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# The Multivariate Gaussian

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1. The Multivariate Gaussian Jian Zhao

2. Outline • What is multivariate Gaussian? • Parameterizations • Mathematical Preparation • Joint distributions, Marginalization and conditioning • Maximum likelihood estimation

3. What is multivariate Gaussian? Where x is a n*1 vector, ∑ is an n*n, symmetric matrix

4. Geometrical interpret • Exclude the normalization factor and exponential function, we look the quadratic form This is a quadratic form. If we set it to a constant value c, considering the case that n=2, we obtained

5. Geometrical interpret (continued) This is a ellipse with the coordinate x1 and x2. Thus we can easily image that when n increases the ellipse became higher dimension ellipsoids

6. Geometrical interpret (continued) Thus we get a Gaussian bump in n+1 dimension Where the level surfaces of the Gaussian bump are ellipsoids oriented along the eigenvectors of ∑

7. Parameterization Another type of parameterization, putting it into the form of exponential family

8. Mathematical preparation • In order to get the marginalization and conditioning of the partitioned multivariate Gaussian distribution, we need the theory of block diagonalizing a partitioned matrix • In order to maximum the likelihood estimation, we need the knowledge of traces of the matrix.

9. Partitioned matrices • Consider a general patitioned matrix To zero out the upper-right-hand and lower-left-hand corner of M, we can premutiply and postmultiply matrices in the following form

10. Partitioned matrices (continued) Define Schur complement of Matrix M with respect to H , denote M/H as the term Since So

11. Partitioned matrices (continued) Note that we could alternatively have decomposed the matrix m in terms of E and M/E, yielding the following expression for the inverse.

12. Partitioned matrices (continued) Thus we get At the same time we get the conclusion |M| = |M/H||H|

13. Theory of traces Define It has the following properties: tr[ABC] = tr[CAB] = tr[BCA]

14. Theory of traces (continued) • We want to show that • Since • Recall • This is equivalent to prove • Noting that

15. Joint distributions, Marginalization and conditioning We partition the n by 1 vector x into p by 1 and q by 1, which n = p + q

16. Marginalization and conditioning

17. Normalization factor

18. Marginalization & Conditioning • Marginalization • Conditioning

19. In another form • Marginalization • Conditioning

20. Maximum likelihood estimation Calculating µ Taking derivative with respect to µ Setting to zero

21. Estimate ∑ • We need to take the derivative with respect to ∑ • according to the property of traces

22. Estimate ∑ • Thus the maximum likelihood estimator is • The maximum likelihood estimator of canonical parameters are

23. THANKS ALL