Assume that we observe \(N\) vectors of attributes, \(\mathbf{x}_{1}, \ldots, \mathbf{x}_{N}\) and associated integer counts \(t_{1}, \ldots, t_{N}\). A Poisson likelihood would be suitable:

\[p\left(t_{n} \mid \mathbf{x}_{n}, \mathbf{w}\right)=\frac{f\left(\mathbf{x}_{n} ; \mathbf{w}\right)^{t_{n}} \exp \left\{-f\left(\mathbf{x}_{n} ; \mathbf{w}\right)\right\}}{t_{n}!}\]

Assuming a Gaussian prior on w, derive the gradient and Hessian needed to use a Newton-Raphson routine to find the MAP solution for the parameters w.