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By Voloshynovskiy, Herrigel, Baumgaertner, Pun

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Additional resources for A Stochastic Approach to Content Adaptive Digital Image Watermarking

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1 ????∑???? ∏???????????? −βλ −????∑???? ???? ???????? ???? ???? Γ(????)∏???????? ! ????????−1+∑???? ???? −λ(β+∑???????? ) ∞ ∫0 ????????−1+∑???? ???? −λ(β+∑???????? ) ???????? Using the identity Γ(????) = ∫???? ???? ????−1 ???? −???? ???????? we can calculate the denominator using the ???? ???????? change of variable ???? = ????(???? + ∑???????? ). This results in ???? = ????+∑???? , and ???????? = ????+∑???? ???? limits of ???? the same as ???? . Substituting back into the posterior equation gives: ????(????|????) = ????????−1+∑???? ???? −λ(β+∑???????? ) ∞ ????−1+∑???? 1 ???? � � � ???? −u ???????? ???? + ∑???????? ???? + ∑???????? 0 ???? with the Parameter Est Introduction Parameter Est 30 Probability Distributions Used in Reliability Engineering = Let ???? = ???? + ∑???? ????????−1+∑???? ???? −λ(β+∑???????? ) ∞ 1 ∫ ????????−1+∑???? ???? −u ???????? (???? + ∑???????? )????+∑???? 0 ????(????|????) = ∞ Using Γ(????) = ∫???? ???? ????−1 ???? −???? ????????: ????(????|????) = Let ???? ′ = ???? + ∑????, ???? ′ = ???? + ∑???????? : ????????−1+∑???? ???? −λ(β+∑???????? ) ∞ 1 ∫ ???? ????−1 ???? −u ???????? (???? + ∑???????? )????+∑???? 0 ????????−1+∑???? (???? + ∑???????? )????+∑???? −λ(β+∑???? ) ???? ???? Γ(???? + ∑????) ′ ????′ ???????? −1 ???? ′ ????(????|????) = Γ(???? ′ ) ′ ???? −β λ As can be seen the posterior is a gamma distribution with the parameters ???? ′ = ???? + ∑????, ???? ′ = ???? + ∑???????? .

1 = ????3 ????23 ⁄2 The kurtosis is a measure of the whether the data is peaked or flat. 3. Covariance ????2 = ????4 ????22 Covariance is a measure of the dependence between random variables. ????????????(????, ????) = ????[(???? − ???????? )(???? − ???????? )] = ????[????????] − ???????? ???????? A normalized measure of covariance is correlation, ????. The correlation has the limits −1 ≤ ???? ≤ 1. e, an increase in X will result in the same increase in Y). e, an increase in X will result in the same decrease in Y). The relationship between covariance and correlation is: ????????????(????, ????) ????????,???? = ????????????????(????, ????) = ???????? ???????? If the two random variables are independent than the correlation is equal to zero, however the reverse is not always true.

1. ∫0 ????(????|????) ???????? Using conjugate relationship (see Conjugate Priors for calculations): • ????~????????????????????(????; 1 + nF , tT ) Principle of Indifference - Proper Uniform Priors. An equal probability is assigned across the values of the parameter within a range defined by the uniform distribution. The uniform distribution is obtained by estimating the far 1 left and right bounds (???? and ????) of the parameter ???? giving ???????? (????) = = ????, ????−???? where c is a constant. When placed in Bayes formula the constant cancels out, however the denominator is integrated over the bound [????, ????].

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