How the t-distribution arises Student's t-distribution

in bayesian statistics, (scaled, shifted) t-distribution arises marginal distribution of unknown mean of normal distribution, when dependence on unknown variance has been marginalised out:

p
(
μ
∣
d
,
i
)
=



∫
p
(
μ
,

σ

2


∣
d
,
i
)

d

σ

2






=



∫
p
(
μ
∣
d
,

σ

2


,
i
)

p
(

σ

2


∣
d
,
i
)

d

σ

2


,






{\displaystyle {\begin{aligned}p(\mu \mid d,i)=&\int p(\mu ,\sigma ^{2}\mid d,i)\,d\sigma ^{2}\\=&\int p(\mu \mid d,\sigma ^{2},i)\,p(\sigma ^{2}\mid d,i)\,d\sigma ^{2},\end{aligned}}}

where d stands data {xi}, , represents other information may have been used create model. distribution compounding of conditional distribution of μ given data , σ marginal distribution of σ given data.

with n data points, if uninformative, or flat, location , scale priors



p
(
μ
∣

σ

2


,
i
)
=

const



{\displaystyle p(\mu \mid \sigma ^{2},i)={\text{const}}}

,



p
(

σ

2


∣
i
)
∝
1

/


σ

2




{\displaystyle p(\sigma ^{2}\mid i)\propto 1/\sigma ^{2}}

can taken μ , σ, bayes theorem gives

p
(
μ
∣
d
,

σ

2


,
i
)



∼
n
(



x
¯



,

σ

2



/

n
)
,




p
(

σ

2


∣
d
,
i
)



∼

s
c
a
l
e
-
i
n
v
-

⁡

χ

2


(
ν
,

s

2


)
,






{\displaystyle {\begin{aligned}p(\mu \mid d,\sigma ^{2},i)&\sim n({\bar {x}},\sigma ^{2}/n),\\p(\sigma ^{2}\mid d,i)&\sim \operatorname {scale-inv-} \chi ^{2}(\nu ,s^{2}),\end{aligned}}}

a normal distribution , scaled inverse chi-squared distribution respectively,



ν
=
n
−
1


{\displaystyle \nu =n-1}

and

s

2


=
∑



(

x

i


−



x
¯




)

2




n
−
1



.


{\displaystyle s^{2}=\sum {\frac {(x_{i}-{\bar {x}})^{2}}{n-1}}.}

the marginalisation integral becomes

p
(
μ
∣
d
,
i
)



∝

∫

0


∞




1


σ

2





exp
⁡

(
−


1

2

σ

2





n
(
μ
−



x
¯




)

2


)

⋅

σ

−
ν
−
2


exp
⁡
(
−
ν

s

2



/

2

σ

2


)

d

σ

2








∝

∫

0


∞



σ

−
ν
−
3


exp
⁡

(
−


1

2

σ

2






(
n
(
μ
−



x
¯




)

2


+
ν

s

2


)

)


d

σ

2


.






{\displaystyle {\begin{aligned}p(\mu \mid d,i)&\propto \int _{0}^{\infty }{\frac {1}{\sqrt {\sigma ^{2}}}}\exp \left(-{\frac {1}{2\sigma ^{2}}}n(\mu -{\bar {x}})^{2}\right)\cdot \sigma ^{-\nu -2}\exp(-\nu s^{2}/2\sigma ^{2})\,d\sigma ^{2}\\&\propto \int _{0}^{\infty }\sigma ^{-\nu -3}\exp \left(-{\frac {1}{2\sigma ^{2}}}\left(n(\mu -{\bar {x}})^{2}+\nu s^{2}\right)\right)\,d\sigma ^{2}.\end{aligned}}}

this can evaluated substituting



z
=
a

/

2

σ

2




{\displaystyle z=a/2\sigma ^{2}}

,



a
=
n
(
μ
−



x
¯




)

2


+
ν

s

2




{\displaystyle a=n(\mu -{\bar {x}})^{2}+\nu s^{2}}

, giving

d
z
=
−


a

2

σ

4






d

σ

2


,


{\displaystyle dz=-{\frac {a}{2\sigma ^{4}}}\,d\sigma ^{2},}

so

p
(
μ
∣
d
,
i
)
∝

a

−



ν
+
1

2





∫

0


∞



z

(
ν
−
1
)

/

2


exp
⁡
(
−
z
)

d
z
.


{\displaystyle p(\mu \mid d,i)\propto a^{-{\frac {\nu +1}{2}}}\int _{0}^{\infty }z^{(\nu -1)/2}\exp(-z)\,dz.}

but z integral standard gamma integral, evaluates constant, leaving

p
(
μ
∣
d
,
i
)



∝

a

−



ν
+
1

2










∝


(
1
+



n
(
μ
−



x
¯




)

2




ν

s

2





)


−



ν
+
1

2




.






{\displaystyle {\begin{aligned}p(\mu \mid d,i)&\propto a^{-{\frac {\nu +1}{2}}}\\&\propto \left(1+{\frac {n(\mu -{\bar {x}})^{2}}{\nu s^{2}}}\right)^{-{\frac {\nu +1}{2}}}.\end{aligned}}}

this form of t-distribution explicit scaling , shifting explored in more detail in further section below. can related standardised t-distribution substitution

t
=



μ
−



x
¯





s

/



n





.


{\displaystyle t={\frac {\mu -{\bar {x}}}{s/{\sqrt {n}}}}.}

the derivation above has been presented case of uninformative priors μ , σ; apparent priors lead normal distribution being compounded scaled inverse chi-squared distribution lead t-distribution scaling , shifting p(μ | d, i), although scaling parameter corresponding s/n above influenced both prior information , data, rather data above.