Multivariate probability distributions and linear regression

Transcription

1 Multivariate probability distributions and linear regression Patrik Hoyer 1

2 Contents: Random variable, probability distribution Joint distribution Marginal distribution Conditional distribution Independence, conditional independence Generating data Multivariate Gaussian distribution Multivariate linear regression Expectation, variance, covariance, correlation Estimating a distribution from sample data Patrik Hoyer 2

3 Random variable - sample space (set of possible elementary outcomes) - probability distribution over sample space Examples: - The throw of a die x P (x) /6 1/6 1/6 1/6 1/6 1/6 - The sum of two dice x P (x) /36 1/18 1/12 1/9 5/36 1/6 5/36 1/9 1/12 1/18 1/36 - Two separate dice (red, blue) x P (x) (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3)... (6,6) 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 1/36 Patrik Hoyer 3

4 Discrete variables: - Finite number of states (e.g. dice examples) - Infinite number of states (e.g. how many heads before one tales in a sequence of coin tosses?) Continuous variables: Each particular state has a probability of zero, so we need the concept of a probability density: P (X x) = x p(t) dt (e.g. how long time until next bus arrives? what will be the price of oil a year from now?) Patrik Hoyer 4

5 A probability distribution satisfies Probabilities are non-negative: P (X = x) = P X (x) = P (x) 0 2. Sum to one: P (x) = 1 x p(x) dx = 1 (discrete) (continuous) [Note that in the discrete case this means that there exists no value of x such that P (x) > 1. However this does not in general hold for a continuous density p(x)!] Patrik Hoyer 5

6 The joint distribution of two random variables: - Let X and Y be random variables. Their joint distribution is P (x, y) = P (X = x and Y = y) - Example: Two coin tosses, X denotes first throw, Y denotes second (note: independence!) P (x, y) : X H T H Y T Example: X: Rain today? Y : Rain tomorrow? P (x, y) : X Y N Y Y N Patrik Hoyer 6

7 Marginal distribution: - Interested in or observing only one of the two variables - The distribution is obtained by summing (or integrating) over the other variable: P (x) = y P (x, y) p(x) = - Example (continued): What is the probability of rain tomorrow? That is, what is P (y)? p(x, y) dy X Y N Y Y N In the same fashion, we can calculate that the chance of rain today is 0.7. P (y) : Patrik Hoyer 7

8 Conditional distribution: - If we observe X = x how does that affect our belief about the value of Y? - Obtained by selecting the appropriate row/column of the joint distribution, and renormalizing it to sum to one: P (x, y) p(x, y) P (y X = x) = P (y x) = p(y x) = P (x) p(x) - Example (continued): What is the probability that tomorrow rains, given that today does not rain? i.e. what is P (y X = no rain )? Y Y N P (y X = no rain ) X Y N / ( ) Y N Patrik Hoyer 8

9 Chain rule: P (x, y) = P (x)p (y x) = P (y)p (x y) p(x, y) = p(x)p(y x) = p(y)p(x y) So the joint distribution can be specified directly, or using the marginal and conditional distribution (can even choose, which way one specifies it) Patrik Hoyer 9

10 Independence: Two random variables are independent, if and only if knowing the value of one does not change our belief about the second: x : P (y x) = P (y) y : P (x y) = P (x) This is equivalent to being able to write the joint distribution as the product of the marginals: P (x, y) = P (x)p (y) We write this as: X Y or, if we want to explicitly specify the distribution: (X Y ) P Example: Two coin tosses... Patrik Hoyer 10

11 Three or more variables: - joint distribution: ( multidimensional array/function ) - marginal distributions: (e.g.) P (x) = P (x, y) = v,w,y,z,... v,w,z,... P (v, w, x, y, z,...) P (v, w, x, y, z,...) P (v, w, x, y, z,...) - conditional distributions: (e.g.) P (x v, w, y, z,...) = P (v, w, x, y, z,...)/p (v, w, y, z,...) P (x, y v, w, z,...) = P (v, w, x, y, z,...)/p (v, w, z,...) P (v, w, y, z,... x) = P (v, w, x, y, z,...)/p (x) P (x y) = P (v, w, x, z,... y) v,w,z,... marginal and conditional Patrik Hoyer 11

12 - Chain rule P (v, w, x, y, z,...) = P (v)p (w v)p (x v, w)p (y v, w, x) P (z v, w, x, y)p (... v, w, x, y, z) - Complete independence between all variables if and only if: P (v, w, x, y, z,...) = P (v)p (w)p (x)p (y)p (z)p (...) - Conditional independence (e.g: if we know the value of z then does not give any additional information about y ): P (x, y z) = P (x z)p (y z) x This is also written: X Y Z or explicitly noting the distribution: (X Y Z) P Patrik Hoyer 12

13 - In general we can say that marginal distributions are conditional on not knowing the value of other variables: P (x) = P (x ) and (marginal) independence is independence conditional on not observing other variables: P (x, y ) = P (x )P (y ) - Example of conditional independence: Drownings and ice-cream sales. These are mutually dependent (both happen during warm weather) but are, at least approximately, conditionally independent given the weather Patrik Hoyer 13

14 Example: conditional dependence: Two coin tosses and a bell that rings whenever they get the same result. The coins are marginally independent but conditionally dependent given the bell! X : Y : Z : First coin toss Second coin toss Bell H Y T P (x, y) = X H T (independent) H Y T P (x, y Z = bell rang ) = X H T (dependent!) Patrik Hoyer 14

15 Data generation, sampling - Given some P (x), how can we draw samples (generate data) from that distribution? Answer: Divide the unit interval [0,1] into parts corresponding to the probabilities, draw a uniformly distributed number in the interval, and select the state into which we fell: X := x 2 P (x 1 ) P (x 2 ) P (x 4 ) P (x 6 ) P (x 3 ) P (x 5 ) Patrik Hoyer 15

16 Given a joint distribution P (x, y, z) (generate data)? - We could list all joint states, then proceed as above, or... - Draw data sequentially from conditional distributions: 1. First draw x from 2. Next y from 3. Finally from P (x) P (y x) z P (z x, y), how can we draw samples Note: We can freely choose any ordering of the variables! Patrik Hoyer 16

17 Example (continued): Two coin tosses and a bell that rings if and only if the two tosses give the same result - can draw all the variables simultaneously by listing all the joint states, calculating their probabilities, placing them on the unit interval, and then draw the joint state - can first independently generate the coin tosses, then assign the bell - can first draw one coin toss and the bell, and then assign the second coin toss Patrik Hoyer 17

18 Numerical random variables - Expectation: E{X} = x xp (x) (discrete) E{X} = x p(x) dx (continuous) - Variance: Var(X) = σx 2 = σ XX = E{(X E{X}) 2 } - Covariance: Cov(X, Y ) = σ XY = E{(X E{X})(Y E{Y })} - Correlation coefficient: ρ XY = σ XY σ 2 X σ 2 Y Patrik Hoyer 18

19 - Multivariate numerical random variables... (random vectors) Expectation: E{V} = E{V 1 } E{V 2 }. E{V N } Covariance matrix ( variance-covariance matrix ) C V = Σ V = E{(V E{V})(V E{V}) T } = Var(V 1 ) Cov(V 1, V N )... Cov(V N, V 1 ) Var(V 2 ) = σ V1 V 1 σ VN V 1... σ V1 V N σ VN V N Patrik Hoyer 19

20 Conditional expectation, variance, covariance, correlation - Conditional expectation (note: function of y!) E{X Y = y} = x xp (x y) (discrete) E{X Y = y} = x p(x y) dx (continuous) - Conditional variance (note: function of y!) Var(X Y = y) = σx y 2 = σ XX y = E{(X E{X}) 2 } P (X Y =y) - Conditional covariance (note: function of z!) Cov(X, Y z) = σ XY z = E{(X E{X})(Y E{Y })} P (X,Y Z=z) - Conditional correlation coefficient (note: function of z!) ρ XY z = σ XY z σ 2 X z σ2 Y z Patrik Hoyer 20

21 - Multivariate Gaussian ( normal ) density: p(x) = N (µ, Σ) = (2π) d/2 Σ 1/2 exp 1 2 (x µ)t Σ 1 (x µ) has the following properties: mean vector (µ, and covariance matrix Σ) as the only parameters x 2 x 1 Patrik Hoyer 21

22 all marginal and conditional distributions are also Gaussian, and the conditional (co)variances do not depend on the values of the conditioning variables: Let x and y be random vectors whose dimensions are n and m. If they are joined together into one random vector z = (x T, y T ), with dimension n + m, then its mean m z and covariance matrix C z are mx Cx C m z =, C z = xy, (1) C yx C y m y where m x and m y are the means of x and y, and C x and C y are the covariance matrices of x and y respectively, and C xy contains the cross covariances. If z is multivariate Gaussian then x and y are also Gaussian. Additionally the conditional distributions p(x y) and p(y x) are Gaussian. The latter s mean and covariance matrix are m y x = m y + C yx C 1 x (x m x ) (2) C y x = C y C yx C 1 x C xy (3) Patrik Hoyer 22

23 The conditional variance, conditional covariance, and conditional correlation coefficient, for the Gaussian distribution, are known as partial variance σx Z 2, partial covariance σ XY Z, and partial correlation coefficient ρ XY Z (respectively) These can of course always be computed directly from the covariance matrix (regardless of whether the distribution actually is Gaussian!)......but they can only be safely interpreted as conditional variance, conditional covariance, and conditional correlation coefficient (respectively) for the Gaussian distribution. Patrik Hoyer 23

24 for Gaussian: zero (partial) covariance zero (conditional) covariance (conditional) independence i.e. (σ XY Z = 0) ( z : σ XY z = 0) (X Y Z) in general: we only have one-way implication: zero (conditional) covariance (conditional) independence i.e. ( z : σ XY z = 0) (X Y Z) Note, however, that conditional independence does not imply zero partial covariance in the completely general case! Patrik Hoyer 24

25 Linear regression: ŷ = r yx x + y y Fit a line through the data, explaining how y varies with x. Minimize sum of squares error between ŷ and y. r yx = σ XY σx 2 Probabilistic interpretation: ŷ E{Y X = x} (note that this is true only for roughly linear relationships) x Patrik Hoyer 25

26 Note the symmetry: We could equally well regress x on y! ˆx = r xy y + x x y Patrik Hoyer 26

27 Multivariate linear regression: ẑ = ax + by + z a = r zx y = σ ZX Y σ 2 X Y x z y Note that the partial regression coefficient r zx y is NOT the same, in general, as one gets from regressing z on x, ignoring y : Note also that r zx y is derived from the partial (co)variances. This holds regardless of the form of the underlying distribution. r zx Patrik Hoyer 27