Anthropics and Biased Models
The Fine-tuned Universe Theory, according to Wikipedia is the belief that, “our universe is remarkably well suited for life, to a degree unlikely to happen by mere chance”. It is typically used to argue that our universe must therefore be the result of Intelligent Design.
One of the most common counter-arguments to this view based on the Anthropic Principle. The argument is that if the conditions were not such that life would be possible, then we would not be able to observe this, as we would not be alive. Therefore, we shouldn’t be surprised that the universe has favourable conditions.
I am going to argue that this particular application of the anthropic principle is in fact an incorrect way to deal with this problem. I’ll begin first by explaining one way to deal with this problem; afterwards I will explain why the other way is incorrect.
Two model approach
We begin with two modes:
Normal universe model: The universe has no bias towards supporting life
Magic universe model: The universe is 100% biased towards supporting life
We can assign both of these models a prior probability, naturally I’d suggest the prior probability for the later should be rather low. We then update based on the evidence that we see.
p(normal universe|we exist) = p(we exist|normal universe)/p(we exist) * p(normal universe)
The limit of p(normal universe|we exist) as p(we exist|normal universe) approaches 0 is 0 (assuming p(normal universe)!=1). This is proven in the supplementary materials at the end of this post. In plain English, as the chance of us existing in the normal universe approaches zero, as long as we assign some probability to the magic universe model we will at some point conclude that the Magic universe model is overwhelming likely to be correct. I should be clear, I am definitely not claiming that the Fine-Tuned Universe argument is correct. I expect that if we come to the conclusion that the Magical model is more likely than the Normal model of the universe, than that is because we have set our prior for the magical model of the universe to be too high or the chances of life inside the normal universe model to be too low. Regarding the former, our exposure to science fiction and fantasy subjects us to the availability bias, which biases our priors upwards. Regarding the later, many scientists make arguments that life can only exist in a very specific form, which I don’t find completely convincing.
Standard anthropic argument
Let’s quote an example of the standard anthropic argument by DanielLC:
Alice notices that Earth survived the cold war. She asks Bob why that is. After all, so much more likely for Earth not to survive. Bob tells her that it’s a silly question. The only reason she picked out Earth is that it’s her home planet, which is because it survived the cold war. If Earth died and, say, Pandora survived, she (or rather someone else, because it’s not going to be the same people) would be asking why Pandora survived the cold war. There’s no coincidence.
This paragraph notes that the answer to the question, “What is the probability that we survived the Cold War given that we can ask this question?” is going to always be 1. It is then implied that since there is no surprise, indeed, this is what must be what happened, the anthropic principle lacks any force.
However, this is actually asking the wrong question. It is right to note that we shouldn’t be surprised to observe that we survived given that it would be impossible to observe otherwise. However, if we were then informed that we lived in a normal, unbiased universe, rather than in an alternate biased universe, if the maths worked out a particular way such that it leaned heavily towards the alternate universe, then we would be surprised to learn we lived in a normal universe. In particular, we showed how this could work out above, when we examined the situation where p(we exist|normal universe) approached 0. The anthropic argument against the alternate hypothesis denies that surprise in a certain sense can occur, however, if fails to show that surprised in another, more meaningful sense can occur.
However, this is actually asking the wrong question. It is right to note that we shouldn’t be surprised to observe that we survived given that it would be impossible to observe otherwise. However, if we were then informed that we lived in a normal, unbiased universe, rather than in an alternate biased universe, if the maths worked out a particular way such that it leaned heavily towards the alternate universe, then we would be surprised to learn we lived in a normal universe. In particular, we showed how this could work out above, when we examined the situation where p(we exist|normal universe) approached 0. The anthropic argument against the alternate hypothesis denies that surprise in a certain sense can occur, however, if fails to show that surprised in another, more meaningful sense can occur.
Reframing this, the problem is that it fails to be comparative. The proper question we should be asking is “Given that we observe an unlikely condition, is it more probable that the normal or magical model of the universe is true?”. Simply noting that we can explain our observations perfectly well within our universe, does not mean that an alternate model wouldn’t provide a better explanation. As an analogy, if we want to determine whether a coin is biased or unbiased, then we must start with (at least) two models—fair and unfair. We assign each a prior probability and then do a Bayesian update on the new information provided—ie. the unusual run or state of the universe.
Coin flip argument
Let’s consider a version of this analogy in more detail. Imagine that you are flipping coins. If you flip a heads, then you live, if you flip a tails, then you are shot. Suppose you get 15 coin flips in a row. You could argue that only the people who got 15 coin flips in a row are alive to ask this question, so there is nothing to explain. However, if there is a 1% chance that the coin you have is perfectly biased towards heads, then the number of people with biased coins who get 15 flips and ask the question will massively outweigh the number of people with unbiased coins who get to 15 flips. Simply stating that that there was nothing surprising about you observing 15 flips given that you would be dead if you hadn’t gotten 15 flips didn’t counteract the fact that one model was more likely than the other.
Edit—Extra Perspective: Null hypothesis testing
Another view comes from the idea of hypothesis testing in statistics. In hypothesis testing, you start with a null hypothesis, ie. a probability distribution based on the Normal universe model and then calculate a p-value representing the chance that you would get this kind of result given that probability distribution. If we get a low p-value, then we generally “reject” the null hypothesis, or at least argue that we have evidence for rejecting it in favour of the alternate hypothesis, which is in this case that there exists at least some bias in the universe towards life. People using the anthropic principle argue that our null hypothesis should be a probability distribution based on the odds of you surviving given that you are asking this question, rather than simply the odds of you surviving fullstop. This would mean that all the probability should be distributed to the survive case, providing a p-value of 1 meaning that we should reject the evidence.
While the p-value may remain fixed as 1 as p(alive|normal universe) → 0, it is worth noting that the prior probability of our null hypothesis, p(alive & normal universe), is actually changing. At some point, the prior probability becomes sufficiently close to 0 that we reject the hypothesis despite the p-value still being stuck at 1. This is, hypothesis testing is not the only situation when we may reject a hypothesis. A hypothesis that perfectly fits the data may be rejected based in a minuscule prior probability.
Summary
This post was originally about the Fine-tuned universe theory, but we also answered the Cold war anthropic puzzle and a Coin Flip Anthropic puzzle. I’m not claiming that all anthropic reasoning is broken in general, only that we can’t use anthropic reasoning on a single side of a model. I think that there are cases where we can use anthropic reasoning, but these are cases where we are trying to determine likely properties of our universe, not ones where we are trying to use it to argue against the existence of a biased model. Future posts will deal with these applications of the anthropic principle.
Edit: After consideration, I have realised that the anthropic principle actually works when combined with the multiple worlds hypothesis as per Luke_A_Somers comment. My argument only works against the idea that there is a single universe with parameters that just happen to be right. If the hypotheses are: a multiverse as per string theory vs. a magical (single) universe, even though each individual universe may only have a small chance of life, the multiverse as a whole can have almost guaranteed life, meaning our beliefs would simply be based on priors. I suppose someone might complain that I should be comparing a Normal multiverse against a Magical multiverse, but the problem is that my priors for a Magical multiverse would be even lower than that of a Magical universe. It is also possible to use the multiple worlds argument without using the anthropic principle at all—you can just deny that the fine tuning argument applies to the multi-verse as a whole.
Supplementary Materials
Limit of p(normal universe|we exist)
The formula we had was:
p(normal universe|we exist) = p(we exist|normal universe)/p(we exist) * p(normal universe)
The extra information that we exist, has led to a factor of p(we exist|normal universe)/p(we exist) being applied.
We note that p(we exist)=p(we exist|normal universe)p(normal universe) + p(we exist|magical universe)p(magical universe)
=p(we exist|normal universe)p(normal universe) + 1 - p(normal universe)
=p(we exist|normal universe)p(normal universe) + 1 - p(normal universe)
The limit of p(we exist) as p(we exist|normal universe) → 0, with p(normal universe) fixed, is 1 - p(normal universe). So long as p(normal universe) != 1, p(we exist) approaches a fixed value greater than 0.
The limit of p(we exist|normal universe)/p(we exist) as p(we exist|normal universe) → 0 is 0.
Meaning that limit of p(normal universe|we exist) as p(we exist|normal universe) → 0 is 0 (assuming p(normal universe)!=1)
Performing Bayesian updates
Again, we’ll imagine that we have a biased universe where we have 100% chance of being alive.
We will use Bayes law:
p(a|b)=p(b|a)p(a)/p(b)
Where:
a = being in a normal universe
b = we are alive
We’ll also use:
p(alive) = p(alive|normal universe)p(normal universe) + p(alive|biased universe)p(biased universe)
Example 1:
Setting:
p(alive|normal universe) = 1⁄100
p(normal universe) = 1⁄2
The results are:
p(we are alive) = (1/100)*(1/2)+1*(1/2) = 101⁄200
p(normal universe|alive) = (1/100)*(1/2)*(200/101) = 1⁄101
Example 2:
Setting:
p(normal universe)=100/101
p(alive|normal universe) = 1⁄100
p(normal universe) = 100⁄101
The results are:
p(we are alive) = 100/101*1/100+1/101*1 = 2⁄101
p(normal universe|alive) = (1/100)*(100/101)* (101/2) = 1⁄2
I think that anthropic reasoning only works when you have a good model of how you could have gotten into the situation in question.
For the beginning of the universe kinds of questions, as I see it, the options boil down to:
1) Is something vaguely like String Theory correct, in which a near-infinite ensemble of universes with different laws is created at the dawn of time, or continuously across time?
2) Are the laws we observe actually perfectly fundamental, and they just happen to be right?
3) Did some entity pick out these laws?
Anthropic reasoning gives us no reason to go for 2, but it is perfectly happy with 1, since it lets us discard all of the parts of the universe with rules that don’t produce life capable of considering the question.
One of the points that I was trying to make is that you can’t apply anthropic reasoning like that. That is, you need to be comparative, to start with at least two models, then update on your anthropic data. As an analogy, I might be able to give you very good reasons for believing that theory A would explain a phenomena, but if theory B explains it better, then we should go with theory B. There are many cases where we can obscure this by talking exclusively about theory A.
So the question is not does 1) explain the situation well, but does 1) explain the situation better than 3), taking into account things such as prior probabilities.
Update: On second thought, multi-worlds is a pretty good answer when combined with the anthropic principle. I suppose that my argument then only shows that case 2) isn’t a very good explanation.
I took it as too-obvious-to-mention that 2 & 3 explain the situation just fine, but have massive complexity penalties.
It is a well estabilished fact in probability that you cannot treat on the same footing a naive notion of surprise and the happening of an event of low probability.
The classical example is the extraction from a set of large cardinality with uniform distribution: one of the occurences is bound to happen, but each has a very low probability.
If you let naive surprise guide your model selection, and you do not have a sound base for model generation, you start falling into a slippery slope that culminates in solipsism.
Case in point: we have three universes, one is normal, one is magical (biased toward our existence), one is solipsistic (biased toward my existence). Clearly, since we exists, the magical is much more probable than the normal, but since I exist, the universe was probably born to allow me to write you in this exact moment (after all, what are the probability that I was born between the trillion of other possible beings?).
So if you generate a number randomly between one and one million, each number has a one in a million chance of being chosen. Like, if I get the number 5, I can say that it is unlikely that it is a coincidence, as there was only a one in a million chance of this happening. However, there is no reason why I wouldn’t have said the same thing if I received a 6 or 335,687. So there isn’t really a coincidence or a surprised, because regardless of result, we could have said something similar.
I don’t believe in the magical universe theory either. My point was simply that the anthropic principle is not an effective counter-argument. If the maths suggests that a magical universe exists or that a sophistic universe exists, I suspect that you’ve probably set the prior probabilities to be too high.
I don’t think there’s actually a problem here.
I think it only looks like there’s a problem because you haven’t separated this into a ‘chance’ hypothesis and an ‘ensemble’ hypothesis, such that we have three initial hypotheses.
Suppose that you are a gambler and a dealer wants you to determine whether or not zer dice are biased. The thing is, ze never lets you observe a dice roll; you’re only allowed to see how the dice landed after they have been rolled. Every time ze goes to roll the dice, you have to leave the room. Over and over again, each time you enter the room, you observe that the dealer has rolled a double six. If our hypothesis space is really limited to these two hypotheses, then the probability of biased dice should skyrocket, which is, as far as I can tell, the point that you’re making in the article above.
But there may be a hypothesis outside of your hypothesis space. Suppose the dealer secretly only lets you into the room if ze rolls a double six. The key here is that our observations of the dice are subject to a selection bias.
It would be really surprising to see a double six if it had been rolled after just one trial (because the probability of rolling at least one double six in one trial is ~0.027), but it would be expected if there had been many trials (because the probability of rolling at least one double six, in say, 200 trials, is ~0.996). And it seems like a sort of explanation to say to the gambler that even though the prior probability of a double six being rolled in any given trial is quite low, it’s not quite so surprising to see it if there have actually been many rolls that you could not observe, since we would either see that outcome or see no outcome at all.
Does that make sense to you?
“Suppose the dealer secretly only lets you into the room if ze rolls a double six”
You seem to be proposing that we should have an alternate hypothesis:
“Our observations are filtered by the requirement of us being alive”
However, this isn’t an alternate hypothesis as in both the Normal universe and the Magical universe it holds.
To make it clearer, if I an examining two hypothesises:
1) “Barrack Obama is human and he is president of the United States” 2) “Barrack Obama is human and he is not president of the United States”
And if your alternate hypothesis is:
3) “Barrack Obama is human”
Then you haven’t actually created a new separate, hypothesis, just a hypothesis that is a superset of the other two.
Anyway, your post seems to just restate the anthropic argument. I explained in my post that this is can’t be applied here because it is necessary to be comparative between the hypothesis, while the way the anthropic argument is being used there only considers a single hypothesis.
Not sure why you’re thinking about these hypotheses as supersets and subsets of one another. If I wanted to get formal with it, I’d describe the hypotheses as programs. The design hypothesis would be an agent program that outputs our local universe by building our universe the hard way. The chance hypothesis would be a set of physical constants with rules determining the time evolution of reality that outputs our universe, and not an entire multiverse of which our universe is a small part. The ensemble hypothesis would be an even simpler and more fundamental set of rules than in the previous program, maybe with some constants as well, and it would output a multiverse, some parts of which are hospitable and even identical to the previous program’s output. It confuses me to think about these hypotheses as subsets of one another, because it makes me think of substrings. These programs would not be substrings of one another. Their output would be though, because they all output us observing our universe. We’re supposed to be talking about hypotheses, not output.
The single unbiased universe, multi-verse and biased universe are not subsets.
I was simply stating that the anthropic principle is a principle that applies to any of these models, not it is not its own separate model.
I think the delivery could be greatly improved by introducing symbols and clarifying the logical environment where the derivation is happening. Allow me to do this, without using LateX I’ll assume /\ stands for logical conjunction and ~ for logical negation.
Proposition symbols
Our universe is normal: N
We exist: E
Our universe is magical: M
Logical environment
[1] The universe is either normal or magical: M = ~N
[2] The magical universe is strongly biased to support our existence: P(E|M) = 1
Derivation
By Bayes theorem:
P(N|E) = P(E|N) P(N) / P(E) <-->
P(N|E) = k P(E) /\ k = P(E|N)/P(E)
By partition of unity and [1]
P(E) = P(E|N)P(N) + P(E|M)P(M) ⇒ (by fact [2])
P(E|N)P(N) + P(~N) ⇒ (by law of probability)
P(E|N)P(N) + 1 - P(N)
If P(E|N) → 0 with P(N) fixed (say c), we get
P(E) → 0/c + 1 - c = 1 - c
From this
P(N|E) = k P(E) → k (1-c) /\ k → 0 / (1-c)
so that
P(N|E) → 0
I believe you can use latex in less wrong. It says so under “show help”. Let me try… $e^{i\pi}+1=0$
EDIT: Forget it. I just reread the help text. You have to use an external website to render the equation into an image...
Thanks for the incredible essay. I have to read it more thoroughly when I find the time. The point about the priors rejecting the hypothesis is a point I never considered and something that should be hammered to those of us working with data (and only consider hypothesis testing using null hypothesis). I wonder what other more mundane problem this might apply to...
Aren’t you simply begging the question? If you have a universe where a fact happens almost surely and another where it almost never happens, but that fact indeed happens, simple logic dictates that the first universe should be favored.
However, I’ve seen no reason to justify that the probability of life should tend to zero given our normal universe nor that the magical universe should be the only other alternative.
How am I begging the question? That isn’t clear.
I don’t believe that the probability of life in our universe is almost zero, but you can read all about the Fine-Tuned Universe argument at Wikipedia which is how this claim is most often made. My post is not targeted at the question of whether life really is that unlikely (I suspect that the parameters needed for life are not as strict as often claimed, only for life as we know it). The question I am trying to examine is whether the anthropic principle is a sufficient counter to the Fine-Tuned Universe (it is not); I’ll leave it to other people to discuss other objections to the Fine-Tuned Universe.