a bonobo humanity?

So I was listening to one of my science podcasts in my usual distracted way, when a segment came on about Bayesian inference, or reasoning, or logic, woteva, and I know I’ve written about this before, but I also know that if someone asked me to explain it, I’d be lost, caught out, shamed and disgraced. Me, an inner-lechal? Come now.

So, once more into the breach – and I’m not even going to look at anything I’ve written on the topic before, nor am I going to avoid the issue by going on about the Reverend Thomas Bayes’ relatively obscure 18th century life in Tunbridge Wells, where he… oh, sorry.

Bayes’ Theorem, or Rule, says that the probability of A, given B, is the probability of B given A, times the probability of A, all divided by the probability of B. Mathematically it looks like this:

P(A|B) = P(B|A).P|(A)/P(B), – which simplifies to

P(A|B) = P(A + B)/P(B)

It’s about statistical inference apparently, and also, to some degree, about common sense and everyday experience. One tutorial puts it this way. Someone tells you, very briefly, that a friend of hers has just been diagnosed with breast cancer. You’ve just recently learned that males too can develop breast cancer. You wonder if that friend is male. What are the chances?

So the probability of maleness, given breast cancer, is equal to the probability of having breast cancer, given maleness, multiplied by the probability of being male, all divided by the probability of having breast cancer. Now, the probability of breast cancer given maleness is about one in a thousand, according to the tutorial (i.e 0.001) And the probability of being male is about one in two, or 50% (0.5), while the overall probability of getting breast cancer is 0.063 (63 in a thousand). So, putting those stats together and doing the maths results in a tiny .79% likelihood of the friend being male, assuming the stats are reliable, and there aren’t any other confounding factors. That’s well under 1%, I have to remind myself. Then again, a more efficient way to find out, as the tutorial points out, is to just ask!

So what’s a more effective scenario for using Bayesian probability? Well…

The ratio of probability of a piece of ‘information’ being true under one hypothesis, compared to it being false, is called a Bayes factor, apparently – representing ‘the amount of information we’ve learned, re our hypothesis, from the available data’. This learning is used, or can be used, to update our prior belief or understanding, to a posterior one. The whole process is one of hypothesis testing, and the belief-changing that should be attendant upon that testing. As if…

So therein lies the problem, that we don’t subject our beliefs to hypothesis-testing, at least not often, and not very much at all when we live in a community that shares those beliefs. If the Reverend Bayes lived in 21st century Adelaide, he might not be so reverential about the putative father-and-son beings he was so reverential about, presumably, in 18th century Tunbridge Wells. But how can you subject your belief in a single, omniscient, ominipotent creator god to hypothesis-testing? That’s when the concept of evidence comes in, and not just evidence about beliefs. So it seems to me that Bayesian probability has rather limited applications.

And yet, it’s pretty obvious that Bayesian woteva – reasoning, inference, probability, priors, theorems etc – has been flavour of the month for months and months and months now (and isn’t ‘month’ a funny word, come to think of it? but I digress…), though there is push-back, and even something of a turf war, between Bayesian and frequentist-type reasoning, and there are articles and videos galore about all this stuff – and I’ve been around for sixty-odd years without giving any of it the slightest thought. Here’s a quick summary from somewhere:

the frequentist approach assigns probabilities to data, not to hypotheses, whereas the Bayesian approach assigns probabilities to hypotheses. Furthermore, Bayesian models incorporate prior knowledge into the analysis, updating hypotheses probabilities as more data become available.

As one nice video puts it, using a coin flip, for those who see the coin land, the datum shows that the coin shows heads, say, and this isn’t probable, it’s a fact. For those who don’t, there’s a 50% probability that the coin will show heads. The Bayesian bases her conjecture on her prior knowledge that coins have two sides, but if she learns that the coin-flipper is a trickster with a double-headed coin (thus updating her prior knowledge) she updates the hypothesis based on this datum. So it just seems to be a difference between data and knowledge of data. I’m not quite sure I understand what all the fuss is about. And yet… As Steven Pinker points out, in his book Rationality: 

In recent decades Bayesian thinking has skyrocketed in prominence in every scientific field. Though few laypeople can name or explain it, they have felt its influence in the trendy term ‘priors’, which refers to one of the variables in the theorem.

I haven’t myself noted the trendiness of priors but I’ve never really been in academia. In any case the term seems pretty basic, and I’m just not sure about the need to ‘mathematise’ it all. Pinker himself first describes Bayes’ Rule in verbal/arithmetical terms  – posterior probability = prior probability x likelihood of the data/commonness of the data – which he then translates into English, and after that into common sense, i.e ‘now that you’ve seen the evidence, how much should you believe the idea?’ So, if you’re an evidence-conscious type, you should generally be fine, methinks. I have heard it said, though, that many people even at high levels of academia trip themselves up because they ‘forget’ to apply Bayes’ Rule. I suspect, though, that it’s not so much forgetting as motivated reasoning or ‘my side bias’, generally a tougher nut to crack…

References

You know I’m all about that Bayes: crash course stats (video)

Are you Bayesian or Frequentist? video, Cassie Kozyrkov

Steven Pinker, Rationality, 2021

https://johnhorgan.org/cross-check/bayes-theorem-and-bullshit