xii MATHEMATICAL NOTATION Er].If the distribution of r is conditioned on another variable z,then the corre- sponding conditional expectation will be written Ef().Similarly,the variance is denoted varf(),and for vector variables the covariance is written covx,y].We shall also use covx as a shorthand notation for covx,x.The concepts of expecta- tions and covariances are introduced in Section 1.2.2. If we have N values x1,...,xN of a D-dimensional vector x=(1,...,p)T we can combine the observations into a data matrix X in which the nth row of X corresponds to the row vector xT.Thus the n,i element of X corresponds to the ith element of the nth observation xn.For the case of one-dimensional variables we shall denote such a matrix by x,which is a column vector whose nth element is n. Note that x(which has dimensionality N)uses a different typeface to distinguish it from x(which has dimensionality D).xii MATHEMATICAL NOTATION E[x]. If the distribution of x is conditioned on another variable z, then the corre￾sponding conditional expectation will be written Ex[f(x)|z]. Similarly, the variance is denoted var[f(x)], and for vector variables the covariance is written cov[x, y]. We shall also use cov[x] as a shorthand notation for cov[x, x]. The concepts of expecta￾tions and covariances are introduced in Section 1.2.2. If we have N values x1,..., xN of a D-dimensional vector x = (x1,...,xD)T, we can combine the observations into a data matrix X in which the nth row of X corresponds to the row vector xT n. Thus the n, i element of X corresponds to the i th element of the nth observation xn. For the case of one-dimensional variables we shall denote such a matrix by x, which is a column vector whose nth element is xn. Note that x (which has dimensionality N) uses a different typeface to distinguish it from x (which has dimensionality D)