1
Appendix S1 Expectation of the reciprocal of a scaled inverse
Chi-square random variable
Given that
γj |σj2 ∼ N (0, σj2 )
and
σj2 ∼
νγ Sγ2
,
χ2νγ
The joint distribution of γj and σj2 is
νγ
1
γj2
νγ Sγ2
2 − 1+ 2
2 −2
σ
p(γj , σj2 ) ∝ exp −
·
σ
exp
−
j
j
2σj2
2σj2
νγ +1
νγ Sγ2 + γj2
2 − 1+ 2
∝ exp −
.
σ
j
2σj2
This is the kernel of the conditional distribution of σj2 given γj , which is a scaled inverse Chi-square
distribution with degrees of freedom νγ + 1 and scale parameter
2
γj2 +νγ Sγ
νγ +1
To show that
Eσ2 |y,γ=bγ (k)
1
σj2
(k)
=
{b
γj }2 + νγ Sγ2
νγ + 1
.
!−1
,
it suffices to show that the expectation of the reciprocal of a scaled inverse Chi-square variable is the
reciprocal of its scale parameter. Suppose X is a scaled inverse Chi-square random variable with degrees
of freedom ν and scale parameter S 2 , the probability density function for X is given by
ν
2
p(x|ν, S ) =
νS 2 2
2 Γ ν2
It follows that the probability density function for Y =
1
X
is
ν
2 exp − νS2 y 1 ·
· − 2
ν
1 1+ 2
y
ν
ν
νS 2
· exp −
y · y 2 −1 .
2
p(y|ν, S ) =
νS 2 2
2 Γ ν2
=
νS 2 2
2 Γ ν2
2
2
exp − νS
2x
·
.
ν
x1+ 2
y
This is the probability density function of Gamma distribution with shape parameter ν2 and rate param2
eter νS2 . The expectation of Gamma distribution is the shape over rate and therefore the expectation of
1
1
X is S 2 .