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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

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Gaussians Pieter Abbeel UC Berkeley EECS

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Gaussians pieter abbeel uc berkeley eecs

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.: AAAAAAAAAAAAA


Outline

Outline

  • 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!


Univariate gaussian

Univariate Gaussian

  • Gaussian distribution with mean , and standard deviation s:


Properties of gaussians

Properties of Gaussians

  • Densities integrate to one:

  • Mean:

  • Variance:


Central limit theorem clt

Central limit theorem (CLT)

  • Classical CLT:

    • Let X1, X2, … be an infinite sequence of independent random variables with E Xi = , E(Xi - )2 = 2

    • Define Zn = ((X1 + … + Xn) – n ) / ( 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.


Multi variate gaussians

Multi-variate Gaussians


Multi variate gaussians1

Multi-variate Gaussians

(integral of vector = vector of integrals of each entry)

(integral of matrix = matrix of integrals of each entry)


Multi variate gaussians examples

Multi-variate Gaussians: examples

  •  = [1; 0]

  •  = [1 0;0 1]

  •  = [-.5; 0]

  •  = [1 0; 0 1]

  •  = [-1; -1.5]

  •  = [1 0; 0 1]


Multi variate gaussians examples1

Multi-variate Gaussians: examples

  •  = [0; 0]

  •  = [1 0 ; 0 1]

  •  = [0; 0]

  •  = [.6 0 ; 0 .6]

  •  = [0; 0]

  •  = [2 0 ; 0 2]


Gaussians pieter abbeel uc berkeley eecs

Multi-variate Gaussians: examples

  •  = [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 examples2

Multi-variate Gaussians: examples

  •  = [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 examples3

Multi-variate Gaussians: examples

  •  = [0; 0]

  •  = [1 -0.5 ; -0.5 1]

  •  = [0; 0]

  •  = [1 -0.8 ; -0.8 1]

  •  = [0; 0]

  •  = [3 0.8 ; 0.8 1]


Partitioned multivariate gaussian

Partitioned Multivariate Gaussian

  • Consider a multi-variate Gaussian and partition random vector into (X, Y).


Partitioned multivariate gaussian dual representation

Partitioned Multivariate Gaussian: Dual Representation

  • Precision matrix

  • Straightforward to verify from (1) that:

  • And swapping the roles of ¡ and §:

(1)


Marginalization p x

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


Marginalization recap

Marginalization Recap

If

Then


Self quiz

Self-quiz


Conditioning p x y y 0

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


Conditioning recap

Conditioning Recap

If

Then


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