st: GLM question

["Cavallo, Alexander" <acavallo@lexecon.com>]

Title: GLM question

I am confused about the GLM model using normal errors and log link.  In Hardin and Hilbe's book, "Generalized Linear Models and Extensions" there is the following quote on page 59:

        "A better approach is to internalize withing the model itself the log transformation of the response. The log link in effect logs the linear predictor, or x*beta, rather than the response to linearize the relationship between the response and predictors. ....  The implementation of a log link within the ML algorithm is straightfoward - simple substitute ln(x*beta) for each instance of x*beta in the log-likelihood function.  ...  Creating a log-normal, or log-linked Gaussian, model using the standard IRLS algorithm is only a bit more complicated ... we must change the link function from eta=mu to eta=ln(mu), and the inverse link function from mu=eta to mu=exp(eta)."

For both ML and IRLS I thought we need to replace each instance of mu or x*beta in the log-likelihood function with exp(x*beta),  I don't understand why to use the link function log for ML but the inverse link function exp for IRLS.

Can anyone explain?

--Alex Cavallo
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