01 Probability Theory and Statistics Review

Transcription

1 NAVARCH/EECS 568, ROB Winter Probability Theory and Statistics Review Maani Ghaffari January 08, 2018

2 Last Time: Bayes Filters Given: Stream of observations z 1:t and action data u 1:t Sensor/measurement model p(z t x t) Action/motion/transition model p(x t x t 1, u t) Wanted: The state x t of dynamical system The posterior of state is called belief bel(x t) = p(x t z 1:t, u 1:t ) 2

3 Bayes Filter: Global Localization Example Monobot World Map is known Door Detector senses presence of door 3

4 Probability Space Probability space: (Ω, F, P ) Sample space Ω: set of all possible outcomes (non-empty) σ-algebra F 2 Ω : a collection of all the events Probability measure P : F [0, 1]: P ( ) = 0 and P (Ω) = 1 State space: X, e.g., X = R Random variable: X : Ω X 4

5 Discrete Random Variable The state space X is finite or countably infinite. A set which can be put in a one-to-one correspondence with the natural numbers is called a countably infinite set. The distribution of X is a probability mass function P (X = x) where x X p(x = x) = 1. Example (Finite) X = {1, 2, 3, 4, 5, 6} Example (Countably infinite) The natural numbers N, the set of all integers Z, and the set of all rational numbers Q are countable. Example (Uncountably infinite) The set of all real numbers R. Example (Fair coin) P (X = head) = P (X = tail) = 1 2 X is a Bernoulli random variable. 5

6 Continuous Random Variable The state space X is continuous. The distribution of X is a probability density function p(x) where p(x = x)dx = 1; and P (X = x) = 0! x X Example (Real-valued random variable) X = R Example (Standard normal distribution) X N (0, 1) p(x) = 1 2π exp( 1 2 x2 ) Remark (Probability of X) P (a X b) = b a p(x)dx 6

7 Warning! In general, we do not distinguish probabilities and probability densities. For the sake of simplicity, it is common to use p(x) instead of p(x = x) and sometimes refer to x as the random variable itself. 7

8 Joint and Conditional Distribution Let X and Y be two random variables. The joint distribution of X and Y is: p(x, y) = p(x = x and Y = y); If X and Y are independent then p(x, y) = p(x)p(y) The conditional probability of X given Y is: p(x y) = p(x,y) p(y) p(y) > 0. 8

9 Marginalization Given the joint distribution of X and Y, the marginalization rule states that the marginal distribution of X can be computed by summing (integration) over Y. The law of total probability is its variant which uses the conditional probability definition and can be written as p(x) = y Y p(x, y) = y Y p(x y)p(y) and for continuous random variables is p(x) = p(x, y)dy = p(x y)p(y)dy y Y y Y 9

10 Bayes Rule p(x, y) = p(x y)p(y) = P (y x)p(x) p(x y) = p(y x)p(x) p(y) = x X p(y x)p(x) Posterior = Likelihood Prior Evidence 10

11 Bayes Rule with Prior Knowledge Given three random variables X, Y, and Z, Bayes rule relates the prior probability distribution, p(x z), and the likelihood function, p(y x, z), as follows. p(x y, z) = p(y x, z)p(x z) p(y z) Given Z, if X and Y are conditionally independent then p(x, y z) = p(x z)p(y z) 11

12 Bayes Rule Example Example (Bayes rule) A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5% of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities: 1 that the test result will be positive; 2 that, given a positive result, the person is a sufferer; 3 that, given a negative result, the person is a non-sufferer; 4 that the person will be misclassified. 12

13 Bayes Rule Example Let us first define the following random variables: T (Test positive), S (Sufferer), M (Misclassified). From given information we have p(t S) = 0.95, p(t S) = 0.1, and P (S) = Then p(t ) = S p(t S)p(S) = p(t S)p(S) + p(t S)p( S) = ( ) = p(s T ) = p(t S)p(S) p(t ) = ( )/ = p( S T ) = p( T S)p( S) p( T ) = (1 0.1) ( )/( ) = p(m) = p(t, S) + p( T, S) = p(t S)p( S) + p( T S)p(S) = 0.1 ( ) + (1 0.95) =

14 Expected Value The expected value of the random variable X is E[X] = Xdp = xp(x)dx Ω X and if X is discrete E[X] = xp(x) X 14

15 Expected Value The expectation operator is linear which follows from linearity of integration and has the following properties: E[a] = a E[aX] = ae[x] E[a + X] = a + E[X] E[aX + by ] = ae[x] + be[y ] where X and Y are two arbitrary random variables and a and b are constants. 15

16 Variance and Covariance The variance of X can be calculated as V[X] = E[(X E[X]) 2 ] = E[X 2 ] E[X] 2 The covariance of a random vector X = x can be calculated as Cov[X] = E[(X E[X])(X E[X]) T ] = E[XX T ] E[X]E[X] T 16

17 Covariance A covariance matrix such as Σ is symmetric, i.e., Σ = Σ T, and positive semi-definite, i.e., x T Σx 0 and all eigenvalues are nonnegative. A symmetric positive definite matrix (x T Σx > 0 and x 0) has positive eigenvalues and a unique Cholesky decomposition, i.e., Σ = LL T, where L is a lower triangular matrix. 17

18 Correlation Coefficient The correlation coefficient is defined as ρ XY = Cov[XY ] V[X]V[Y ] ρ XY 1 where the bound follows from the Cauchy-Schwarz inequality which asserts E[XY ] 2 E[X 2 ]E[Y 2 ] 18

19 Uncorrelated Independent Remark In general, E[XY ] E[X]E[Y ], unless X and Y are uncorrelated. Now it is clear that if X and Y are independent, i.e., X Y, then Cov[XY ] = E[XY ] E[X]E[Y ] = 0 and X and Y are uncorrelated (ρ XY = 0). However, uncorrelated random variables are not necessarily independent. 19

20 Univariate Normal Distribution The univariate (one-dimensional) Gaussian (or normal) distribution with mean µ and variance σ 2 has the following Probability Density Function (PDF). p(x) = 1 2πσ exp( 1 (x µ) 2 ) 2 2 σ 2 We often write x N (µ, σ 2 ) or N (x; µ, σ 2 ) to imply that x follows a Gaussian distribution with mean µ = E[x] and variance σ 2 = V[x]. 20

21 Multivariate Normal Distribution The multivariate Gaussian (normal) distribution of an n-dimensional random vector x N (µ, Σ), with mean µ = E[x] and covariance Σ = Cov[x] = E[(x µ)(x µ) T ], can be written as follows. p(x) = (2π) n 2 Σ 1 2 exp( 1 2 (x µ)t Σ 1 (x µ)) 21

22 Marginalization and Conditioning of Normal Distribution Let x and y be jointly Gaussian random vectors [ ] [ ] x µx A C N ([, y] µ y C T ) B then the marginal distribution of x is x N (µ x, A) and the conditional distribution of x given y is x y N (µ x + CB 1 (y µ y ), A CB 1 C T ) 22

23 Affine Transformation of a Multivariate Gaussian Suppose x N (µ, Σ) and y = Ax + b. Then y N (Aµ + b, AΣA T ). 23

24 Canonical Parametrization of a Multivariate Gaussian The canonical parametrization of a multivariate Gaussian N (µ, Σ) can be identified by the information (or precision) matrix Λ = Σ 1, and the information vector η = Σ 1 µ. Then we can write p(x) = N (µ, Σ) = N 1 (η, Λ). N 1 (x; η, Λ) = γ exp( 1 2 xt Λx + x T η) 24

25 Visualizing Multivariate Gaussian Example (Visualizing multivariate Gaussian) Let x = vec(x 1, x 2 ) and x N (µ, Σ) where [ ] [ ] µ =, Σ = See mvn plots.m for code and reproducing the plots. 25

26 Visualizing Multivariate Gaussian 26

27 0.1 Visualizing Multivariate Gaussian

28 Sampling from Gaussian Distributions We wish to draw samples from Y N (µ, Σ); If Cov[Y ] is not degenerate (is positive definite), then Σ = LL T. L is a lower triangular matrix computed using Cholesky decomposition; Let X N (0 n, I n ) be a vector of standard normal random variables where n is the dimension (length) of Y. Define Z LX + µ; We have E[Z] = µ and Cov[Z] = LL T. Therefore, using samples from N (0, 1), we are able to draw samples from N (µ, Σ). 28

29 Sample Mean and Covariance Sometimes we do not know the distribution of data, but instead we have access to samples or observations. Let X = x be a random vector and x 1,..., x n be n independent samples (realization) of X. The sample or empirical mean, x, and (unbiased) covariance, Σ, can be computed as follows. x = 1 n x i n i=1 Σ = 1 n 1 n (x i x)(x i x) T i=1 Note that the sample mean is a random variable. 29

30 Chi-Square Distribution Let X N (µ, Σ) be an n-dimensional Gaussian random vector. The scalar random variable, q, defined by the quadratic form q = (x µ) T Σ 1 (x µ) is the sum of the squares of n independent zero-mean, unity-variance Gaussian random variables. We say q has a chi-square distribution with n degrees of freedom, i.e., q χ 2 n. It can be shown that E[q] = n and V[q] = 2n. If q 1 χ 2 n 1 and q 2 χ 2 n 2, then q 3 = q 1 + q 2 χ 2 n 1 +n 2. 30

31 Chi-Square Goodness-of-Fit Chi-square goodness-of-fit is a statistical test that determines if an observation sample comes from a specified probability distribution. In particular, we would like to test the null hypothesis that the data in x comes from a normal distribution such as N (µ, Σ). This test is useful for data association. 31

32 Chi-Square Distribution Remark The weighted sum of independent identically distributed (i.i.d.) chi-square random variables does not follow a chi-square distribution. 32

33 Mahalanobis Distance The distance (x µ) T Σ 1 (x µ) is known as Mahalanobis distance. 33

34 Confidence Ellipsoid Given N (µ, Σ), illustrate the confidence region that the mentioned distribution covers with a certain probability. Geometrically, (x b) T A(x b) = 1, where A is positive definite and x, b R n, corresponds to an ellipsoid in R n centered at b. Since A is positive definite, we have A = LL T. It can be shown that L corresponds to a linear transformation that rotates and scale a sphere to the desired ellipsoid. 34

35 Confidence Ellipsoid q = (x µ) T Σ 1 (x µ) also corresponds to an ellipsoid. The chi-square value, q, for a desired significance level (p-value) can be found using the pre-calculated chi-square table. to map a point from a unit sphere, x s, to the desired ellipsoid, x e, we can use the following formula: x e = qlx s + µ L is the Cholesky factor of covariance matrix, i.e., Σ = LL T. 35

36 Confidence Ellipsoid See confidence ellipsoid plots.m for code and reproducing the plots. 36

37 Next Time Estimation and Information Theory Readings: Probabilistic Robotics Ch. 2, Understand Example Lecture note 2 Bar-Shalom Ch. 2 Cover and Thomas Ch. 2 (skim) 37