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1、#2. PROBABILITY DISTRIBUTIONS2.3.2 Marginal Gaussian distributionsWe have seen that if a joint distribution p(xa,Xb) is Gaussian, then the conditional distribution p(xa|xf) will again be Gaussian. Now we turn to a discussion of the marginal distribution given by(2.83)/ p(X,X6)dX6which, as we shall s
2、ee, is also Gaussian. Once again, our strategy for evaluating this distribution efficiently will be to focus on the quadratic form in the exponent of the joint distribution and thereby(o identify the mean and covariance of the marginal distribution p(xj.The quadratic form for the joint distribution
3、can be expressed, using the partitioned precision matrix, in the form (2.70). Because our goal is to integrate out xs this is most easily achieved by first considering the terms involving Xb and then completing the square in order to facilitate integration. Picking out just those terms that involve
4、we have一*xf Aqq,十xfm = (Xb Am)T人冊仕厂人Am (2.84)where we have definedm = A”他 一 A血(Xa - Ma)-(2.85)We see that the dependence on has been cast into the standard quadratic form of a Gaussian distribution corresponding to the first term on the right-hand side of (2.84), plus a term that does not depend on
5、(but that does depend on xa). Thus, when we take the exponential of this quadratic form, we see that the integration over x& required by (2.83) will take the form/ oxp -*(Xb A1m)TA66(X6 一 A赢Un) dxt.(2.86)This integration is easily performed by noting that it is the integral over an unnormalized Gaus
6、sian, and so the result will be the reciprocal of the normalization coefficient. We know from the form of the normalized Gaussian given by (2.43), that this coefficient is independent of the mean and depends only on the determinant of the covariance matrix. Thus, by completing the square with respec
7、t to we can integrate out Xb and the only term remaining from the contributions on the left-hand side of (2.84) that depends on xa is the last term on the right-hand side of (2.84) in which m is given by (2.85). Combining this term with the remaining terms from2.3. The Gaussian Distribution#(2.70) t
8、hat depend on xa, we obtain2 AbbDb Aba(Xa Ma ) A&b AbUb Aba(Xa Ma)AX 十 x(Aaap,a + A十 const十 const(2.87)xf(Ag AabAbb Aba)Xa 十Xq (Aaa AabAb Aba) Mawhere const denotes quantities independent of xu. Again, by comparison with(2.71), we see that the covariance of the marginal distribution of p(xa) is give
9、n bysa = (Aaa 一 A“bA计Aba)l.(2.88)Similarly, the mean is given bySa(Aaa - A必A臨i人韵如=Ma(2.89)x =2.3. The Gaussian Distribution#x =2.3. The Gaussian Distribution#where we have used (2.88). The covariance in (2.88) is expressed in tenns of the partitioned precision matrix given by (2.69). We can rewrite
10、this in terms of the corresponding partitioning of the covariance matrix given by (2.67), as we did for the conditional distribution. These partitioned matrices are related by(2.90)x =2.3. The Gaussian Distribution#x =2.3. The Gaussian Distribution#Making use of (2.76), we then have(AGU 一 AabAAba) 1
11、 = Saa (2.91)x =2.3. The Gaussian Distribution#x =2.3. The Gaussian Distribution#Thus we obtain the intuitively satisfying result that the marginal distribution p(xj has mean and covariance given by(2.92)(2.93)叫X=%covxrt = Saa.We see that for a marginal distribution, the mean and covariance are most
12、 simply expressed in tenns of the partitioned covariance matrix, in contrast to the conditional distribution for which the partitioned precision matrix gives rise to simpler expressions.Our results for the marginal and conditional distributions of a partitioned Gaussian are summarized belowPartition
13、ed GaussiansGiven a joint Gaussian distribution53) with A 三 S-1 and(2.94)x =2.3. The Gaussian Distribution#2.3. The Gaussian Distribution#Figure 2.9 The plot on the left shows the contours of a Gaussian distribution p(xa,xb) over two variables, and the plot on the right shows the marginal distributi
14、on p(xa) (blue curve) and the conditional distribution p(xaxb) for Xb = 0.7 (red curve).(2.95)(2.96)(2.97)(2.98)yi _ (人 _ ( Aq。Aab= Ab “ Abb,Conditional distribution:P(Xa|Xb) = (X|“a|b，A爲)= Ma - A；Aab(Xb - Mb)-Marginal distribution:P(Xa) =Af(Xala,Saa).We illustrate the idea of conditional and margin
15、al distributions associated with a multivariate Gaussian using an example involving two variables in Figure 2.9.2.3.3 Bayes theorem for Gaussian variablesIn Sections 2.3.1 and 2.3.2, we considered a Gaussian p(x) in which we partitioned the vector x into two subvectors x = (xa. x/J and then found expressions for the conditional distribution p(xei |xb) and the marginal distribution p(xa ). We noted that the mean of the conditional distribution p(xXb) was a linear function of x. Here we shall suppose that we are given a Gaussia
