P\left\{Y\left|X ight| ight\}=\prod \:j\frac{e^{-j}\cdot \:^{yj}}{yj!}

Tasks to be accomplished in the case study a) Show that the likelihood that the total emissions received by all detector pairs is y given that the expected number of emissions from all voxels is x is given by: P(y | x) = II e i Vi Vi where y ; = >Gijx i (Use hints given at the end of this document.) Prove the final expression for the maximum likelihood estimation objective function fuzz- Hint: Take the logarithm of the likelihood and omit the constant term that does not depend on x. b) Derive the formulas for the gradient and Hessian matrix of fin. c) Show that the Hessian of fizz may be dense, even when its matrix factors are sparse. Do this by considering C = (I I en) and y = y = e, . where I is the identity matrix, and ex is a vector of ones of size k. Show that every element of the Hessian is nonzero