Prosecutor’s fallacy — Brexit style
When it comes to Brexit, I’m finding harder and harder to concentrate and either pretend that nothing bad is happening or actually acknowledging all the madness going around (which, when I do, almost invariably makes me rather angry)…
Anyway: the other day, this exchange caught my eye. Now, I am fully aware I am biased towards the people who think Brexit is possibly the most stupid idea ever. But, I don’t really know much about Will Self (other than in this story he represents the “Remainers”); and I don’t know much (although, arguably, a lot more than I wish I had to) about Mark Francois (the “Brexiteer”). Basically, their sort of scary encounter was based on the following exchange:
Will Self (WS): Every racist and anti-Semite in the country pretty much probably voted for Brexit.
Mark Francois (MF): You’ve basically tried to slur anybody who voted Leave as a bigot.
Now, this is a classical case of “prosecutor’s fallacy” — what WS is making is an assessment of
But these two are quite different! So: first things first. The point here is a classic application of Bayes’ theorem
(I suppose, to be precise, I should clarify that this isn’t necessarily WS’s subjective probability — rather is my own subjective interpretation of WS’s subjective probability…).
So in order to conclude what MF has we really need another bit of information — that’s an estimate of the proportion of racists and anti-Semites in the general population,
That’s easily done in a sort of Monte Carlo simulation framework.
# Simulates from the subjective distribution of leavers among racists/anti-Semites
= rbeta(10000,99,4)
lmidr # Defines the marginal probability of voting leave
= 17410742/(17410742+16141241)
l # Defines a range of potential values for the probability of somebody being racist/anti-Semite
= seq(0,.5,.001)
r # Computes the posterior probability (what MF wants)
=matrix(NA,length(lmidr),length(r))
rmidlfor(i in 1:length(r)) {
=(lmidr*r[i])/l
rmidl[,i] }
Interestingly, because of how the problem is set up and the assumed relationships among the variables of interest, basically we have a natural upper limit for the probability that a random individual is racist or anti-Semite. This is of course because by necessity the ratio on the right-hand side of Bayes’ theorem has to be bounded by 1 (as it represents a probability). And so I don’t need to consider all possible scenarios for
The code above computes, for each assumed value of
The graph above shows a summary of the underlying probability distribution for
There’s of course a lot more to the story than the simple prosecutor’s fallacy — I honestly have no idea whether all racists/anti-Semites would in fact be Leavers. I too suspect the proportion might be very high, but that’s besides the point. I think what really matters is that politicians have no excuse for ignorance of these things, particularly when they try and score political points off them…