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The Maximum Entropy Method

Generalizing MEM is quite easy: one maximizes the entropy of $\vec{\cal I}\,$ subject to $\chi^{2}_{}$ taking some specified value, and possibly that the integral of $\vec{\cal I}\,$ be equal to some value. Nityananda and Narayan (1982) show that for an entropy measure H(), the entropy of polarized radiation is given the sum of the entropies for the two independent polarization states of the radiation (i.e. eigenvalues of the coherence matrix $\cal {B}$ .)

H $\displaystyle \left(\vphantom{\vec{\cal I}}\right.$ $\displaystyle \vec{\cal I}\,$ $\displaystyle \left.\vphantom{\vec{\cal I}}\right)$ = H $\displaystyle \left(\vphantom{I+\sqrt{Q^2+U^2+V^2}}\right.$ I + $\displaystyle \sqrt{Q^2+U^2+V^2}$ $\displaystyle \left.\vphantom{I+\sqrt{Q^2+U^2+V^2}}\right)$ + H $\displaystyle \left(\vphantom{I-\sqrt{Q^2+U^2+V^2}}\right.$ I - $\displaystyle \sqrt{Q^2+U^2+V^2}$ $\displaystyle \left.\vphantom{I-\sqrt{Q^2+U^2+V^2}}\right)$

(11)

One then calculates the gradient of H with respect to $\vec{\cal I}\,$ and in combination with the gradient of $\chi^{2}_{}$ , generates a search direction with which to update the current estimate of $\vec{\cal I}\,$ (see Holdaway, 1990 for an algorithm).

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