EX 4.6. Assume that we observe N vectors of attributes X,...,xN and associ- ated integer counts...

 

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EX 4.6. Assume that we observe N vectors of attributes X,...,xN and associ- ated integer counts t,...,tN. A Poisson likelihood would be suitable: p(tn/xn, w) = V" exp{-Vn} tn! where Vn = exp(wxn) 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 pa- rameters w. EX 4.7. Derive the Laplace approximation for the model in Exercise EX 4.6. EX 4.8. Implement a Metropolis-Hastings sampling scheme for the model of Ex- ercise EX 4.6 and compare the posterior with the Laplace approximation derived in Exercise EX 4.7. EX 4.6. Assume that we observe N vectors of attributes X,...,xN and associ- ated integer counts t,...,tN. A Poisson likelihood would be suitable: p(tn/xn, w) = V" exp{-Vn} tn! where Vn = exp(wxn) 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 pa- rameters w. EX 4.7. Derive the Laplace approximation for the model in Exercise EX 4.6. EX 4.8. Implement a Metropolis-Hastings sampling scheme for the model of Ex- ercise EX 4.6 and compare the posterior with the Laplace approximation derived in Exercise EX 4.7.