05_Normal_Distribution
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1.Computer vision: models, learning and inference Chapter 5 The Normal Distribution
2.Univariate Normal Distribution For short we write: Univariate normal distribution describes single continuous variable. Takes 2 parameters m and s 2 >0 2 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
3.Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Multivariate Normal Distribution For short we write: Multivariate normal distribution describes multiple continuous variables. Takes 2 parameters a vector containing mean position, m a symmetric “positive definite” covariance matrix S Positive definite: is positive for any real 3
4.Types of covariance Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 4 Symmetric
5.Diagonal Covariance = Independence Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 5
6.Consider green frame of reference: Relationship between pink and green frames of reference: Substituting in: Conclusion: Full covariance can be decomposed into rotation matrix and diagonal Decomposition of Covariance Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 6
7.Transformation of Variables If and we transform the variable as The result is also a normal distribution: Can be used to generate data from arbitrary Gaussians from standard one Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 7
8.Marginal Distributions Marginal distributions of a multivariate normal are also normal If then Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 8
9.Conditional Distributions If then Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 9
10.Conditional Distributions Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 10 For spherical / diagonal case, x 1 and x 2 are independent so all of the conditional distributions are the same.
11.Product of two normals (self conjugacy w.r.t mean) The product of any two normal distributions in the same variable is proportional to a third normal distribution Amazingly, the constant also has the form of a normal! Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 11
12.Change of Variables where If the mean of a normal in x is proportional to y then this can be reexpressed as a normal in y that is proportional to x Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 12
13.Conclusion 13 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince Normal distribution is used ubiquitously in computer vision Important properties: Marginal dist. of normal is normal Conditional dist. of normal is normal Product of normals prop. to normal Normal under linear change of variables

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