# LKJ Cholesky Density derivation

The density of the LKJ cholesky is

\text{LkjCholesky}(L \mid \eta) \propto |J|\det(LL^T)^{\eta - 1} = \prod_{k = 2}^K L_{kk}^{K - k + 2\eta - 2}

which is from the Stan manual ( 24.2 Cholesky LKJ correlation distribution | Stan Functions Reference (mc-stan.org)). I get where the 2\eta - 2 comes from, it is

\det(LL^T)^{\eta - 1} = [\det(L)^2]^{\eta - 1} = \prod_{k=2}^K L_{kk}^{2 \eta - 2}

which implies that |J| = \prod_{k=2}^K L_{kk}^{K - k}. However, the Edelman handout ( see top of page 13 z_handout2.dvi (mit.edu))
has the jacobian for the Cholesky transform as

\det J = 2^K \prod_{k = 1}^K L_{kk}^{K + 1 - k} .

So where am I going wrong with this?

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Ok, the panic of posting jarred my stupidity into clarity. I guess that power becomes K - k since when k = 1 L is 1. Then in the log that 2^K is constant so we can drop it. Then we get the above.

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Thanks for sharing, that might be useful for others who will wonder the same. Do I get it correctly that it is resolved? If so, could you mark your own answer as solution?