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?

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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.

Dacyn

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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)).

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I'm not sure how to do all of the math in context. but what if your dentist had been named Steve?