What is the philosophy behind the logarithm as link function for Poisson regression? What is it about count variables that modelling them as a GLM with the natural logarithm function is appropriate?
I suppose it has something to do with the definition of the Poisson distribution, for which the probability function is exponential in lambda?
The point is that in Poisson regression you are dealing with count variables, which are non-negative by definition. Using the log-link assures that the modelled means on the count scale are strictly positive. This has nothing to do with the exponential function in the Poisson density function. In theory you could use any link function, but e.g. an identity link may yield nonsensical, negative means for some combinations of predictor variables.
