Poisson Log-likelihood Function Sample Clauses

Poisson Log-likelihood Function. ‌ With different energy levels, the forward model is nonlinear and it is not possible to transform it into an equivalent linear model. Xxxxx-Xxxxxxxxxx et al. [7] use the assumption that each entry in y follows a Poisson distribution,

Poisson Log-likelihood Function. ‌ With different energy levels, the forward model is nonlinear and it is not possible to transform it into an equivalent linear model. Xxxxx-Xxxxxxxxxx et al. [7] use the assumption that each entry in y follows a Poisson distribution, y ∼ Poisson exp −AW CT s . (3.10) With this assumption, one can formulate a maximum likelihood estimator (MLE) based on Poisson distribution. Before we show the expressions of objective function, gradient and Hessian, we first note that if the weights (unknowns) are stored as a matrix, then differentiation results in tensors, requiring tedious bookkeeping in the computations. In this work we instead rewrite the function to put the unknowns in vector form, and differentiate the objective function with respect to this vector. Notice that vec(AW CT ) = (C ⊗ A) vec (W ) , (3.11) where vec(·) reshapes a given matrix into a vector by stacking the columns on top of each other. Therefore, we can rewrite Equation (3.9) as y = (sT ⊗ I) exp [− (C ⊗ A) vec (W )] + η. (3.12) If we let w = vec (W ) and K (w) = exp {− (C ⊗ A) w} , (3.13) then Equation (3.12) is equivalent to y = (sT ⊗ I)K (w) + η. (3.14) In this problem, we assume that each element of measured data, yi, follows a Poisson distribution with mean sT ⊗ I K(w) . That is to say, i yi ∼ Poisson sT ⊗ I K(w) . (3.15) Based on this assumption, the corresponding probability density function can be expressed as fy (y; w) = NYd×Np [(sT ⊗ I)K(w)]yi exp [− sT ⊗ I K(w)] y !