Consider the GLM for the Poisson exponential family. Assume for simplicity that n=1. What is the log-likelihood function \ell (\mathbf Y,\mathbb {X},{\boldsymbol \beta }) with the canonical link function?
Use X for X_1, Y for Y_1, and + c for the constant term. Do not use \phi and instead use the actual value of \phi for the Poisson exponential family. To input a dot product a^ T b, write it as a*b.
\ell (\mathbf Y,\mathbb {X},{\boldsymbol \beta })=
Y*log(X*beta)-X*beta-log(Y!)+Y*log(1)-log(1)
The log-likelihood function for the Poisson exponential family with the canonical link function is given by:
\ell(\mathbf{Y}, \mathbb{X}, \boldsymbol{\beta}) = \sum_{i=1}^{n} \left( Y_i \cdot \boldsymbol{\beta}^T \mathbb{X}_i - e^{\boldsymbol{\beta}^T \mathbb{X}_i} - \log(Y_i!) \right)