## On Probability and Cosmology

I just noticed a potentially interesting paper by Martin Sahlén on the arXiv. I haven’t actually read it yet, so don’t know if I agree with it, but thought I’d point it out here for those interested in cosmology and things Bayesian.

Here is the abstract:

Modern scientific cosmology pushes the boundaries of knowledge and the knowable. This is prompting questions on the nature of scientific knowledge. A central issue is what defines a ‘good’ model. When addressing global properties of the Universe or its initial state this becomes a particularly pressing issue. How to assess the probability of the Universe as a whole is empirically ambiguous, since we can examine only part of a single realisation of the system under investigation: at some point, data will run out. We review the basics of applying Bayesian statistical explanation to the Universe as a whole. We argue that a conventional Bayesian approach to model inference generally fails in such circumstances, and cannot resolve, e.g., the so-called ‘measure problem’ in inflationary cosmology. Implicit and non-empirical valuations inevitably enter model assessment in these cases. This undermines the possibility to perform Bayesian model comparison. One must therefore either stay silent, or pursue a more general form of systematic and rational model assessment. We outline a generalised axiological Bayesian model inference framework, based on mathematical lattices. This extends inference based on empirical data (evidence) to additionally consider the properties of model structure (elegance) and model possibility space (beneficence). We propose this as a natural and theoretically well-motivated framework for introducing an explicit, rational approach to theoretical model prejudice and inference beyond data.

You can download a PDF of the paper here.

As usual, comments are welcome below. I’ll add my thoughts later, after I’ve had the chance to read the article!

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December 12, 2018 at 5:06 pm

Before you apply probability to anything – let alone the universe – you have to understand it correctly. I was concerned when I reached the first equation in this paper (Bayes’ theorem) and saw the symbol partitioning the argument of the probability and the conditioning information written in two different ways, a solidus and a semi-colon.

Forget ‘probability’ for a moment and consider a number p(A|B) that represents how strongly B implies A, where A and B are binary propositions (true or false) – this number measures how strongly A is implied to be true upon supposing that B is true, according to relations known between their referents. Degree of implication is actually what you want in every problem involving uncertainty. Now, from the Boolean algebra obeyed by propositions it is possible to derive algebraic relations for the degrees of implication of the propositions. These relations turn out to be the sum and product rules – the ‘laws of probability.’

So let’s call degree of implication ‘probability’. But if anybody objects then bypass them; simply calculate the degree of implication in each problem, because it’s what you want in order to solve it, and call this quantity whatever name you like. The concept is what matters, not its name. In this viewpoint there are no worries over ‘belief’ or imaginary ensembles, and all probabilities are automatically conditional. The confusing word ‘random’ is downplayed. The confusion over ‘physical’ and ‘inductive’ probabilities which mars the present paper never comes into play.

The sum and product rules were first derived from Boolean algebra by RT Cox in 1946. (Cox did not use the rhetorical device advocated here about the name of the quantity.) I consider this to be a more compelling derivation than Kolmogorov’s, because it starts from far more compelling axioms, but Cox and Kolmogorov reach exactly the same equations and tinkering with their axioms and consequently worrying about negative probabilities and ‘pseudo-probabilities’ is futile. Knuth and Skilling (both of whom I know well) also run from the calculus of propositions to the calculus of probabilities, but use neater properties of the former than Cox (although Skilling doesn’t emphasise propositions as much as Knuth in their separate papers). All of these authors, though, end with the sum and product rules

and no alternative.Bayes’ theorem then follows immediately from the sum and product rules.Bayes’ theorem shows how to incorporate new information (phrased propositionally) into a probability. You want to do this whenever you get experimental data relating to a question of interest. In parameter estimation the propositions are something like “the parameter takes value between z and z+dz”. But how to assign the probability that you update, before the experiment? This is the issue of ‘prior probabilities’, which is called the ‘measure problem’ in the present paper. The point is that you always have prior physical information, and the question is how to encode it into a prior probability distribution. There is no one answer to that – it depends on the physical situation. It does not depend, however, on the sampling distribution, as is asserted in the second complete sentence on p4. Symmetry is one important principle here.

Only if all of this is understood is there any hope of shedding light using probability theory on the questions which underlie the cosmological notion of the multiverse.

December 13, 2018 at 7:53 am

From the arXiv “abstract”:

Slightly expanded version of contributed chapter.

Journal reference: K. Chamcham, J. Silk, J. Barrow, & S. Saunders (Eds.), The Philosophy of Cosmology (pp. 429-446), Cambridge University Press 2017

DOI: 10.1017/9781316535783.022

I reviewed this book (though not, in particular, this chapter) about a year ago.