This Week's Brain Bender:

How to Count Cosmologists - the Anthropic Measure Problem

Few simple ideas have been so confused as the Anthropic Principle (AP) introduced by Brandon Carter more than 30 years ago. The original formulation of the AP is as follows:

What we can expect to observe must be restricted by the conditions necessary for our presence as observers. - Brandon Carter, 1974

This principle is extreemly important. Unfortunately philosophers and phisicists still struggle with its complete interpretation and implimentation.

The Anthropic Principle represents an obviously necessary departure from the once-profound Copernican idea that we live in a very average part of the universe - not at the center. Pushed to it's extremities, a purely Copernican philosophy leads to eroneous predictions. For example, one might expect our astronomical neighbourhood to be a cold empty void since those overwhelming dominate the volume of the universe. The Anthropic Principle reconsiles the fact that we find ourselves in a rare stellar-rich, metal-rich, energy-rich environment because only those environments can be expected to host observers.

In the past, some have argued against the use of the Anthropic Principle on the count that it doesn't make predictions, and is therefore not a testable scientific theory. The Anthropic Principle IS NOT a scientific theory... it is a principle demanding that observational selection effects are accounted for when comparing theories with observations. The importance of the AP varies from application to application, but must not be a-priori ignored.

So, how does one apply the anthropic principle quantitatively? Most people, including myself, subscribe to what is known the Self Sampling Assumption (SSA):

One should reason as if one were a random sample from the set of all observers in one's reference class.

from which follows the Bayesian conditional probability statement

The distribution of outcomes of some measurement is given by the prior probability distribution of possible outcomes, weighted by the relative likelihood of a measurement being taken for that outcome. The first term on the right is called the "prior" and is predicted by theory. The second term is the "observation selection effect". In most fields of physics (and the sciences in general) the act of experimentation is not correlated with the outcome of the experiment. In this simplified senario the distribution of experimental results is directly indicative of the prior; observation selection effects are irrelevant. But in cosmology, observation selection effects may dominate the above equation, as we have seen in the example above whereby only dense regions of the universe are observed dispite their relative rarity. Another example is given below.

EXAMPLE 1
In this example we use the AP and Bayesian conditional probability equation to draw predictions from a simple multiverse model. This multiverse model is made-up, but is similar (in principle) to several serious cosmological theories. The multiverse in this example consists of 101 parallel universes. One of these is what we will call a "type A" universe and the others are all the same - "type B". The type A universe is large and hospitable and is host to 10000 observer civilisations. The type B universes are small and hostile and typically host just 1 observer civilisation each. The following table summarized this information in terms of the quantities in the previous equation.

Therefore, although type B universes outnumber type A universes 100-to-1, the correct prediction from this multiverse theory is that the parameter x is ~99% certain to be observed to have a value x=x(type A).

THE REAL DIFFICULTIES WITH THE ANTHROPIC PRINCIPLE START NOW. What constitutes the set of observers in my reference class? Am I to proceed assuming that I am a randomly selected human? Or that I am a randomly selected member of some larger set of "observers"? Are apes part of my reference class? Are non-physicists? What should we count? (This is what I refer to as the anthropic measure problem) Do we count measurements? Do we count individuals? If 2 individuals are not independent do they count as 1 or 2? If an individual is told the answer to the measurement, does that count as a measurement? Should we count civilisations with equal weighting regardless of their size because the observations of their subjects are not independent?

I'm think we should be counting individuals. Dr Lineweaver would disagree. To be continued...