Anthropics and Biased Models

casebash

6 Anthropics and Biased Models

15th Apr 2016

7 min read

6

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.

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)

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

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6 Anthropics and Biased Models

by casebash

15th Apr 2016

7 min read

6

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)

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.

Coin flip argument

Edit - Extra Perspective: Null hypothesis testing

Summary

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)

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

Anthropics

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