Thursday, 3 September 2015

Bayes' Rule and Political Inference

Consider a question such as: Would politician A be a good leader or not?  Now consider some information that might arrive that might inform our answer to that question in the form of the recommendation of an opinion former, maybe a newspaper makes a decision as to whether to endorse A or not.
Suppose an individual’s prior belief that A is a good leader is p, and their prior belief that the newspaper endorses a politician who is a good leader is q.  Once the news arrives, the newspaper will either endorse A or not.  If they do not endorse A, then the individual’s posterior probability that A is a good leader will be:
p'= p(1-q)/[p(1-q)+q(1-p)]
But the posterior belief about the accuracy of the newspaper’s endorsements will also have changed, and will indeed be the complement of the probability above:
q'= q(1-p)/[p(1-q)+q(1-p)]
Afterall, once the newspaper does not endorse A, either A is not a good leader or the newspaper does not make good endorsements. 
Suppose that the newspaper is a source that we would normally trust reasonably well, q=0.75.  It might, for example, be The Economist. But suppose that we are convinced on a level approaching religious fervour that politician A would be a good leader, and p=0.99.  Then as a result of the newspaper’s lack of endorsement for A, a bit of doubt will creep in, and  p'=0.97, but by far the biggest movement is in the probability that the newspaper endorses good candidates and doesn’t endorse bad ones as q'=0.03.
What if the newspaper endorses the politician?  Then the posterior probabilities in this case will be:
p''= pq/[pq+(1-p)(1-q)]
q''= pq/[pq+(1-p)(1-q)]
Similar results obtain if the individual detests the politician in question, and believes, with semi-religious fervour, that they are not a good leader.  Suppose that q=0.75, and p=0.01, and the newspaper endorses the politician.  Then the posterior probabilities will be p''= q''= 0.03.  So, once again, by far the biggest update  from the news of the recommendation is on the reliability of the recommender, rather than the subject of the recommendation.

The lesson?  When a group of people are absolutely convinced that a politician is the right/wrong person to lead, endorsements, or failures to endorse will be treated more as evidence of the accuracy of the person giving the endorsements rather than as evidence about the potential leader in question.  This offers some explanation as to why "marmite politicians" people who are either really liked or really hated, are able to maintain their core support in the face of such ferocious attacks as they frequently receive.  To their supporters, the attacks contain more information about the attackers than about their target.  To their detractors, any defence of them contains more information about the defenders than about the subject of  the defence.