Algorithms for Uncertainty Quantification
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1 Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017
2 Lecture 2: Repetition of probability theory and statistics
3 Example: coin flip
4 Example Experiment 1 given a fair coin (heads and tails) flip it once Q: what is the probability of getting head/tail? Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
5 Example Experiment 1 given a fair coin (heads and tails) flip it once Q: what is the probability of getting head/tail? Answer possible outcomes: head, tail P(head) = P(tail) = 1/2 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
6 Example (cont d) Experiment 2 given two fair coins flip them twice Q: what is the probability of getting {(head, head),(tail, head)}? What about {(head,head),(head,tail)}? Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
7 Example (cont d) Experiment 2 given two fair coins flip them twice Q: what is the probability of getting {(head, head),(tail, head)}? What about {(head,head),(head,tail)}? Answer possible outcomes: (head, head), (tail, head), (head, tail), (tail, tail) P({(head,head),(tail,head)}) = 1/2 P({(head,head),(head,tail)}) = 1/2 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
8 Formal definition of probability
9 Two perspectives to probability Frequentist probability = frequency with which an event occurs if the experiment is repeated a large number of times ( ) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
10 Two perspectives to probability Frequentist probability = frequency with which an event occurs if the experiment is repeated a large number of times ( ) Bayesian probability = distribution of subjective values constructed or updated as data is observed Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
11 Probability space A probability space is a triple (Ω,F,P), where Ω: sample space; set of all possible outcomes F : σ algebra; set of events s.t. each event is a set containing zero or more outcomes P : F [0,1] probability measure that satisfies: 1. P( ) = 0 2. P(Ω) = 1 3. if A i F and A i A j =, then P( i=1 A i) = i=1 P(A i) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
12 Coins flip example revised Let h := head, t:= tail Experiment 1 sample space Ω = {h,t} σ algebra F = {,{h},{t},{h,t}} events of interest A = {h}, B = {t} P(A ) = P(B) = 1/2 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
13 Coins flip example revised Let h := head, t:= tail Experiment 1 sample space Ω = {h,t} σ algebra F = {,{h},{t},{h,t}} events of interest A = {h}, B = {t} P(A ) = P(B) = 1/2 Experiment 2 sample space Ω = {(h,h),(t,h),(h,t),(t,t)} σ algebra F = {,(h,h),(h,t),(t,h),(t,t),{(h,h),(h,t),...},ω} events of interest A = {(h,h),(t,h)}, B = {(h,h),(h,t)} P(A ) = P(B) = 1/2 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
14 Univariate concepts
15 Univariate concepts: random variables A random variable is a function X : Ω R s.t. {ω Ω X(ω) x} F Example If, in Experiment 2 (two coins flip), X(ω) counts the number of tails, 0, ω = (h,h) 1, ω = (h,t) X(ω) = 1, ω = (t,h) 2, ω = (t,t) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
16 Univariate concepts: random variables A random variable is a function X : Ω R s.t. {ω Ω X(ω) x} F Example If, in Experiment 2 (two coins flip), X(ω) counts the number of tails, 0, ω = (h,h) 1, ω = (h,t) X(ω) = 1, ω = (t,h) 2, ω = (t,t) X is said to be discrete if it takes values in a countable subset {x 1,x 2,...} R; otherwise, it is said to be continuous Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
17 Univariate concepts: CDF Every random variable X has an associated cumulative distribution function (CDF) F X : R [0,1] F X = P(ω Ω X(ω) x) Often, the CDF is expressed as F X (x) = P{X x} Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
18 Univariate concepts: CDF Every random variable X has an associated cumulative distribution function (CDF) F X : R [0,1] F X = P(ω Ω X(ω) x) Often, the CDF is expressed as F X (x) = P{X x} Example 0, ω = (h,h) 1, ω = (h,t) If X(ω) = 1, ω = (t,h) 2, ω = (t,t) 0, x < 0 1/4, 0 x < 1, then F X = 3/4, 1 x < 2 1, x 2 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
19 Univariate concepts: PDF A random variable X is continuous if its CDF is absolutely continuous, i.e. F X (x) = x f X (s)ds, x R (1) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
20 Univariate concepts: PDF A random variable X is continuous if its CDF is absolutely continuous, i.e. F X (x) = x f X (s)ds, x R (1) From Equation 1, the derivative is called the probability density function (PDF) of X f X (x) = df X(x), f X : R [0, ) dx Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
21 Univariate concepts: PDF Properties of the PDF Let supp(f X ) = {x R : f X (x) 0} f X (x) 0, x supp(f X ) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
22 Univariate concepts: PDF Properties of the PDF Let supp(f X ) = {x R : f X (x) 0} f X (x) 0, x supp(f X ) supp(f X ) f X(x)dx = 1 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
23 Univariate concepts: expectation, variance Let supp(f X ) = {x R : f X (x) 0} The expectation (mean value, first statistical moment) of a continuous random variable X with PDF f X is defined as µ := E[X] = xf X (x)dx, supp(f X ) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
24 Univariate concepts: expectation, variance Let supp(f X ) = {x R : f X (x) 0} The expectation (mean value, first statistical moment) of a continuous random variable X with PDF f X is defined as µ := E[X] = xf X (x)dx, supp(f X ) The variance (density s variability, second central statistical moment) of a continuous random variable X with PDF f X is defined as σ 2 := Var(X) = (x E[X]) 2 f X (x)dx = E[X 2 ] E[X] 2 supp(f X ) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
25 Univariate concepts: discrete random variables The probability mass function (PMF) of a discrete random variable X is given by f X (x) = P(X = x) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
26 Univariate concepts: discrete random variables The probability mass function (PMF) of a discrete random variable X is given by f X (x) = P(X = x) The (sample) mean ˆX and variance S 2 of a discrete random variable X with equiprobable realizations X 1,...,X n are ˆX = 1 n n X i, S 2 = 1 n 2 i=1 n 1 (X i ˆX) i=1 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
27 Example distributions Discrete distributions Binomial Poisson Bernoulli geometric... Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
28 Example distributions Discrete distributions Binomial Poisson Bernoulli geometric... Continuous distributions Normal (Gaussian) Uniform Beta Gamma... Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
29 Example continuous distributions: normal distribution The PDF of the normal distribution is f X : R [0, ), f X (x) = 1 σ 2π exp( (x µ)2 /2σ 2 ) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
30 Example continuous distributions: normal distribution The PDF of the normal distribution is f X : R [0, ), f X (x) = 1 σ 2π exp( (x µ)2 /2σ 2 ) The notation X N (µ,σ 2 ) means that X is normally distributed with mean µ and variance σ Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
31 Example continuous distributions: normal distribution The PDF of the normal distribution is f X : R [0, ), f X (x) = 1 σ 2π exp( (x µ)2 /2σ 2 ) The notation X N (µ,σ 2 ) means that X is normally distributed with mean µ and variance σ When µ = 0, σ = 1, X N (0,1) is a standard normal random variable Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
32 One dimensional normal distribution Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
33 Example continuous distributions: uniform distribution The PDF of the uniform distribution is f X : [a,b] {0, 1 b a }, where I [a,b] (x) = { 1, x [a,b] 0, otherwise f X (x) = 1 b a I [a,b](x), Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
34 Example continuous distributions: uniform distribution The PDF of the uniform distribution is f X : [a,b] {0, 1 b a }, where I [a,b] (x) = { 1, x [a,b] 0, otherwise f X (x) = 1 b a I [a,b](x), The notation X U (a,b) means that X is uniformly distributed on the interval [a, b] Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
35 Example continuous distributions: uniform distribution The PDF of the uniform distribution is f X : [a,b] {0, 1 b a }, where I [a,b] (x) = { 1, x [a,b] 0, otherwise f X (x) = 1 b a I [a,b](x), The notation X U (a,b) means that X is uniformly distributed on the interval [a, b] Expectation and variance of the uniform distribution If X U (a,b), E[X] = a+b (b a)2 2, Var(X) = 12 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
36 One dimensional uniform distribution Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
37 Multivariate concepts
38 Multivariate concepts: random vectors, covariance, correlation X : Ω R n, X = [X 1,X 2,...,X n ] is called an n-dimensional random vector, where X 1,X 2,...,X n are random variables Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
39 Multivariate concepts: random vectors, covariance, correlation X : Ω R n, X = [X 1,X 2,...,X n ] is called an n-dimensional random vector, where X 1,X 2,...,X n are random variables The covariance of two random variables X and Y is cov(x,y ) = E[(X E[X])(Y E[Y ])] = E[XY ] E[X]E[Y ] Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
40 Multivariate concepts: random vectors, covariance, correlation X : Ω R n, X = [X 1,X 2,...,X n ] is called an n-dimensional random vector, where X 1,X 2,...,X n are random variables The covariance of two random variables X and Y is cov(x,y ) = E[(X E[X])(Y E[Y ])] = E[XY ] E[X]E[Y ] The Pearson correlation coefficient of two random variables X and Y is ρ XY = cov(x, Y) Var(X)Var(Y ), ρ XY [ 1,1] Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
41 Multivariate concepts: expectation, variance of a sum Let a 1,a 2,...,a n R and X 1,X 2,...,X n be random variables E [ n ] n a i X i = a i E[X i ] i=1 i=1 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
42 Multivariate concepts: expectation, variance of a sum Let a 1,a 2,...,a n R and X 1,X 2,...,X n be random variables E [ n ] n a i X i = a i E[X i ] i=1 i=1 Var ( n ) n a i X i = ai 2 Var(X i) + 2 a i a j cov(x i,x j ) i=1 i=1 i<j If X i,x i+1 are uncorrelated, i.e. ρ Xi X i+1 = 0, 1 i n, Var ( n ) n a i X i = ai 2 Var(X i) i=1 i=1 Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
43 Multivariate concepts: independence Let A, B denote two events. The joint probability of A and B, P(A,B), is defined as P(A,B) = P(A B) = P(A B)P(B), where P(A B) is the probability of A given that B already happened Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
44 Multivariate concepts: independence Let A, B denote two events. The joint probability of A and B, P(A,B), is defined as P(A,B) = P(A B) = P(A B)P(B), where P(A B) is the probability of A given that B already happened A and B are independent if P(A,B) = P(A)P(B), i. e. P(A B) = P(A) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
45 Multivariate concepts: independence Let A, B denote two events. The joint probability of A and B, P(A,B), is defined as P(A,B) = P(A B) = P(A B)P(B), where P(A B) is the probability of A given that B already happened A and B are independent if P(A,B) = P(A)P(B), i. e. P(A B) = P(A) The random variables X and Y are independent if their joint PDF (PMF) is f XY (x,y) = f X (x)f Y (y) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
46 Multivariate concepts: i.i.d. random variables Random variables X 1,X 2,...,X n are called independent and identically distributed (i.i.d.) with PDF f X if they are mutually independent and, if f Xi is the PDF of X i, 1 i n, f X1 = f X2 =... = f Xn := f X and f X (x 1,x 2,...x n ) = n i=1 f X (x i ) Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
47 Example multivariate distribution: multivariate normal The n-dimensional random vector X is normally distributed with mean vector µ = [µ 1, µ 2,..., µ n ] T and covariance matrix V,V ij = cov(x i,x j ), written X N (µ,v ), if f X (x) = where V is the determinant of V 1 (2π)n V exp[ 1 2 (x µ)v 1 (x µ) T ] Standard multivariate normal: µ = [0,0,...,0] T, V = I n Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
48 Standard bivariate normal distribution Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
49 Summary
50 Summary Coin flip experiment Probability space (Ω,F,P) Univariate concepts Random variables: discrete and continuous Cumulative distribution function (CDF) Continuous random variables Discrete random variables Examples: normal, uniform distributions Multivariate concepts Random vectors Covariance, correlation Independent and identically distributed random variables Example: bivariate normal distribution Dr. rer. nat. Tobias Neckel Algorithms for Uncertainty Quantification Summer Semester
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