Category: Bayesian

Junk (filter) science

This is a mirror of a PLOS blogpost. Formatting is usually nicer there.

This is part 4 of a series of introductory posts about the principles of climate modelling. Others in the series: 1 | 2 | 3

In the previous post I said there will always be limits to our scientific understanding and computing power, which means that “all models are wrong.” But it’s not as pessimistic as this quote from George Box seems, because there’s a second half: “… but some are useful.” A model doesn’t have to be perfect to be useful. The hard part is assessing whether a model is a good tool for the job. So the question for this post is:

How do we assess the usefulness of a climate model?

I’ll begin with another question: what does a spam (junk email) filter have in common with state-of-the-art predictions of climate change?

Wall of SPAM
Modified from a photo by freezelight

The answer is they both improve with “Bayesian learning”. Here is a photo of the grave of the Reverend Thomas Bayes, which I took after a meeting at the Royal Statistical Society (gratuitous plug of our related new book, “Risk and Uncertainty Assessment for Natural Hazards”):

Bayes' grave

Bayesian learning starts with a first guess of a probability. A junk email filter has a first guess of the probability of whether an email is spam or not, based on keywords I won’t repeat here. Then you make some observations, by clicking “Junk” or “Not Junk” for different emails. The filter combines the observations with the first guess to make a better prediction. Over time, a spam filter gets better at predicting the probability that an email is spam: it learns.

The filter combines the first guess and observations using a simple mathematical equation called Bayes’ theorem. This describes how you calculate a “conditional probability”, a probability of one thing given something else. Here this is the probability that a new email is spam, given your observations of previous emails. The initial guess is called the “prior” (first) probability, and the new guess after comparing with observations is called the “posterior” (afterwards) probability.

The same equation is used in many state-of-the-art climate predictions. We use a climate model to make a first guess at the probability of future temperature changes. One of the most common approaches for this is to make predictions using many different plausible values of the model parameters (control dials): each “version” of the model gives a slightly different prediction, which we count up to make a probability distribution. Ideally we would compare this initial guess with observations, but unfortunately these aren’t available without (a) waiting a long time, or (b) inventing a time machine. Instead, we also use the climate model to “predict” something we already know, to make a first guess at the probability of something in the past, such as temperature changes from the year 1850 to the present. All the predictions of the future have a twin “prediction of the past”.

We take observations of past temperature changes – weather records – and combine them with the first guess from the climate model using Bayes’ theorem. The way this works is that we test which versions of the model from the first guess (prior probability) of the past are most like the observations: which are the most useful. We then apply those “lessons” by giving these the most prominence, the greatest weight, in our new prediction (posterior probability) of the future. This doesn’t guarantee our prediction will be correct, but it does mean it will be better because it uses evidence we have about the past.

Here’s a graph of two predictions of the probability of a future temperature change (for our purposes it doesn’t matter what) from the UK Climate Projections:

The red curve (prior) is the first guess, made by trying different parameter values in a climate model. The predicted most probable value is a warming of about three degrees Celsius. After including evidence from observations with Bayes’ theorem, the prediction is updated to give the dark blue curve (posterior). In this example the most probable temperature change is the same, but the narrower shape reflects a higher predicted probability for that value.

Probability in this Bayesian approach means “belief” about the most probable thing to happen. That sounds strange, because we think of science as objective. One way to think about it is the probability of something happening in the future versus the probability of something that happened in the past. In the coin flipping test, three heads came up out of four. That’s the past probability, the frequency of how often it happened. What about the next coin toss? Based on the available evidence – if you don’t think the coin is biased, and you don’t think I’m trying to bias the toss – you might predict that the probability of another head is 50%. That’s your belief about what is most probable, given the available evidence.

My use of the word belief might trigger accusations that climate predictions are a matter of faith. But Bayes’ theorem and the interpretation of “probability” as “belief” are not only used in many other areas of science, they are thought by some to describe the entire scientific method. Scientists make a first guess about an uncertain world, collect evidence, and combine these together to update their understanding and predictions. There’s even evidence to suggest that human brains are Bayesian: that we use Bayesian learning when we process information and respond to it.

The next post will be the last in the introductory series on big questions in climate modelling: how can we predict our future?