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 ID: 617825
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GaussiansPieter AbbeelUC Berkeley EECSMany slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.:
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Univariate GaussianMultivariate GaussianLaw of Total ProbabilityConditioning (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!OutlineSlide3
Univariate GaussianGaussian distribution with mean , and standard deviation s:Slide4
Densities integrate to one: Mean:Variance:Properties of GaussiansSlide5
Central limit theorem (CLT)Classical CLT:Let X1, X2, … be an infinite sequence of independent random variables with E Xi = , E(Xi - )2 = 2Define Zn = ((X1 + … +
X
n
) – 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.Slide6
Multi-variate GaussiansSlide7
Multi-variate Gaussians
(integral of vector = vector of integrals of each entry)
(integral of matrix = matrix of integrals of each entry) Slide8
= [1; 0]
= [1 0;
0 1]
= [-.5; 0]
= [1 0; 0 1]
= [-1;
-1.5
]
= [
1
0
; 0 1]
Multi-
variate
Gaussians: examplesSlide9
Multi-variate Gaussians: examples = [0; 0] = [1 0 ; 0 1]
= [0; 0]
= [.6 0 ; 0 .6]
= [0; 0]
= [2 0 ; 0
2
]Slide10
= [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: examplesSlide11
= [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: examplesSlide12
= [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: examplesSlide13
Gaussians
-
s
s
m
Univariate
m
MultivariateSlide14
Partitioned Multivariate Gaussian
Consider a multi-
variate
Gaussian and partition random vector into (X, Y).Slide15
Partitioned Multivariate Gaussian: Dual Representation
Precision matrix
Straightforward to verify from (1) that:
And swapping the roles of
¡
and
§
:
(1)Slide16
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 throughSlide17
If
Then
Marginalization RecapSlide18
Self-quizSlide19
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
y
0Slide20
If
Then
Conditioning Recap