# Judge Bans Thinking

Hello!

I just saw this news story on singingbanana’s Tumblr:

The footwear expert made what the judge believed were poor calculations about the likelihood of the match, compounded by a bad explanation of how he reached his opinion. The conviction was quashed. But more importantly, as far as mathematicians are concerned, the judge also ruled against using [Bayes' theorem] in the courts in future.

*What*?

In case you don’t know, Bayes’ Theorem is a formula from the days when mathematicians could get away with just naming bits of common sense after themselves, and states:

P(A|B) = P(B|A) × P(A) ÷ P(B)

That is,

(probability A is true given evidence B) = (probability of finding B* if A is true*) × (probability of A being true) ÷ (probability of finding B)

Obviously that’s pretty useful in court, and it’s also pretty easy to understand: clearly, (probability of finding B) = (probability of finding B if A is true) × (probability of A being true) + (probability of finding B if A is* false*) × (probability of A being *false*). That’s all the options: A must be true or false. All Bayes’ theorem does is work out in what the proportion of the cases in which B is found, A is true. It’s almost obvious, in an informal kind of way, if you have the right kind of brain.

Point is, you’re now *not allowed* to use that reasoning in court, which means if your defence depends on any evidence more subtle than “fifteen people saw me across town” then you’d better find another way of phrasing it than this. It’s basically maths, so you ought to be able to work round it and find a more circuitous way of showing the same thing.

Strictly, the ruling was that you can’t use Bayes’ theroem “unless the underlying statistics are firm” but that doesn’t help a lot. In qualitative terms, all the theory says is that something is more likely if you have evidence for it – so I think we can safely assume it’s the use of maths that the judge objected to. It’s the same reasoning behind the Drake equation: people are bad at guessing big, complicated things like “how many aliens might there be” or “how likely is this guy to be guilty given we found this footprint”, but pretty good at estimating simple things like “how long might an alien race beam signals into space” or “how likely is it there’d be a footprint like this just here”. You can do the same thing if you get to the tie-break in a pub quiz: come up with a way to work out the answer from easier to estimate quantities and crunch the numbers. You’ll almost certainly do better than the team that guesses the final answer directly.

And *that* is why this ruling is worrying: because not only has a judge fallen foul of the natural but wrong tendency for humans to overestimate their own judgement and distrust logic and reason, but they’ve ruled that all other judges and lawyers have to make the same error. It’s a shame, because while any large group tends to be a bit rubbish at thinking, judges are usually pretty good – as, I think, is anyone impartial with the time and inclination to look into things.

I didn’t know they could do that. (I actually suspect they can’t and the story is overblown, but I wouldn’t know enough to decide.) What’s next – a judge commits the prosecutor’s fallacy and then rules that everyone else has to do it as well? Idunno, seems a bit dangerous to me.

It just seems like it’d be fairly easy to set someone up if there’s a big list of thought processes their defence lawyer isn’t allowed to invoke. Or design a crime that could be easily proven but not with the limited methods of thinking allowed in the courtroom.

There are some interesting mental exercises there – coming up with crimes or frames for differently handicapped justice systems – but not ones actual lawyers should have to bother with.