Pattern Recognition and Machine Learning

132 2. PROBABILITY DISTRIBUTIONS

Mahalanobis distance∆is given by

VD|Σ|^1 /^2 ∆D (2.286)

whereVDis the volume of the unit sphere inDdimensions, and the Mahalanobis

distance is defined by (2.44).

2.24 ( ) www Prove the identity (2.76) by multiplying both sides by the matrix

(

AB

CD

)

(2.287)

and making use of the definition (2.77).

2.25 ( ) In Sections 2.3.1 and 2.3.2, we considered the conditional and marginal distri-

butions for a multivariate Gaussian. More generally, we can consider a partitioning

of the components ofxinto three groupsxa,xb, andxc, with a corresponding par-

titioning of the mean vectorμand of the covariance matrixΣin the form

μ=

(

μa

μb

μc

)

, Σ=

(

Σaa Σab Σac

Σba Σbb Σbc

Σca Σcb Σcc

)

. (2.288)

By making use of the results of Section 2.3, find an expression for the conditional

distributionp(xa|xb)in whichxchas been marginalized out.

2.26 ( ) A very useful result from linear algebra is theWoodburymatrix inversion

formula given by

(A+BCD)−^1 =A−^1 −A−^1 B(C−^1 +DA−^1 B)−^1 DA−^1. (2.289)

By multiplying both sides by(A+BCD)prove the correctness of this result.

2.27 ( ) Letxandzbe two independent random vectors, so thatp(x,z)=p(x)p(z).

Show that the mean of their sumy=x+zis given by the sum of the means of each

of the variable separately. Similarly, show that the covariance matrix ofyis given by

the sum of the covariance matrices ofxandz. Confirm that this result agrees with

that of Exercise 1.10.

2.28 ( ) www Consider a joint distribution over the variable

z=

(

x

y

)

(2.290)

whose mean and covariance are given by (2.108) and (2.105) respectively. By mak-

ing use of the results (2.92) and (2.93) show that the marginal distributionp(x)is

given (2.99). Similarly, by making use of the results (2.81) and (2.82) show that the

conditional distributionp(y|x)is given by (2.100).