Sunday, June 12, 2011

Bayesian Updating about a Politician During a Sex Scandal

Every once in a while, Corrections hears the erroneous argument that we shouldn't care about politicians having sex scandals.  Another version of the argument goes that we should only care about Republicans having sex scandals, for various reasons.  Corrections argues that even when one does not care at all about a politican's personal life, they should care and often condemn a politician in light of a sex scandal.

Why?  We posit that all politicians are unknowns.  We can imagine they have two traits: degree to which they engage in extramarital affairs and the degree to which they are willing to steal from the taxpayer.  Further, we argue these two traits are correlated.  So, we might have a multivariate normal with a covariance matrix with non-zero entries, as depicted in 3D in Figure 1 below (click to enlarge).
So, most likely (on average) the unconditional politician is at (0,0).  But what happens if we find out he's a philanderer?  Then, looking at this image from the perhaps more clear "birds eye view", we see that our new belief about him is that not only is he a philanderer, but he's also dishonest (click to enlarge).  The first, left hand circle is what we believe about a politician unconditionally, e.g. before the scandal.  Conditional on discovering he is a philanderer pushes us to the right of the graph.  Updating about his dishonesty, we find ourselves at a higher point there, as well!
There are, of course, more complex ways to do this, but the point is made: when distributions are not independently distributed, then information on a trait that you may not care about at all should influence your beliefs about something you do care about.

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