Probability theory. References:

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1 Reasoning Under Uncertainty References: Probability theory Mathematical methods in artificial intelligence, Bender, Chapter 7. Expert systems: Principles and programming, g, Giarratano and Riley, pag Artificial intelligence: A modern approach, Russell and Norvig, pg

2 Review of Probability and Statistics Probability Definition of probability Axioms and properties Conditional probability Bayes theorem Random Variables Definition Probability density function Statistical characterization of random variables Random Vectors Mean vector Covariance matrix The Gaussian random variable

3 Experiment An experiment is a process for which the outcome is not known with certainty. Examples Rolling a fair six-sided sided die. Randomly choosing 10 transistors from a lot of Selecting a newborn child at a known hospital.

4 Event An event is an outcome or combination of outcomes from a statistical experiment. Examples Obtaining a 6 when a die is rolled. Obtaining an even number when a die is rolled. Selecting a newborn child in a hospital weighing more than 8 pounds.

5 Sample Space The sample space is the event consists of all possible outcomes of a statistical experiment Sample space Composite event {5,6} Simple event {4}

6 Relations between events in a sample space Interseccion of events A B A B U Union of events A B A B U Complement of an event A A A U

7 Theory of Probability The 3 axioms of probability Axiom 1: 0 P(E) 1 Axiom 2: P(Universe)=1 P( )=0 P(True)=1 P(False)=0 Axiom 3: If {A B= } then P(A B)=P(A)+P(B) Corollary (Axiom 2 and 3): P(A)+P(~A)=1

8 Probability of Composite events U={1,2,3,4,5,6} P{1}=P{2}=P{3}=P{5}=P{6}=1/6 A={2,4,6} B={3,6} A B={6} P(A B)=P({6})=1/6 Independent Events P( Y)=P()P(Y)

9 Conditional Probability P(A B)=P(A B)/P(B) P(B) 0 P(A B)=P(A B)P(B) General Rule P(A 1 A 2 A N )=P(A 1 A 2 A N )* P(A 2 A 3 A N )* P(A N-1 A N )*P(A N ) Independent d Events P(A B)=P(A) P(B A)=P(B) P(B) P(A B)=P(A)*P(B)

10 Diagram: conditional probability

11 Theorem of total probability

12 Bayes Theorem

13 Bayes Theorem and Statistical Pattern Recognition Bayes decision making choosing the most likely class, given the value of the feature or features. Feature value x, Class C P(x) probability distribution for x in the entire population (normalization constant that does not affect the decision) P(C) prior probability that a random sample is from class C (prior) P(x C) conditional probability of obtaining feature value x given sample is from class C (likelihood) P(C x) have to estimate that given a sample has feature value x, it is from class C (posterior) We know probability of the joint event that a sample comes from class C and has feature value x is P(C x)=p(c)p(x C)=P(x)P(C x) Rearranging P(C x)=p(c)p(x C) / P(x)

14 Law of Total Probability and Bayes Theorem Law of total probability states that if an event A can occur in m different ways: B1,B2,,Bm (m subevents are mutually exclusive-cannot occur at the same time), the probability of y occuring is the sum of the probabilities of the subevents xi P( y) = P( x y) x From definition of conditional probability P(A B) (Bj A)=P(A Bj)P(Bj) Then P(Bj A)=P(A Bj)P(Bj)/ΣP(A Bj)P(Bj) Posterior = likelihood x prior / evidence

15 Example

16 Solution

17 Solution using Bayes Theorem

18 Exercise in group (supplement problem) As an example in which two events are not independent, suppose that a company plans to test two new techniques for improving the extraction of oil from the ground. The first technique consists fo setting off an explosion at the bottom of a well to fracture the strata and then testing seismically to determine the extent of the fracturing. The second technique consists of injecting hot brine into the well to loosen the oil and then pumping to measure the oil recovery. Let E be the event that the explosion successfully fractures the strate within a radius of 100 meters, and let R be the event that oil can be recovered at a rate of 50 barrels per day after pumping in hot brine. In a certain region, P(E) is estimated to be 0.8. If E occurs, the probability of R occuring is estimated ed to be P(R E)=0.9, but if the explosion o is not a success, the estimated probability of recovery isonly P(R ~E)=0.3. These 3 assumptions are sufficient to define the probabilities of any combination of outcomes, given any set of constraints.

19 What is the probability that the explosion and the brine tests are both successful or both fail? What is the probability that the explosion test was successful, give the constraint that only one of the two tests was successful? What is the probability that the explosion was successful, given that recovery using brine was successful? If one or more of the tests was successful, what is the probability that the other one was successful?

20 Random Variables Example: A fair coin is tossed 3 times. S={(TTT),(TTH),(THT),(HTT),(HHT),(HTH),(THH), (HHH)} 20

21 Random Variables Let be the number of heads tossed in 3 tries. (TTT)= (HHT)= (TTH)= (HTH)= (THT)= (THH)= (HTT)= (HHH)= So P(=0)= P(=1)= P(=2)= P(=3)= 21

22 Random Variable as a Measurement Think of much more complicated experiments A chemical reaction. A laser emitting photons. A packet arriving at a router. Sample space here is a set here with noncountable infinite number of elements 22

23 Random Variable as a Measurement Thus a random variable can be thought of as a measurement on an experiment 23

24 Random Variable as a Measurement Thus a random variable can be thought of as a measurement on an experiment 24

25 Probability Density Function (pdf) f ( x) df ( x ) = dx F x ( x) f ( x) = dx 25

26 Properties of the pdf Positivity f ( x) 0 Integral over all x (unit area) f ( x) dx =1 26

27 More properties for the pdf f (x) can be used to calculate probabilities of events { } ( ) + < x 2 dx x f x x P{ } ( ) { } ( ) + < = < 2 1 x x 2 1 dx x f x x P dx x f x x P { } ( ) { } ( ) + = < 2 1 x x 2 1 dx x f x x P dx x f x x P { } ( ) { } ( ) < < = 2 1 x x 2 1 dx x f x x P dx x f x x P 27 { } ( ) + = < < x dx x f x x P

28 PMF and CDF for the 3 Coin Toss Example: = # of heads What would be the pdf for this example? 28

29 Types of Random Variables Discrete: pdf consist only of impulses. CDF has the shape of a staircase Continuous: CDF is a continuous function or, equivalently, pdf has no impulses. Mixed: neither continuous nor discrete 29

30 Types of Random Variables 30

31 CMF, PMF, pdf p(i) F(i) i Discrete i Continuous f(x) F(x) x x 31

32 Statistical Characterizations Expectation (Mean Value, First Moment): 32

33 Statistical Characterizations Variance of : σ 2 x = E {( ) } 2 ( ) 2 η = x η f ( x) { 2 } ( η ) 2 = E dx Standard d Deviation of : σ 2 x x σ = 33

34 Higher Order Statistics k-th moment of m k { k} = = E x f (x)dx k k-th central moment of : μ k = E { ( k k η ) } = ( x η ) f ( x)dx) 34

35 Conditional Cumulative Functions Cumulative distribution function F ( x/c) = P{ x/c} = P { x,c } P { C } Probability density function ( x/c ) f = df ( x/c) dx 35

36 Uniform pdf A R.V. that is uniformly distributed between x 1 and x 2 has density function: 1 2 y

37 Gaussian pdf A R.V. that is normally distributed has density function: μ 37

38 Transformations of Random Variables Y = g() 38

39 Transformations of Random Variables Y = g() The basic question is: Knowing the statistical ti ti characterization ti of f, partial or total, what are the resulting characterizations of Y? 39

40 Two Random Variables Defined as real value functions, (ζ) and Y(ζ), over the same sample space S 40

41 Total Characterization: Joint CDF The joint cumulative distribution function (JCDF) for a random variables and Y is F Y ( x, y ) = P { x, Y y } 41

42 Properties: Joint CDF Note that F Y (x,y) is bounded from the above and below 0 F Y (x,y) 1 Note that F Y( (x,y) is non-decreasing in x and y, i.e. ( x, y ) F ( x, y ) for all x > x, and all F > F Y Y 2 ( x, y ) F ( x, y ) for all y > y, and all x 2 Y Y y 42

43 Properties: Joint CDF F Y (x,y) is continuous from the right lim ε 0 δ F Y ( x + ε, y + δ) = F ( x, y) F Y (x,y) can be used to calculate probabilities of rectangular events P { x < x, y < Y y } = F ( x, y ) F ( x, ) y1 Y 43

44 Properties of the Joint pdf 44

45 Partial Characterization Marginal densities Conditional densities Joint moments Conditional means Correlation, covariance Higher order moments 45

46 Marginal Densities f ( x ) f ( x, y ) = y dy Y f ( y ) f ( x, ) Y = y dx Y 46

47 f f f Conditional pdf ( ) f x/y = Y = f ( y/x) f Y ( x, y ) f ( x, y ) Y ( y) ( x, y) ( x ) = Y = Y f f Y f f Y Y ( x, y )dx ( x, y) ( x, y)dy ( x, y) f ( y/x) f ( x) f ( x/y) f ( y) = = Y Y Y 47

48 Statistical Independence f ( x, y) ( y ) ( ) f x/y = Y = f f Y f ( x) f f ( x, y ) ( x ) ( ) f y/x = Y = Y f Y ( x, y ) = f ( y ) f ( x ) Y f Y ( y) 48

49 Correlation Coefficient ρ Y (cont.) 1 ρ +1 1 Y The closer ρ Y is to -1 or +1 the more the random variables and Y are said to be linearly related 49

50 Correlation Between 2 Variables

51 ρ 15 =-0.098

52 ρ 13 =0.694

53 Correlation Coefficient ρ Y Useful relations ρ = Y σ Y σ σσ Y σ Y = ρ Y σ σ 2 ry = 1 ρ Y σ Y 53

54 54

55 Orthogonal, Uncorrelated, Independent Two random variables are defined as uncorrelated if R = E[] E[Y] or σ = 0 or ρ = Y Y Y Notice that independence uncorrelated (Why?) Two random variables are called orthogonal if R Y=0 but it does not imply that σ Y =0 0or equivalently that ρ Y =0 Need one mean =0 and the other finite. 0 55

56 Orthogonal, Uncorrelated, Independent 56

57 Jointly Gaussian Random Variables (JGRV) and Y are jointly Gaussian if their joint pdf f Y (x,y) has the form: Y(,y) f Y 2 2 ( x η ) 2ρ ( x η )( y η ) ( y η ) 1 1 Y Y Y (x, y) = ( ) exp π σ 2( 1 ρ ) 2 Y σ σσy σy σy 1 ρy A very important model in statistical signal processing, communications, and control. 57

58 Properties of JGRV Marginal distributions are Gaussian Conditional pdf are Gaussian (Show this!) 58

59 f Multidimensional Gaussian pdf 1 ( ) ( ) T -1 x = exp ( ) ( ) n/2 1/ 2 x η K x η 2π K 2 ( x ) ~ N ( η, K ) f η, 1 η K [ ] = [ E[ ] [ ] [ ] 1 E 2 E n [ ( η )( η ) T ] = E L = E T 59

60 Multidimensional Gaussian Density 60

61 Properties of the Multivariate Gaussian pdf All marginal functions are Gaussian Joint pdf of any subset of components of are JGRV Conditional pdfs of a subset of variables on a different subset are Gaussian Any linear transformation of, Z=A+b, the RV Z has a multidimensional Gaussian pdf 61

62 Uniform pdf A R.V. that is uniformly distributed between x 1 and x 2 has density function: 1 2 y a b f () (x) = b 1 a 0 for a x otherwise b 62

63 Gaussian pdf A R.V. that is normally distributed has density function: f (x) exp ( ) 2 x 1 η = 2 2 2πσ 2σ η 63

64 Detection of a Signal Is there some signal around?

65 Detection Concept Signal Signal + Noise

66 Bit 1 or t (s) 0 0 t (s)

67 Two Signals + Noise = S i + n S 0 = Bit 0 0 S 1 = Bit 1

68 What is this? t (s)

69 Need a Threshold & Sampling t (s)