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-

Integral of a product of 3 Gaussians.pdf

IBM Research – Haifa, Haifa, Israel. hagaia@il.ibm.com ... We first define lemma 2: Lemma 2: Given two ... Integral of a product of 3 Gaussians.pdf. Integral of a ...

28KB Sizes 3 Downloads 243 Views

Recommend Documents

No documents