According to some psychology research (eg Pelham, Mirenberg, and Jones, people are slightly more likely to choose occupations that sound like their names: “people named Dennis or Denise are overrepresented among dentists”. Other research (eg Simonsohn (2011)) claims this result is spurious. I haven’t read these papers and I probably don’t have enough expertise to judge them, so my prior on this hypothesis is somewhere around 50%.
But my own dentist is named Dennis.
Dennis isn’t that common of a name. Should this be a strong update in favor of the hypothesis? Or does it not matter that much?
The Pelham paper found that Dennises were only about 1% more likely than the base rate to go into dentistry, so even if the hypothesis is true, it’s improbable that my dentist would be named Dennis. So perhaps I should believe that the true effect size is even larger than what Pelham found?
I would imagine that if there is an effect it works throught something like alliterative options invoking a stronger sense of identity by being cooler thus making them able to overcome more contrary influences.
Let’s limit our attention to the three hypotheses (a) there is no correlation between names and occupations, (b) the Pelham paper is right that Dennises are about 1% more likely to go into dentistry, and (c) the effect is much larger, e.g. Dennises are 100% more likely to go into dentistry. Then Bayes’ theorem says observing a Dennis in dentistry increases the odds ratio P(b)/P(a) by a factor of 1% and the odds ratio P(c)/P(a) by a factor of 100%. You say you consider (a) and (b) to each have prior probability of 50%, which presumably means (c) has negligible prior probability. Applying Bayes’ theorem means (a) has a posterior probability of slightly less than 50%, (b) slightly more than 50%, and (c) still negligible.
So no, observing a Dennis in dentistry does not produce a strong update in favor of the hypothesis that there is a correlation between names and occupations (i.e. the union of (b) and (c)).