The Azimuth Project Separable function (Rev #2, changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

Separable function

Details

Multiplicatively separable

(1) F(\Theta,\mathbf{x})=\prod_{i=1:n} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x})

Then

(2) \frac{\partial F(\Theta,\mathbf{x})}{\partial \theta_j} =\frac{\partial f_j(\theta_j,x_j)}{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} \prod_{i=1:n,i \ne j} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x})
(3) \frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j^2} =\frac{\partial^2 f_j(\theta_j,x_j)}{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j^2} \prod_{i=1:n,i \ne j} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x})
(4) \frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j \partial \theta_k} =\frac{\partial f_j(\theta_j,x_j)}{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} \frac{\partial f_k(\theta_k,x_k)}{\partial f_k(\theta_k,\mathbf{x})}{\partial \theta_k} \prod_{i=1:n,i \ne j,k} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x})

Special case:

(5) F(\Theta,\mathbf{x})=\exp \sum_{i=1:n} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x})

Then

(6) \frac{\partial F(\Theta,\mathbf{x})}{\partial \theta_j} =\exp (\frac{\partial f_j(\theta_j,x_j)}{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} +\sum_{i=1:n,i \ne j} f_i(\theta_i,x_i)) f_i(\theta_i,\mathbf{x}))
(7) \frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j^2} =\exp (\frac{\partial^2 f_j(\theta_j,x_j)}{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j^2} +\sum_{i=1:n,i \ne j} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x}) )
(8) \frac{\partial^2 F(\Theta,\mathbf{x})}{\partial \theta_j \partial \theta_k} =\exp (\frac{\partial f_j(\theta_j,x_j)}{\partial f_j(\theta_j,\mathbf{x})}{\partial \theta_j} \frac{\partial f_k(\theta_k,x_k)}{\partial f_k(\theta_k,\mathbf{x})}{\partial \theta_k} +\sum_{i=1:n,i \ne j,k} f_i(\theta_i,x_i) f_i(\theta_i,\mathbf{x}) )