Transformations - University of Arizona
Hierarchical Models
Convolution
Chapter 4
Examples of Mass Functions and Densities
Transformations
1 / 18
Hierarchical Models
Convolution
Outline
Convolution
Hierarchical Models
Bernoulli Censored Poisson
Normal Random Variables
Bayes Formula Revisited
2 / 18
Hierarchical Models
Convolution
Convolution
Let
s = x1 + x2 ,
y = x1 .
Then,
x2 = s ? y
x1 = y ,
J(s, y ) = det
0 1
1 ?1
= ?1
This yields
fS,Y (s, y ) = fX1 ,X2 (y , s ? y ).
The marginal distribution for S = X1 + X2 can be found by taking an integral
Z ¡Þ
fX1 +X2 (s) =
fX1 ,X2 (x1 , s ? x1 ) dx1 .
?¡Þ
3 / 18
Hierarchical Models
Convolution
Gamma Distribution
If X1 and X2 are independent with sum S, then
Z ¡Þ
fS (s) =
fX1 (x1 )fX2 (s ? x1 ) dx1 .
?¡Þ
This is called the convolution and is often written fX1 +X2 = fX1 ? fX2 . For Xi ¡« ¦£(¦Ái , ¦Â)
fS (s)
=
=
=
=
s
¦Â ¦Á1 ¦Á1 ?1 ?¦Âx1 ¦Â ¦Á2
x1
e
(s ? x1 )¦Á2 ?1 e ?¦Â(s?x1 ) dx1 .
¦£(¦Á2 )
0 ¦£(¦Á1 )
Z s
¦Â ¦Á1 +¦Á2
?¦Âs
e
x1¦Á1 ?1 (s ? x1 )¦Á2 ?1 dx1 .
¦£(¦Á1 )¦£(¦Á2 )
0
Z 1
¦Â ¦Á1 +¦Á2
s ¦Á1 +¦Á2 ?1 e ?¦Âs
u ¦Á1 ?1 (1 ? u)¦Á2 ?1 du, u = x1 /s.
¦£(¦Á1 )¦£(¦Á2 )
0
¦Â ¦Á1 +¦Á2
¦£(¦Á1 )¦£(¦Á2 )
¦Â ¦Á1 +¦Á2
s ¦Á1 +¦Á2 ?1 e ?¦Âs
=
s ¦Á1 +¦Á2 ?1 e ?¦Âs
¦£(¦Á1 )¦£(¦Á2 )
¦£(¦Á1 + ¦Á2 )
¦£(¦Á1 + ¦Á2 )
Z
and X1 + X2 ¡« ¦£(¦Á1 + ¦Á2 , ¦Â)
4 / 18
Hierarchical Models
Convolution
Gamma Distribution
Continuing,
fS,X1 (s, x1 )
fX ,X (x1 , s ? x1 )
= 1 2
.
fS (s)
fS (s)
For the example on the sum of gamma random variables
fX1 |S (x1 |s) =
fX1 |S (x1 |s) =
=
=
¦Â ¦Á1 +¦Á2
?¦Âs x ¦Á1 ?1 (s ? x )¦Á2 ?1
1
1
¦£(¦Á1 )¦£(¦Á2 ) e
,
¦Á
+¦Á
1
2
¦Â
¦Á1 +¦Á2 ?1 e ?¦Âs
s
¦£(¦Á1 +¦Á2 )
¦£(¦Á1 +¦Á2 ) ¦Á1 ?1
(s ? x1 )¦Á2 ?1
¦£(¦Á1 )¦£(¦Á2 ) x1
s ¦Á1 +¦Á2 ?1
0 ¡Ü x1 ¡Ü s
¦£(¦Á1 + ¦Á2 ) x1 ¦Á1 ?1
x1 ¦Á2 ?1 1
1?
¦£(¦Á1 )¦£(¦Á2 ) s
s
s
Thus, X1 |S ¡« S ¡¤ Beta(¦Á1 , ¦Á2 ) and E [X1 |S] = ¦Á1 S/(¦Á1 + ¦Á2 )
5 / 18
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