## 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.