Gaussians. Pieter Abbeel UC Berkeley EECS. Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics

Transcription

1 Gaussians Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics

2 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!

3 Univariate Gaussian Gaussian distribution with mean µ, and standard deviation σ:

4 Properties of Gaussians Densities integrate to one: Mean: Variance:

5 Central limit theorem (CLT) Classical CLT: Let X 1, X 2, be an infinite sequence of independent random variables with E X i = µ, E(X i - µ) 2 = σ 2 Define Z n = ((X X n ) n µ) / (σ n 1/2 ) Then for the limit of n going to infinity we have that Z n 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.

6 Multi-variate Gaussians

7 Multi-variate Gaussians: examples µ = [1; 0] Σ = [1 0; 0 1] µ = [-.5; 0] Σ = [1 0; 0 1] µ = [-1; -1.5] Σ = [1 0; 0 1]

8 Multi-variate Gaussians: examples µ = [0; 0] Σ = [1 0 ; 0 1] Σ = [.6 0 ; 0.6] Σ = [2 0 ; 0 2]

9 Multi-variate Gaussians: examples Σ = [1 0; 0 1] Σ = [1 0.5; 0.5 1] Σ = [1 0.8; 0.8 1]

10 Multi-variate Gaussians: examples Σ = [1 0; 0 1] Σ = [1 0.5; 0.5 1] Σ = [1 0.8; 0.8 1]

11 Multi-variate Gaussians: examples Σ = [1-0.5 ; ] Σ = [1-0.8 ; ] Σ = [3 0.8 ; 0.8 1]

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

13 Partitioned Multivariate Gaussian: Dual Representation Precision matrix (1) Straightforward to verify from (1) that: And swapping the roles of and :

14 Marginalization Recap If Then

15 Conditioning Recap If Then

16 Optimal estimation in linear-gaussian systems Consider the partially-observed system x k+1 = Ax k + Cω k y k = Hx k + Dε k with hidden state x k, measurement y k, and noise ε k, ω k N (0, I). Given a Gaussian prior x 0 N (bx 0, Σ 0 ) and a sequence of measurements y 0, y 1, y k, we want to compute the posterior p k+1 (x k+1 ). We can show by induction that the posterior is Gaussian at all times. Let p k (x k ) be N (bx k, Σ k ). This will act as a prior for estimating x k+1. Now x k+1 and y k are jointly Gaussian, with mean and covariance xk+1 Abxk E = y k xk+1 Cov y k = Hbx k CC T + AΣ k A T HΣ k A T AΣ k H T DD T + HΣ k H T Emo Todorov (UW) AMATH/CSE 579, Winter 2014 Winter / 11

17 Kalman filter Lemma If u, v are jointly Gaussian with means bu, bv and covariances Σ uu, Σ vv, Σ uv = Σ T vu, then u given v is Gaussian with mean and covariance E [ujv] = bu + Σ uv Σvv 1 (v bv) Cov [ujv] = Σ uu Σ uv Σvv 1 Σ vu Applying this to our problem with u = x k+1 and v = y k yields Theorem (Kalman filter) The mean bx and covariance Σ of the Gaussian posterior satisfy bx k+1 = Abx k + K k (y k Hbx k ) Σ k+1 = CC T + (A K k H) Σ k A T K k, AΣ k H T DD T + HΣ k H T 1 Emo Todorov (UW) AMATH/CSE 579, Winter 2014 Winter / 11

18 Duality of LQG control and Kalman filtering LQG controller State dynamics: Kalman filter Estimated state dynamics: x k+1 = (A BL k ) x k + Cε k bx k+1 = (A K k H) bx k + K k y k Gain matrix: 1 L k = R + B T V k+1 B B T V k+1 A Backward Riccati equation: Gain matrix: K k = AΣ k H T DD T + HΣ k H T 1 Forward Riccati equation: V k = Q + A T V k+1 (A BL k ) Σ k+1 = CC T + (A K k H) Σ k A T Emo Todorov (UW) AMATH/CSE 579, Winter 2014 Winter / 11

19 Duality of LQG control and Kalman filtering LQG controller State dynamics: Kalman filter Estimated state dynamics: x k+1 = (A BL k ) x k + Cε k bx k+1 = (A K k H) bx k + K k y k Gain matrix: 1 L k = R + B T V k+1 B B T V k+1 A Backward Riccati equation: Gain matrix: K k = AΣ k H T DD T + HΣ k H T 1 Forward Riccati equation: V k = Q + A T V k+1 (A BL k ) Σ k+1 = CC T + (A K k H) Σ k A T This form of duality does not generalize to non-lqg systems. However there is a different duality which does generalize (see later). It involves an information filter, computing Σ 1 instead of Σ. Emo Todorov (UW) AMATH/CSE 579, Winter 2014 Winter / 11