Deriving a Marginal Student-t Distribution
Aaron A. D’Souza
Assume that x is a Gaussian distributed random variable with a known mean µ but unknown
precision (inverse variance) ψ. This unknown precision is distributed according to a Gamma
distribution with order and scale a and b respectively:
1/2
ψ
ψ
2
p(x|ψ) =
exp − (x − µ)
2π
2
ba a−1
ψ
exp (−bψ)
p(ψ) =
Γ(a)
The marginal distribution of x is derived as follows:
p(x) =
Z
p(x|ψ)p(ψ)dψ
ba
=
Γ(a)
ba
Γ(a)
1
2π
ψ
a+1/2−1
1/2
1
2
exp − b + (x − µ) ψ dψ
2
Γ (a + 1/2)
ia+1/2
h
2
b + 21 (x − µ)
1/2
Γ (a + 1/2) 1
ba
=
ia+1/2
h
Γ(a)
2π
2
b + 21 (x − µ)
#
1/2 "
2 −(a+1/2)
1
(x − µ)
Γ (a + 1/2)
1+
=
Γ(a)
2πb
2b
=
1
2π
1/2 Z
Hence the marginal distribution of x is a Student-t distribution with shape parameter a and
scale parameter b. Figure 2 compares the conditional and marginal distributions of x under
several Gamma distributions.
1
ψ ∼ Gamma(0.001, 0.001)
0.01
conditional and marginal
0.009
10
log p(x|ψ)
log p(x)
−1
10
0.008
0.007
log conditional and marginal
0.009
0.008
0.007
0.3
−2
10
0.006
p(ψ)
0.006
p(ψ)
0
0.01
p(x|ψ)
p(x)
0.4
0.005
0.2
0.004
−3
0.005
10
0.004
−4
0.003
0.002
0.002
0.001
0
10
0.003
0.1
−5
10
0.001
0
2
4
ψ
6
8
10
0
−5
ψ ∼ Gamma(0.1, 0.1)
0.7
0
x
0
5
conditional and marginal
0.5
1
10
0
x
5
log conditional and marginal
log p(x|ψ)
log p(x)
−1
10
0.5
0.3
−5
10
0.6
−2
10
p(ψ)
0.4
p(ψ)
0.4
10
0
0.7
p(x|ψ)
p(x)
0.4
0.6
−6
0
10
log ψ
0.2
0.3
−3
10
0.3
−4
0.2
10
0.2
0.1
−5
0.1
0
0
2
4
ψ
6
8
10
0
−5
ψ ∼ Gamma(1, 1)
1
0
x
0
5
conditional and marginal
0.9
0.4
−6
0
10
log ψ
1
10
−5
10
0
x
5
log conditional and marginal
log p(x|ψ)
log p(x)
0.9
−1
10
0.8
0.7
10
0
1
p(x|ψ)
p(x)
0.8
0.7
0.3
−2
10
0.6
p(ψ)
0.6
p(ψ)
10
0.1
0.5
0.2
0.4
−3
0.5
10
0.4
−4
0.3
0.1
0.2
0.2
0.1
0
10
0.3
−5
10
0.1
0
2
4
ψ
6
8
10
0
−5
ψ ∼ Gamma(10, 10)
1.4
0
x
0
5
conditional and marginal
1
1
10
0
x
5
log conditional and marginal
log p(x|ψ)
log p(x)
−1
10
1
−2
10
p(ψ)
0.8
p(ψ)
0.8
−5
10
1.2
0.3
10
0
1.4
p(x|ψ)
p(x)
0.4
1.2
−6
0
10
log ψ
0.2
0.6
−3
10
0.6
−4
0.4
10
0.4
0.1
−5
0.2
0
10
0.2
0
2
4
ψ
6
8
10
0
−5
0
x
5
0
−6
0
10
log ψ
1
10
10
−5
0
x
5
Figure 1: For the given Gamma distribution over ψ, we can compare the conditional and
marginal distributions. Note that a higher variance Gamma distribution causes a greater
smearing, resulting in a heavier tailed marginal distribution.
2