of your mind or the accuracy of your tools - now this would be a quite literal psychic power, contradicting the Second Law of Thermodynamics [? · GW]. The ability to apply any map to any territory and get an accurate result. 

Conclusion

So, it turns out, rare events are indeed rare. Quite unsurprisingly so. In this particular case our naive intuitions seem to be quite on point. Where our intuitions are leading us astray, however, is in assumption that event space always contains a set, consisting of any particular elementary outcome from the sample space:

$\forall (\Omega ,F,P),E\in \Omega \Rightarrow \left\{E\right\}\in F$

This assumption is wrong. The definition of probability space doesn't prevent us from using a less rich σ-algebra, for example, for two coin tosses we can have this set up:

$\Omega =\{HH,TT,HT,TH\},F=\{\varnothing ,\{HT,TH\},\{HH,TT\},\{HH,TT,HT,TH\left\}\right\}$

$P(HT,TH)=1/2,P(HH,TT)=1/2$

And yet, when we see outcome $HH$ we intuitively assume that it necessarily has to be its own event with probability 1/4, instead of just part of a larger event with probability 1/2. As if rareness of event is the sole property [LW · GW] of the sequence of coin tosses.

This is also related to the way humans use words. We often use "event" and "outcome" as synonyms. But mathematically, they are quite different. Event is a set of outcomes. And as the domain of probability function $P$ is event space $F$, not sample space $\Omega$, there is no such thing as probability of an outcome.^[4] 

This is not a mistake that often reminds us about itself. And so, it's easy to keep making it without noticing. Even now when specifically highlighted it may look like a completely niche thing, an irrelevant nitpick. Most of the time we don't need to rigorously construct probability space to find a correct answer to a decision theory problem.

But then, occasionally, we do need to be rigorous. And if we are not used to it - we fail to arrive to a correct answer. And, in case we are particularly unlucky, also create a decades long philosophical dispute with different new schools of thought, persuading otherwise reasonable people to believe in ridiculous things. And if you know anything about philosophical disputes then you understand that untangling them is much more complicated task than not making the initial mistake in the first place.

But, as this ship has long sailed, my next couple of posts will be dedicated to this untangling procedure as we've finally done all the groundwork to solve the infamous Sleeping Beauty Paradox.

The next post in the series is Why Two Valid Answers Approach is not Enough for Sleeping Beauty [LW · GW].

 

1. ^{^}

   I do not actually mean that probability theory itself is unsound and should be abandoned. By "paradox" here I mean an apparent contradiction, that people are confused about, because they misinterpret probability theory. As always, it's not the art that has failed us, but it is us who has failed the art.

2. ^{^}

   However, math can be unapplicable to a specific real-world situation or be applicable differently than we originally thought.

3. ^{^}

   We do not even need to postulate a weird alien psyche here. These creatures can very well be humans - more on that below.

4. ^{^}

   Sadly, this confusion is so pervasive that, at the moment of writing, Wikipedia article on outcome talks about their probabilities, despite specifically mentioning that outcomes shouldn't be confused with events. The article on probability space, however, correctly states that probability function assigns values to events.

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