By Morris J.S.

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**Extra resources for A Bayesian analysis of colonic crypt structure and coordinated response to carcinogen exposure incor**

**Example text**

1 r-variate standard t vector with degrees of freedom v.. +r)/2 K ft 1 + ST (vk/vk) BkBk k=1 k=1 K (vk/vk) BkBT dv1 ... 41) k=1 where r ((v. r/2r (v1/ 2) ... r (vK/2) and as above K or = {(vi. ,VK) : Y Ckvk = 1, Vk > 0 k=1 This identity has applications in Behrens-Fisher problems. +p V. 2 2 , , 2 , 2 +A Z1, z2 where C- r((v. +p)/2) 7r n/2r (v1/2) ... 1 II S II2 = 0. 38). 41) is a generalization of Ruben's (1960) integral representation (in the univariate case) for the usual Behrens-Fisher densities.

1 II S II2 = 0. 38). 41) is a generalization of Ruben's (1960) integral representation (in the univariate case) for the usual Behrens-Fisher densities. 1) have been studied in the literature with great detail. 1) when µ = 0 and R is given by the equicorrelation matrix 1 -1/p R = -1/p ... -1/p ... -1/p 1 -1/p -1/p ... 1 / The following interesting properties were established XTR-1X has the noncentral F distribution with degrees of freedom p and v. XTR-1 X has the Fisher's z distribution with degrees of freedom p - q and v - when X is subject to the linearly independent homogeneous conditions represented by the equation SX = 0, where S is of order gxpandrank q

30) for the multi- variate t distribution. 31) , where 1 is given by Q= (i tog r (p/2) BP xp/2 B (p+v a) 2 ' 2P_) + Al + v) 2 p2v{0(p2v)- \2/}. 32) It is easy to see that fl -4 0 as v -4 no. The mutual information for the multivariate normal distribution with correlation matrix R is given by -(1/2) log I R I (Kullback, 1968). 31) for v = 1 gives the mutual information for the multivariate Cauchy distribution with c taking the simpler form Il = log 8P r(p+2) }-12pt,,(12p)- (2)}. 32) for a range of v and p.

### A Bayesian analysis of colonic crypt structure and coordinated response to carcinogen exposure incor by Morris J.S.

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