Even though I studied it at university (long, long ago, in a universe far, far away), I have recently become re-acquainted with Bayesian probability (BP) as a result of some of the reading I’ve been doing on the subjects explored in this blog. It turns out that what seemed like an abstract mathematical concept at uni is actually a tremendously useful tool – both in a quantitative sense and in a conceptual/philosophical sense. Hopefully as a result of the brief discussion below it will become obvious how it relates to all the various things I’ve been ranting about over the past few months. And, don’t worry, there’s no maths involved unless you want to pursue it yourself.
In a nutshell, BP provides us a way to consider the likelihood of something, but in the context of our prior experience. In contrast, standard probability ignores anything we might already know which could affect the probability. Let’s look at two simplistic (admittedly extreme) examples which illustrate the differences:
- We are testing our theory that gravity accelerates all objects at the same rate, and we do 100 drops of a bowling ball and a tennis ball and get 100/100 correspondence in the acceleration (for the pedants out there: we’ve corrected for atmospheric drag).
- We are testing our theory that a test subject can read minds, and we do 100 reads of an experimenter drawing from a deck of playing cards, and get 100/100 correct reads by the subject.
Now, using standard probability both results are equally valid, and the researchers are equally happy to publish.
Using BP, we have a different view altogether. In the first experiment, the scientific community is inclined to accept the outcome, given what is already known about the natural world. In other words, our prior experience tells us, or at least hints, that the finding is consistent with what we already know.
In the second experiment, although the numerical results are strong, that same community is entitled to be sceptical, since the outcome is not consistent with anything known of the natural world (other than similar claims of course). Our prior experience tells us to be suspicious of the outcome, and hints to us that we should look for errors in the experimenters’ assumptions, experimental setup or other confounding factors. This consideration would in effect lead us to search for an alternative hypothesis which is more consistent with our knowledge of the natural world. For example, things like self-delusion, or cheating, are far more likely given our prior knowledge, than the discovery of mind-reading.
This, by the way, is the basis of one of Carl Sagan’s more famous quotes:
“Extraordinary claims require extraordinary evidence”
In other words, if the finding of the experiment is way beyond our existing understanding of the world, then the burden of evidence is accordingly greater.
The other interesting thing about BP, is that it is has learning built into it. That is, as our experience grows, it is possible to update the probability calculation to reflect that new knowledge. In effect we are constantly evaluating our hypotheses about the world, and updating our knowledge based on observation and experiment.
In my opinion, BP suggests to us why it is people believe in magic, or religion for that matter. In those cases, people assume zero prior knowledge of the natural world and are persuaded by what is placed before their eyes. I am reminded of Arthur C. Clarke’s third law:
“Any sufficiently advanced technology is indistinguishable from magic”
What this says to me is that if we have insufficient prior knowledge to evaluate what we are seeing we can come to the wrong conclusion about how it works. We are told it’s magic, and we just accept it. What Bayesianism says instead is that rather than accept an implausible explanation, let’s assess what we are told on the basis of that prior probability. In this case, where we are shown advanced technology and told it is magic, we would say – “I doubt it – I need more data/evidence and will update my belief after I get it”.
And for me, this whole subject explains a lot about how those who passionately believe in some pseudo-science came to their conclusions.If you’d like a nice discussion and worked example, I’d recommend Pages 276-278 of ‘Nonsense on Stilts’, the excellent book by Massimo Pigliucci http://www.rationallyspeaking.org/.