One of the questions most frequently asked of me is what are
the chances of a no-deal Brexit? In common with most analysts I tend to give an
answer couched in subjective probability terms. This has the advantage that it
is my view – nobody can say it is definitively wrong. Others may not agree with
it, but they have the right to apply their own subjective assessments. However,
this does strike me as a bit imprecise and led me to wonder whether it might be
possible to derive a more accurate data-driven assessment of the odds of no
deal.
One way to approach the problem is in terms of Bayesian statistics. In simple terms, Bayesian statistics assesses probability in terms of the degree of belief in an event. This contrasts with the more traditional frequentist school of statistics which represents probability as the number of times an event will occur based on an infinite number of representations of the process (I looked at this issue in more detail here). Bayesian statistics has always struck me as a sensible way to approach problems like Brexit where belief plays a big role and where the frequentist approach, which is based on the assumption that an event can be repeated, is unrealistic.
Bayesian statistics is based on Bayes’ Theorem which describes the conditional probability of an event based upon prior beliefs (or information) and which is written thus:
One way to approach the problem is in terms of Bayesian statistics. In simple terms, Bayesian statistics assesses probability in terms of the degree of belief in an event. This contrasts with the more traditional frequentist school of statistics which represents probability as the number of times an event will occur based on an infinite number of representations of the process (I looked at this issue in more detail here). Bayesian statistics has always struck me as a sensible way to approach problems like Brexit where belief plays a big role and where the frequentist approach, which is based on the assumption that an event can be repeated, is unrealistic.
Bayesian statistics is based on Bayes’ Theorem which describes the conditional probability of an event based upon prior beliefs (or information) and which is written thus:
P(A | B) = P(B | A) . P(A)
P(B)
This says that P(A | B), which is the probability of event A
conditional on event B, depends on the probability of the event B given that A
is true (the so-called posterior probability), and the prior probabilities of A
and B.
To put this into Brexit terms, let us assume that event A is a no-deal Brexit and B is the event Remain. We are interested in the probability of a no-deal Brexit, P(A). We can rewrite this:
To put this into Brexit terms, let us assume that event A is a no-deal Brexit and B is the event Remain. We are interested in the probability of a no-deal Brexit, P(A). We can rewrite this:
P(A) = P(A | B) . P(B)
P(B
| A)
or
P(No-deal Brexit) = P(No-deal Brexit | Remain) . P(Remain)
P(Remain | No-deal Brexit)
Using data from the website What UK Thinks,
we can use survey data to provide some of the evidence. Based on the question
of how people would be likely to vote in a second referendum, we can derive a
value for P(Remain) which is currently 44% (or 51% once we strip out those
voters who either do not intend to vote or who have reported as “don’t know”).
We proxy the value for P(Remain | No deal Brexit) by looking at the survey
evidence which asks respondents to choose between no-deal and Remain. Faced
with this choice, 44% of respondents indicated that they would vote Remain (54%
once we strip out don’t knows). Similarly, we estimate a value for P(No deal |
Remain) by looking at the proportion who would vote for no-deal if the
alternative is Remain (38%, or 46% after adjusting for “don’t knows”).
Putting these numbers together gives a constrained
probability of no-deal of 44% (where the constrained probability is derived by
stripping out “don’t knows” and constraining the remaining categories to sum to
100%). The unconstrained probability, where we do not adjust for the “don’t
knows”, comes out around 36%. The good news is that however we slice it, the
numbers come out at less than 50%; the bad news is that they are higher than my
subjective probability of 30%.
To the extent that politicians are looking at the polling numbers to assess whether a no-deal policy makes sense, this Bayesian interpretation of the evidence suggests that there is more support for a no-deal Brexit than is often supposed. However, it is not high enough to push through with it in the event that it proves to be economically disastrous, since more than 50% of voters do not support it and the backlash from the disaffected half is likely to be severe. One caveat is that some of the surveys have not been updated for a few weeks and therefore do not reflect any possible change of stance since Boris Johnson became prime minister. Moreover, Bayesian probabilities change as more information becomes available, so these are not set in stone by any means.
Bayesian statistics are the big thing these days but as the pollster Nate Silver pointed out, “under Bayes' theorem, no theory is perfect. Rather, it is a work in progress, always subject to further refinement and testing.” The prime minister’s new chief adviser, Dominic Cummings, pointed out in a blog post in 2017 “Rationality is more than ‘Bayesian updating’”. But cold hard statistics do catch up with you eventually and the evidence suggests that support for a no-deal Brexit is limited. If I were a politician it would not be the ground on which I would want to fight a battle – irrespective of what the prime minister says these days.
To the extent that politicians are looking at the polling numbers to assess whether a no-deal policy makes sense, this Bayesian interpretation of the evidence suggests that there is more support for a no-deal Brexit than is often supposed. However, it is not high enough to push through with it in the event that it proves to be economically disastrous, since more than 50% of voters do not support it and the backlash from the disaffected half is likely to be severe. One caveat is that some of the surveys have not been updated for a few weeks and therefore do not reflect any possible change of stance since Boris Johnson became prime minister. Moreover, Bayesian probabilities change as more information becomes available, so these are not set in stone by any means.
Bayesian statistics are the big thing these days but as the pollster Nate Silver pointed out, “under Bayes' theorem, no theory is perfect. Rather, it is a work in progress, always subject to further refinement and testing.” The prime minister’s new chief adviser, Dominic Cummings, pointed out in a blog post in 2017 “Rationality is more than ‘Bayesian updating’”. But cold hard statistics do catch up with you eventually and the evidence suggests that support for a no-deal Brexit is limited. If I were a politician it would not be the ground on which I would want to fight a battle – irrespective of what the prime minister says these days.
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