THE INTEGRAL OF A PRODUCT OF THREE GAUSSIANS Hagai Aronowitz IBM Research – Haifa, Haifa, Israel
[email protected] Lemma 1
Given three multivariate normal distributions f~N(µ f, Σf), g~N(µ g, Σg), and h~N(µ h, Σh) the integral of the product of the three distributions is:
∫ f (x )g (x )h(x )dx =N (µ
f
(
(
; µ g , , Σ f + Σ g )N µ h ; µ fg , , Σ −f1 + Σ −g1
)
−1
+ Σh
)
( 1)
Proof of lemma 1: We first define lemma 2: Lemma 2: Given two multivariate normal distributions a~N(µ a, Σa) and b~N(µ b, Σb), the product of the two distributions is
(
(
a ( x )b( x ) = N x; µ ab , , Σ a−1 + Σ b−1
(
with µ ab = Σ a−1 + Σ b−1
) (Σ −1
−1 a
)
−1
)N (µ ; µ a
b,
, Σa + Σb )
(2)
µ a + Σ b−1µ b ) , and the integral of the product of the two distributions is
∫ a(x )b(x )dx =N (µ
a
; µb , Σ a + Σ b ) .
(3)
The proof for lemma 2 can be found in [1]. Using lemma 2 with a(x)=f(x) and b(x)=g(x) we get
∫ f (x )g (x )h(x )dx = ∫ N (x; µ , (Σ = N (µ ; µ , Σ fg ,
f
(
with µ fg = Σ −f1 + Σ −g1
) (Σ −1
−1 f
g,
−1 f
f
) )N (µ ; µ , Σ + Σ )h(x )dx . )∫ N (x; µ , (Σ + Σ ) )h(x )dx
+ Σ −g1 + Σg
−1
f
g,
fg ,
f
g
(4)
−1 −1 g
−1 f
(
µ f + Σ −g1 µ g ). We apply again Lemma 2 with a(x)= N x; µ fg , , (Σ −f1 + Σ −g1 )
−1
) and b(x)=h(x). We get:
∫ f (x )g (x )h(x )dx =N (µ ; µ , Σ + Σ )∫ N (x; µ , (Σ + Σ ) )h(x )dx . = N (µ ; µ , Σ + Σ )N (µ ; µ , (Σ + Σ ) + Σ ) g,
f
f
f
g,
fg ,
g
f
g
h,
fg
−1 −1 g
−1 f
−1 f
−1 −1 g
(5)
h
□
REFERENCE [1] S. Vinga, "Convolution integrals of Normal id.pt/~svinga/renyi/convolution_normal.pdf
distribution
functions",
2004.
[Online].
Available:
http://kdbio.inesc-
