## Cosmology and the Born-Again Bayesians!

The other day, via Twitter, I came across an interesting blog post about the relatively recent resurgence of Bayesian reasoning in science. That piece had triggered a discussion about why cosmologists seem to be largely Bayesian in outlook, so I thought I’d share a few thoughts about that. You can find a lot of posts about various aspects of Bayesian reasoning on this blog, e.g. here.

When I was an undergraduate student I didn’t think very much about statistics at all, so when I started my DPhil studies I realized I had a great deal to learn. However, at least to start with, I mainly used frequentist methods. Looking back I think that’s probably because I was working on cosmic microwave background statistics and we didn’t really have any data back in 1985. Or actually we had data, but no firm detections. I was therefore taking models and calculating things in what I would call the *forward * direction, indicated by the up arrow. What I was trying to do was find statistical descriptors that looked likely to be able to discriminate between different models but I didn’t have the data.

Once measurements started to become available the inverse-reasoning part of the diagram indicated by the downward arrow came to the fore. It was only then that it started to become necessary to make firm statements about which models were favoured by the data and which weren’t. That is what Bayesian methods do best, especially when you have to combine different data sets.

By the early 1990s I was pretty much a confirmed Bayesian – as were quite a few fellow theorists -but I noticed that most observational cosmologists I knew were confirmed frequentists. I put that down to the fact that they preferred to think in “measurement space” rather than “theory space”, the latter requiring the inductive step furnished by Bayesian reasoning indicated by the downward arrow. As cosmology has evolved the separation between theorists and observers in some areas – especially CMB studies – has all but vanished and there’s a huge activity at the interface between theory and measurement.

But my first exposure to Bayesian reasoning came long before that change. I wasn’t aware of its usefulness until 1987, when I returned to Cambridge for a conference called *The Post-Recombination Universe* organized by Nick Kaiser and Anthony Lasenby. There was an interesting discussion in one session about how to properly state the upper limit on CMB fluctuations arising from a particular experiment, which had been given incorrectly in a paper using a frequentist argument. During the discussion, Nick described Anthony as a “Born-again Bayesian”, a phrase that stuck in my memory though I’m still not sure whether or not it was meant as an insult.

It may be the case for many people that a relatively simple example convinces them of the superiority of a particular method or approach. I had previously found statistical methods – especially frequentist hypothesis-testing – muddled and confusing, but once I’d figured out what Bayesian reasoning was I found it logically compelling. It’s not always easy to do a Bayesian analysis for reasons discussed in the paper to which I linked above, but it least you have a clear idea in your mind what question it is that you are trying to answer!

Anyway, it was only later that I became aware that there were many researchers who had been at Cambridge while I was there as a student who knew all about Bayesian methods: people such as Steve Gull, John Skilling, Mike Hobson, Anthony Lasenby and, of course, one Anthony Garrett. It was only later in my career that I actually got to talk to any of them about any of it!

So I think the resurgence of Bayesian ideas in cosmology owes a very great deal to the Cambridge group which had the expertise necessary to exploit the wave of high quality data that started to come in during the 1990s and the availability of the computing resources needed to handle it.

But looking a bit further back I think there’s an important Cambridge (but not cosmological) figure that preceded them, Sir Harold Jeffreys whose book *The Theory of Probability* was first published in 1939. I think that book began to turn the tide, and it still makes for interesting reading.

P.S. I have to say I’ve come across more than one scientist who has argued that you can’t apply statistical reasoning in cosmology because there is only one Universe and you can’t use probability theory for unique events. That erroneous point of view has led to many otherwise sensible people embracing the idea of a multiverse, but that’s the subject for another rant.

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May 10, 2021 at 4:00 pm

In Cambridge, Steve Gull was the pioneer, having been dripfed maximum entropy by Geoff Daniell, a former research student of Ken Budden (as was I), who took a position at Southampton. By reading Ed Jaynes, Steve then became a convinced Bayesian, and passed it on to John Skilling via St Johns College.

Steve and Anthony Lasenby also kicked off the Clifford Algebra revolution in Cambridge a decade later, but I think this one started with Anthony rather than Steve. The ‘Jaynes of Clifford Algebra’ is David Hestenes.

In those days there was much discussion of “objective Bayesian” and “subjective Bayesian”. And some people use ‘Bayesian’ to denote a way of finding a sufficient statistic – a concept which is as un-Bayesian as you can get. All of this verbiage, including the argument about the definition of probability, is got round by arguing that a number representing how strongly one binary proposition (“The data take value x”; “the hypothesis is true”)) implies another is what you actually want in all problems of this sort, and using consistency arguments to set up functional equations for this quantity – whose solution leads to the sum and product rules. Call the quantity whatever you like; it is what you need, and it obeys the sum and product rules.

Sir Harold Jeffreys was a Bayesian at a far harder time to be one, and it is a pity that he is remembered today more for his opposition to plate tectonics.

May 10, 2021 at 10:37 pm

This reminds me that I must invite John Skilling to Maynooth for a seminar…

May 11, 2021 at 8:24 am

I have fond memories of the Post-Recombination Universe conference, and can recall that particular jibe. I think the conference was the first time I met you. So long ago now…

May 23, 2021 at 2:02 pm

I think the failure to understand that quantum theory is (a generalisation of) probability theory has done worse. I remember feeling quite annoyed after reading Jaynes etc. and reflecting on what I’d been taught of probability and statistics as an undergraduate. I was [am] furious about the way quantum mechanics was [is] taught.