Explaining anthropic reasoning

I realize that I’ve been a little unclear in my last few posts. I presupposed a degree of familiarity with anthropic reasoning that most people don’t have. I want to remedy that by providing a short explanation of what anthropic reasoning is and why it is useful.

First of all, one thing that may be confusing is that the term ‘anthropic reasoning’ is used in multiple very different ways. In particular, its most common usage is probably in arguments about the existence of God, where it is sometimes presented as an argument against the evidential force of fine tuning. I have no interest in this, so please don’t take me to be using the term this way. My usage is identical with that of Nick Bostrom, who wrote a fantastic book about anthropic reasoning. You’ll see precisely what this usage entails shortly, but I just want to plant a flag now in case I use the word ‘anthropic’ in a way that you are unfamiliar with.

Good! Now, let’s start with a few thought experiments.

  1. Suppose that the universe consists of one enormous galaxy, divided into a central region and an outer region. The outer region is densely populated with intelligent life, containing many trillions of planetary civilizations at any given moment. The inner region is hostile to biology, and at any given time only has a few hundred planetary civilizations. It is impossible for life to develop beyond the outer region of the galaxy.

    Now, you are a member of a planetary civilization that knows all of this, but doesn’t know its location in the galaxy. You reason that it is:

    (a) As likely that you are in the central region as it is that you are in the the outer region
    (b) More likely
    (c) Less likely

  2. Suppose that the universe consists of one galaxy that goes through life phases. In its early phase, life is very rare and the galaxy is typically populated by only a few hundred planetary civilizations. In its middle phase, life is plentiful and the galaxy is typically populated by billions of planetary civilizations. And in its final phase, which lasts for the rest of the history of the universe, it is impossible for life to evolve.

    You are born into a planetary civilization that knows all of this, but doesn’t know what life phase the galaxy is in. You reason that it is:

    (a) As likely that you are in the early phase as the middle phase
    (b) More likely
    (c) Less likely

  3. You are considering two competing theories of cosmology. In Cosmology X, 1% of life exists in Region A and 99% in Region B. In Cosmology Y, 99% of life is in Region A and 1% in Region B. You currently don’t know which region you are in, and have equal credence in Cosmology X and Cosmology Y.

    Now you perform an experiment that locates yourself in the universe. You find that you are in Region A. How should your beliefs change?

    (a) They should stay the same
    (b) Cosmology X becomes more likely than Cosmology Y
    (c) Cosmology Y becomes more likely than Cosmology X

If you answered (c) for all three, then congratulations, you’re already an expert anthropic reasoner!

What we want to do is explain why (c) was the right answer in all three cases, and see if we can unearth any common principles. You might think that this is unnecessary; after all, aren’t we just using a standard application of Bayes’ theorem? Sort of, but there’s a little more going on here. Consider, for instance the following argument:

1. Most people have property X,
2. Therefore, I probably have property X.

Ignoring the base rate fallacy here, there is an implicit assumption involved in the jump from 1 to 2. This assumption can be phrased as follows:

I should reason about myself as if I am randomly sampled from the set of all people.

A similar principle turns out to be implicit in the reasoning behind our answers to the three starting questions. For question 1, it was something like

I should reason about myself as if I am randomly sampled from the set of all intelligent organisms in the universe at this moment.

For 2, it might be

I should reason about myself as if I am randomly sampled from the set of all intelligent organisms in the history of the universe.

And for 3, it is pretty much the same as 1:

I should reason about myself as if I am randomly sampled from all intelligent organisms in the universe.

These various sampling assumptions really amount to the notion that we should reason about ourselves the same way we reason about anything else. If somebody hands us a marble from an urn that contains 99% black marbles, (and we have no other information) we should think this marble has a 99% chance of being black. If we learn that 99% of individuals like us exist in Region A rather than Region B (and we have no other information), then we should think that we have a 99% chance of being in Region A.

In general, we can assert the Self-Sampling Assumption (SSA):

SSA: In the absence of more information, I should reason about myself as if I am randomly sampled from the set of all individuals like me.

The “individuals like me” is what gives this principle the versatility to handle all the various cases we’ve discussed so far. It’s slightly vague, but will do for now.

And now we have our first anthropic principle! We’ve seen how eminently reasonable this principle is in the way that it handles the cases we started with. But at the same time, accepting this basic principle pretty quickly leads to some unintuitive conclusions. For instance:

  1. It’s probably not the case that there are other intelligent civilizations that have populations many times larger than ours (for instance, galactic societies).
  2. It’s probably not the case that we exist in the first part of a long and glorious history of humanity in which we expand across space and populate the galaxy (this is called the Doomsday argument).
  3. On average, you are probably pretty average in most ways. (Though there might be a selection effect to be considered in who ends up regularly reading this blog.)

These are pretty dramatic conclusions for a little bit of armchair reasoning! Can it really be that we can assert the extreme improbability of a glorious future and the greater likelihood of doomsday from simply observing our birth order in the history of humanity? Can we really draw these types of conclusions about the probable distributions of intelligent life in our universe from simply looking at facts about the size of our species?

It is tempting to just deny that this reasoning is valid. But to do so is to reject the simple and fairly obvious-seeming principle that justified our initial conclusions. Perhaps we can find some way to accept (c) as the answer for the three questions we started with while still denying the three conclusions I’ve just listed, but it’s not at all obvious how.

Just to drive the point a little further, let’s look at (2) – the Doomsday argument – again. The argument is essentially this:

Consider two theories of human history. In Theory 1, humans have a brief flash of exponential growth and planetary domination, but then go extinct not much later. In this view, we (you and me) are living in a fairly typical point in the history of humanity, existing near its last few years when its population is greatest.

In Theory 2, humans continue to expand and expand, spreading civilization across the solar system and eventually the galaxy. In this view, the future of humanity is immense and glorious, and involves many trillions of humans spread across hundreds or thousands of planets for many hundreds of thousands of years.

We’d all like Theory 2 to be the right one. But when we consider our place in history, we must admit that it seems incredibly less likely for us to be in the very tiny period of human history in which we still exist on one planet, than it is for us to be in the height of human history where most people live.

By analogy, imagine a bowl filled with numbered marbles. We have two theories about the number of marbles in the bowl. Theory 1 says that there are 10 marbles in the bowl. Theory 2 says that there are 10,000,000. Now we draw a marble and see that it is numbered 7. How should this update our credences in these two theories?

Well, on Theory 2, getting a 7 is one million times less likely than it is on Theory 1. So Theory 1 gets a massive evidential boost from the observation. In fact, if we consider the set of all possible theories of how many marbles there are in the jar, the greatest update goes to the theory that says that there are exactly 7 marbles. Theories that say any fewer than 7 are made impossible by the observation, and theories that say more than 7 are progressively less likely as the number goes up.

This is exactly analogous to our birth order in the history of humanity. The self-sampling assumption says that given that you are a human, you should treat yourself as if you are randomly sampled from the set of all humans there will ever be. If you are, say, the one trillionth human, then the most likely theory is that there are not many more than a trillion humans that will ever exist. And theories that say there will be fewer than a trillion humans are ruled out definitively by the observation. Comparing the theory that says there will be a trillion trilllion humans throughout history to the theory that says there will be a trillion humans throughout history, the first is a trillion times less likely!

In other words, applying the self-sampling assumption to your birth order in the history of humanity, we update in favor of a shortly upcoming doomsday. To be clear, this is not the same as saying that doomsday soon is inevitable and that all other sources of evidence for doomsday or not-doomsday are irrelevant. This is just another piece of evidence to be added to the set of all evidence we have when drawing inferences about the future of humanity, albeit a very powerful one.

Okay, great! So far we’ve just waded into anthropic reasoning. The self-sampling assumption is just one of a few anthropic principles that Nick Bostrom discusses, and there are many other mind boggling implications of this style of reasoning. But hopefully I have whetted your appetite for more, as well as given you a sense that this style of reasoning is both nontrivial to refute and deeply significant to our reasoning about our circumstance.

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