gaussians pieter abbeel uc berkeley eecs
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Gaussians Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun , Burgard and Fox, Probabilistic Robotics. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A. Outline. Univariate Gaussian Multivariate Gaussian - PowerPoint PPT PresentationTRANSCRIPT
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Gaussians
Pieter AbbeelUC Berkeley EECS
Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics
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Univariate Gaussian Multivariate Gaussian Law of Total Probability Conditioning (Bayes’ rule)
Disclaimer: lots of linear algebra in next few lectures. See course homepage for pointers for brushing up your linear algebra.
In fact, pretty much all computations with Gaussians will be reduced to linear algebra!
Outline
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Univariate Gaussian Gaussian distribution with mean , and standard
deviation s:
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Densities integrate to one:
Mean:
Variance:
Properties of Gaussians
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Central limit theorem (CLT) Classical CLT:
Let X1, X2, … be an infinite sequence of independent random variables with E Xi = , E(Xi - )2 = s2
Define Zn = ((X1 + … + Xn) – n ) / (s n1/2) Then for the limit of n going to infinity we have
that Zn is distributed according to N(0,1) Crude statement: things that are the result of the
addition of lots of small effects tend to become Gaussian.
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Multi-variate Gaussians
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Multi-variate Gaussians
(integral of vector = vector of integrals of each entry)
(integral of matrix = matrix of integrals of each entry)
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= [1; 0] = [1 0; 0 1]
= [-.5; 0] = [1 0; 0 1]
= [-1; -1.5] = [1 0; 0 1]
Multi-variate Gaussians: examples
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Multi-variate Gaussians: examples
= [0; 0] = [1 0 ; 0 1]
= [0; 0] = [.6 0 ; 0 .6]
= [0; 0] = [2 0 ; 0 2]
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= [0; 0] = [1 0; 0 1]
= [0; 0] = [1 0.5; 0.5
1]
= [0; 0] = [1 0.8; 0.8
1]
Multi-variate Gaussians: examples
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= [0; 0] = [1 0; 0 1]
= [0; 0] = [1 0.5; 0.5
1]
= [0; 0] = [1 0.8; 0.8
1]
Multi-variate Gaussians: examples
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= [0; 0] = [1 -0.5 ; -0.5 1]
= [0; 0] = [1 -0.8 ; -0.8
1]
= [0; 0] = [3 0.8 ; 0.8 1]
Multi-variate Gaussians: examples
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Partitioned Multivariate Gaussian
Consider a multi-variate Gaussian and partition random vector into (X, Y).
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Partitioned Multivariate Gaussian: Dual Representation
Precision matrix
Straightforward to verify from (1) that:
And swapping the roles of ¡ and §:
(1)
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Marginalization: p(x) = ?
We integrate out over y to find the marginal:
Hence we have:
Note: if we had known beforehand that p(x) would be a Gaussian distribution, then we could have found the result more quickly. We would have just needed to find and , which we had available through
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If
Then
Marginalization Recap
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Self-quiz
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Conditioning: p(x | Y = y0) = ?
We have
Hence we have:
• Mean moved according to correlation and variance on measurement
• Covariance §XX | Y = y0 does not depend on y0
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If
Then
Conditioning Recap