# “weight” input in glm and lm functions in R

I am confused with the definition of the weights in glm and lm.

Using the McCullagh and Nelder (1989)'s notation, If random variable $y_i$ is from the Generalized Linear Model (GLM), then its density is modelled in the form:

$$f(y_i) = exp\Big(\frac{m_i}{\phi} [\theta_i y_i - b(\theta_i) ] + c(y_i;\phi)\Big)$$

where $\theta_i$ is the canonical parameter, $\kappa$ is the dispersion parameter and $m$ is the known prior "weight". I would like to know that this prior "weight" is NOT the weight specified in glm. help(glm) says that:

Non-NULL weights can be used to indicate that different observations have different dispersions (with the values in weights being inversely proportional to the dispersions); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations. For a binomial GLM prior weights are used to give the number of trials when the response is the proportion of successes: they would rarely be used for a Poisson GLM.

Therefore, in my understanding, what "weight" $w_i$ does is to re-parameterize the dispersion parameter as

$$\phi=\frac{\phi^*}{w_i},$$

where $\phi^*$ is the redefined dispersion parameter. This means that for example, when $y_i$ is modelled only with an intercept term $\beta_0$, lm function with non NULL "weight" specification maximizes the sum of the weighted likelihood of $y_i$ with respect to $\phi^*$ and $\beta_0$ where:

$$f(y_i)=\sqrt{ \frac{w_i}{2\pi \phi^*} } \exp\Big(-\frac{1}{2}\frac{w_i (y-\beta_0)^2}{\phi^*}\Big),$$ where the identity link is used $\beta_0=\theta_i$.

Similarly, glm function with family = "poisson" with non NULL "weight" maximizes the sum of the weighted likelihood of $y_i$ with respect to $\beta_0$ where:

$$f(y_i)=\frac{\beta_0^{w_i y_i}}{y_{i}!} exp(-w_i \beta_0),$$

where the log link is used $\beta_0=exp(\theta_i)$.

Similarly, glm function with family = "binomial" with non NULL "weight" maximizes the sum of the weighted likelihood of $y_i$ with respect to $\phi^*$ and $\beta_0$ where:

$$f(y_i)= \begin{pmatrix} m\\ y_i \end{pmatrix} \beta_0^{w_iy_i}(1-\beta_0)^{w_i(m-y_i)}$$

where logit link is used $\beta_0 = logit^{-1}(\theta_i)$.

Is my understanding correct?

Reference:

C.E. McCulloch and J.A. Nelder. Generalized Linear Models. Chapman and Hall, London, 1989.

The book "Modern Applied Statics with S" written by W.N Venables and B.D Ripley (Fourth edition) defines GLM model for $y_i$ as:
$$f(y_i;\theta_i, \phi)=\exp \Big( \frac{A_i (y_i\theta_i-b(\theta_i))}{\phi}+c(y_i,\phi/A_i)\Big)$$
"Prior weights $A_i$ may be specified using weight argument."