## Posts Tagged ‘**Bayes’ Theorem**’

## Maths With Morals

The S I is proud to present Bayes’ theorem, a mathematical treasure and ethical dilemma all in one:

This equation gives us a mathematical formalism for updating old opinions with new evidence.

We begin with our initial hypothesis, *H*, and the probability, in our opinion, that *H* is true, *p(H)*. Say you’re sitting next to a young man on a bus. Does this man want to kill you? Probably not, you think. Most people are not killers, so the chance of having one sitting next to you is very slight – *p(H)* is low. Note that this number *p(H)* is a knowable *fact*: it comes from crime statistics.

But then he reaches into his briefcase and pulls out a sharpened axe. This provides new evidence, *E*: our man has an axe.

Now, not everyone who carries an axe is a killer. Some are, some are not. But it certainly changes your assessment of the situation. What is your new assessment? In the equation, it is written as *p(H|E)* _{}– the probability of hypothesis *H* being true *given *new evidence *E*. We base the calculation on one more term, *p(E)*, which basically translates to ‘how often killers carry axes with them’. This number is also a fact, and can be found from case studies of murders.

Putting these numbers together allows you to determine, with mathematical exactness, just how worried you should be when your scary-looking travel companion starts to grin at you and make suggestive slashing motions.

Bayes’ theorem is used all the time in science, finding uses in artificial intelligence, drugs testing, even searches for archaeological ruins. So why did I say it was an ethical dilemma? The answer is easy enough to see when you repeat the story given above, but change *E*. You are sitting by yourself on the bus when someone sits next to you. Probably not dangerous. But you look up and learn new information: the man is – well, pick your prejudice. Black? White? Muslim? Christian? Homeless?

Isn’t it the case that Bayes theorem takes knowable *facts* about the world, and turns them into a kind of statistically valid and logically justified racism?

No. Yes, the reasoning is sound, and in some cases when time and resources are extremely scarce it is unfortunately necessary to treat people as representatives of groups rather than individuals.* But this is always evil – sometimes necessary, but always evil. And what makes the racists different is that they are *content* with it. If, facing a selection of candidates for a job, with all the time in the world do make a decision, you don’t look at their CVs *because you are satisfied with what race alone tells you* – that is racism, and rotten to the core.

As is so often the case, Hitchens says it best:

“It especially annoys me when racists are accused of ‘discrimination.’ The ability to discriminate is a precious facility; by judging all members of one ‘race’ to be the same, the racist precisely shows himself incapable of discrimination.”

REFERENCES

Christopher Hitchens – *Letters to a Young Contrarian*

Also Richard Dawkins’s chapter on racism in *The Ancestor’s Tale* is highly recommended

* Blood donation is a good example. In the UK, people from South America are not allowed to give blood because of their higher probability of carrying Chagas disease. This is a decision to treat everyone from one ethnic group in the same way. Of course the ideal solution would be to test everybody for the disease wherever they come from; but in a world of scarce resources, time spent testing is time that could also be spent collecting more blood from ‘safe’ groups. So high-risk groups are eliminated out of hand, ignoring the individuals, so that lives can be saved. Evil, but necessary.