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Continuous probability distribution
Lewandowski-Kurowicka-Joe distribution Notation
LKJ
(
η
)
{\displaystyle \operatorname {LKJ} (\eta )}
Parameters
η
∈
(
0
,
∞
)
{\displaystyle \eta \in (0,\infty )}
Support
R
{\displaystyle \mathbf {R} }
Mean
the identity matrix
In
probabilistic programming language and as a library linked to the
Turing.jl probabilistic programming library in
Julia .
The distribution has a single shape parameter
η
{\displaystyle \eta }
probability density function for a
d
×
d
{\displaystyle d\times d}
R
{\displaystyle \mathbf {R} }
p
(
R
;
η
)
=
C
×
[
det
(
R
)
]
η
−
1
{\displaystyle p(\mathbf {R} ;\eta )=C\times [\det(\mathbf {R} )]^{\eta -1}}
with normalizing constant
C
=
2
∑
k
=
1
d
(
2
η
−
2
+
d
−
k
)
(
d
−
k
)
∏
k
=
1
d
−
1
[
B
(
η
+
(
d
−
k
−
1
)
/
2
,
η
+
(
d
−
k
−
1
)
/
2
)
]
d
−
k
{\displaystyle C=2^{\sum _{k=1}^{d}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left[B\left(\eta +(d-k-1)/2,\eta +(d-k-1)/2\right)\right]^{d-k}}
Beta functions . For
η
=
1
{\displaystyle \eta =1}
References External links
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle p(\mathbf {R} ;\eta )=C\times [\det(\mathbf {R} )]^{\eta -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/789875bdb13d8e410ffa6e3e515879a1de6d4e20)
![{\displaystyle C=2^{\sum _{k=1}^{d}(2\eta -2+d-k)(d-k)}\prod _{k=1}^{d-1}\left[B\left(\eta +(d-k-1)/2,\eta +(d-k-1)/2\right)\right]^{d-k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/291a0f65e2e811840d404932a6b586bb4d622ef8)
