Let's say we have multinomial counts $y_{jp}$ (corresponding to observations $j$ over categories $p=1,...P$) that are arranged in a table of $n$ rows and $P$ columns. Then say we have the log-linear model:
$log(\mu_{jp}) = \alpha_j + x_i \beta_p,$
and $y_{jp} \sim Poisson(\mu_{jp})$
Note here that $\alpha_j, j=1,....,n$ are observation specific parameters, and we can assume that $\beta_K = 0$.
I need to show that the MLE for $\beta_1,...\beta_{p-1}$ for this log-linear model is equivalent to the multinomial logit model with probabilities:
$p_{jp} = \frac{exp(x_j \beta_p)}{\Sigma_{w=1}^{P} exp(x_j \beta_w)}$
I think this should be a straightforward question, but I'm having trouble getting the form of the log-likelihoods right. The multinomial MLE is what really messed me up, although I'm not 100% sure what the correct form of MLE actually is, which is why I feel like I'm currently running in circles. Can anyone see how they're equivalent? I'd really appreciate any help.
