xii MATHEMATICAL NOTATION

`E[x]. If the distribution ofxis conditioned on another variablez, then the corre-`

sponding conditional expectation will be writtenEx[f(x)|z]. Similarly, the variance

is denotedvar[f(x)], and for vector variables the covariance is writtencov[x,y].We

shall also usecov[x]as a shorthand notation forcov[x,x]. The concepts of expecta-

tions and covariances are introduced in Section 1.2.2.

If we haveNvaluesx 1 ,...,xNof aD-dimensional vectorx=(x 1 ,...,xD)T,

we can combine the observations into a data matrixXin which thenthrow ofX

corresponds to the row vectorxTn. Thus then, ielement ofXcorresponds to the

ithelement of thenthobservationxn. For the case of one-dimensional variables we

shall denote such a matrix byx, which is a column vector whosenthelement isxn.

Note thatx(which has dimensionalityN) uses a different typeface to distinguish it

fromx(which has dimensionalityD).