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Statistics and Steganalysis CSM25 Secure Information Hiding Dr Hans Georg Schaathun University of Surrey Spring 2007 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 1 / 52 Learning Outcomes After this sessions, everyone

  1. Probabilities Probabilities Statistical Steganalysis The goal of Statistical Steganalysis: Wendy observes an image X sent from Alice to Bob. Let x be the observed image. Is x a stegogramme (from S ) or a pure image (from C )? Unfortunately, most of the time x ∈ C and x ∈ S are both possible. . . Compare Pr ( S = x ) and Pr ( C = x ) . If Pr ( S = x ) > Pr ( C = x ) , then x is most likely to be a stegogramme. How confident can we be about the conclusion? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 13 / 52

  2. Probabilities Probabilities Statistical Steganalysis The goal of Statistical Steganalysis: Wendy observes an image X sent from Alice to Bob. Let x be the observed image. Is x a stegogramme (from S ) or a pure image (from C )? Unfortunately, most of the time x ∈ C and x ∈ S are both possible. . . Compare Pr ( S = x ) and Pr ( C = x ) . If Pr ( S = x ) > Pr ( C = x ) , then x is most likely to be a stegogramme. How confident can we be about the conclusion? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 13 / 52

  3. Probabilities Probabilities Statistical Steganalysis The goal of Statistical Steganalysis: Wendy observes an image X sent from Alice to Bob. Let x be the observed image. Is x a stegogramme (from S ) or a pure image (from C )? Unfortunately, most of the time x ∈ C and x ∈ S are both possible. . . Compare Pr ( S = x ) and Pr ( C = x ) . If Pr ( S = x ) > Pr ( C = x ) , then x is most likely to be a stegogramme. How confident can we be about the conclusion? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 13 / 52

  4. Probabilities Probabilities Statistical Steganalysis The goal of Statistical Steganalysis: Wendy observes an image X sent from Alice to Bob. Let x be the observed image. Is x a stegogramme (from S ) or a pure image (from C )? Unfortunately, most of the time x ∈ C and x ∈ S are both possible. . . Compare Pr ( S = x ) and Pr ( C = x ) . If Pr ( S = x ) > Pr ( C = x ) , then x is most likely to be a stegogramme. How confident can we be about the conclusion? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 13 / 52

  5. Probabilities Probabilities Statistical Steganalysis If P S = P C , i.e. if covertexts and stegogrammes have the same distributions then no steganalysis is possible. If P S � = P C , then statistics can help us. Remark The perfect stego-system gives P S = P C . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 14 / 52

  6. Probabilities Probabilities Statistical Steganalysis If P S = P C , i.e. if covertexts and stegogrammes have the same distributions then no steganalysis is possible. If P S � = P C , then statistics can help us. Remark The perfect stego-system gives P S = P C . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 14 / 52

  7. Probabilities Probabilities Statistical Steganalysis If P S = P C , i.e. if covertexts and stegogrammes have the same distributions then no steganalysis is possible. If P S � = P C , then statistics can help us. Remark The perfect stego-system gives P S = P C . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 14 / 52

  8. Probabilities Random variables Outline Probabilities 1 Experiments and Events Probabilities Random variables Independence Probability Distributions 2 The uniform distribution Some other distributions Hypothesis tests 3 Introduction Pairs of Values Steganalysis 4 Pairs of Values Generalised χ 2 test 5 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 15 / 52

  9. Probabilities Random variables Random variable Definition (Bhattacharyya and Johnson) A random variable X is a numerical valued function defined on a sample space. (. . . ) I.e. a number X ( e ) is assigned to each simple event. The random variable X is always a number. The event e could be anything (colour, multi-dimensional vector). Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 16 / 52

  10. Probabilities Random variables Random variable Definition (Bhattacharyya and Johnson) A random variable X is a numerical valued function defined on a sample space. (. . . ) I.e. a number X ( e ) is assigned to each simple event. The random variable X is always a number. The event e could be anything (colour, multi-dimensional vector). Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 16 / 52

  11. Probabilities Random variables Random variable Definition (Bhattacharyya and Johnson) A random variable X is a numerical valued function defined on a sample space. (. . . ) I.e. a number X ( e ) is assigned to each simple event. The random variable X is always a number. The event e could be anything (colour, multi-dimensional vector). Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 16 / 52

  12. Probabilities Random variables Random variable Definition (Bhattacharyya and Johnson) A random variable X is a numerical valued function defined on a sample space. (. . . ) I.e. a number X ( e ) is assigned to each simple event. The random variable X is always a number. The event e could be anything (colour, multi-dimensional vector). Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 16 / 52

  13. Probabilities Random variables Discrete random variable A discrete random variable X is characterised by The discrete set Ω = { x 1 , x 2 , . . . } from which it is drawn, The Probability Distribution p X : Ω → [ 0 , 1 ] , where p X ( x ) = Pr ( X = x ) . Definition The cumulative density function (CDF) f X : Ω → [ 0 , 1 ] of X is defined by F X ( x ) = Pr ( X ≤ x ) . Remark � F X ( x ) = p X ( x ) . y ≤ x Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 17 / 52

  14. Probabilities Random variables Discrete random variable A discrete random variable X is characterised by The discrete set Ω = { x 1 , x 2 , . . . } from which it is drawn, The Probability Distribution p X : Ω → [ 0 , 1 ] , where p X ( x ) = Pr ( X = x ) . Definition The cumulative density function (CDF) f X : Ω → [ 0 , 1 ] of X is defined by F X ( x ) = Pr ( X ≤ x ) . Remark � F X ( x ) = p X ( x ) . y ≤ x Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 17 / 52

  15. Probabilities Random variables Discrete random variable A discrete random variable X is characterised by The discrete set Ω = { x 1 , x 2 , . . . } from which it is drawn, The Probability Distribution p X : Ω → [ 0 , 1 ] , where p X ( x ) = Pr ( X = x ) . Definition The cumulative density function (CDF) f X : Ω → [ 0 , 1 ] of X is defined by F X ( x ) = Pr ( X ≤ x ) . Remark � F X ( x ) = p X ( x ) . y ≤ x Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 17 / 52

  16. Probabilities Random variables Discrete random variable A discrete random variable X is characterised by The discrete set Ω = { x 1 , x 2 , . . . } from which it is drawn, The Probability Distribution p X : Ω → [ 0 , 1 ] , where p X ( x ) = Pr ( X = x ) . Definition The cumulative density function (CDF) f X : Ω → [ 0 , 1 ] of X is defined by F X ( x ) = Pr ( X ≤ x ) . Remark � F X ( x ) = p X ( x ) . y ≤ x Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 17 / 52

  17. Probabilities Random variables Continuous random variable A continuous random variable Y is characterised by The continuous set Ω from which it is drawn, The probability density function (PDF) p Y : Ω → [ 0 , 1 ] . The CDF is � y F Y ( y ) = Pr ( Y ≤ y ) = p Y ( y ) dy 0 Clearly, Pr ( Y = y ) = 0 for all y . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 18 / 52

  18. Probabilities Random variables Continuous random variable A continuous random variable Y is characterised by The continuous set Ω from which it is drawn, The probability density function (PDF) p Y : Ω → [ 0 , 1 ] . The CDF is � y F Y ( y ) = Pr ( Y ≤ y ) = p Y ( y ) dy 0 Clearly, Pr ( Y = y ) = 0 for all y . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 18 / 52

  19. Probabilities Random variables Continuous random variable A continuous random variable Y is characterised by The continuous set Ω from which it is drawn, The probability density function (PDF) p Y : Ω → [ 0 , 1 ] . The CDF is � y F Y ( y ) = Pr ( Y ≤ y ) = p Y ( y ) dy 0 Clearly, Pr ( Y = y ) = 0 for all y . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 18 / 52

  20. Probabilities Random variables Continuous random variable A continuous random variable Y is characterised by The continuous set Ω from which it is drawn, The probability density function (PDF) p Y : Ω → [ 0 , 1 ] . The CDF is � y F Y ( y ) = Pr ( Y ≤ y ) = p Y ( y ) dy 0 Clearly, Pr ( Y = y ) = 0 for all y . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 18 / 52

  21. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  22. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  23. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  24. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  25. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  26. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  27. Probabilities Random variables Example (random variables) Consider a (random) M × N Grayscale Image X . X = [ x xy ] 0 < x ≤ M , 0 < y ≤ N , where each x ij ∈ { 0 , 1 , 2 , . . . , 255 } . The sample space is all M × N matrices of 8-bit integers. Let k = 0 , . . . , 255 Define random variable: Y k = # { x xy | x xy = k } . We shall see later that the distribution of Y k , varies according to whether X is a stegogramme or a natural image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 19 / 52

  28. Probabilities Random variables Expectation Definition The expectation of a discrete random variable X is � E ( X ) = x Pr ( X = x ) , x ∈X where X is the set of possible values for X . For a continuous variable, this becomes � ∞ E ( X ) = xp X ( x ) dx . −∞ Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 20 / 52

  29. Probabilities Random variables Expectation Definition The expectation of a discrete random variable X is � E ( X ) = x Pr ( X = x ) , x ∈X where X is the set of possible values for X . For a continuous variable, this becomes � ∞ E ( X ) = xp X ( x ) dx . −∞ Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 20 / 52

  30. Probabilities Random variables Variance Definition The variance of a discrete random variable X is � ( x − E ( X )) 2 p X ( x ) , Var ( X ) = x ∈X where X is the set of possible values for X . For a continuous variable, it becomes � ∞ ( x − E ( X )) 2 p X ( x ) dx . Var ( X ) = −∞ Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 21 / 52

  31. Probabilities Random variables Variance Definition The variance of a discrete random variable X is � ( x − E ( X )) 2 p X ( x ) , Var ( X ) = x ∈X where X is the set of possible values for X . For a continuous variable, it becomes � ∞ ( x − E ( X )) 2 p X ( x ) dx . Var ( X ) = −∞ Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 21 / 52

  32. Probabilities Random variables Example: Expectation Consider a stegogramme X as a grayscale image, Y k = # { x xy | x xy = k } , as discussed earlier. ( 2 l , 2 l + 1 ) is a pairs of values The embedding changes 2 l → 2 l + 1 and vice versa. Suppose the hidden message is a uniformly random bit string. Then E ( Y 2 k + 1 ) = E ( Y 2 k ) . This is not the case if X is a pure image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 22 / 52

  33. Probabilities Random variables Example: Expectation Consider a stegogramme X as a grayscale image, Y k = # { x xy | x xy = k } , as discussed earlier. ( 2 l , 2 l + 1 ) is a pairs of values The embedding changes 2 l → 2 l + 1 and vice versa. Suppose the hidden message is a uniformly random bit string. Then E ( Y 2 k + 1 ) = E ( Y 2 k ) . This is not the case if X is a pure image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 22 / 52

  34. Probabilities Random variables Example: Expectation Consider a stegogramme X as a grayscale image, Y k = # { x xy | x xy = k } , as discussed earlier. ( 2 l , 2 l + 1 ) is a pairs of values The embedding changes 2 l → 2 l + 1 and vice versa. Suppose the hidden message is a uniformly random bit string. Then E ( Y 2 k + 1 ) = E ( Y 2 k ) . This is not the case if X is a pure image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 22 / 52

  35. Probabilities Random variables Example: Expectation Consider a stegogramme X as a grayscale image, Y k = # { x xy | x xy = k } , as discussed earlier. ( 2 l , 2 l + 1 ) is a pairs of values The embedding changes 2 l → 2 l + 1 and vice versa. Suppose the hidden message is a uniformly random bit string. Then E ( Y 2 k + 1 ) = E ( Y 2 k ) . This is not the case if X is a pure image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 22 / 52

  36. Probabilities Random variables Example: Expectation Consider a stegogramme X as a grayscale image, Y k = # { x xy | x xy = k } , as discussed earlier. ( 2 l , 2 l + 1 ) is a pairs of values The embedding changes 2 l → 2 l + 1 and vice versa. Suppose the hidden message is a uniformly random bit string. Then E ( Y 2 k + 1 ) = E ( Y 2 k ) . This is not the case if X is a pure image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 22 / 52

  37. Probabilities Random variables Example: Expectation Consider a stegogramme X as a grayscale image, Y k = # { x xy | x xy = k } , as discussed earlier. ( 2 l , 2 l + 1 ) is a pairs of values The embedding changes 2 l → 2 l + 1 and vice versa. Suppose the hidden message is a uniformly random bit string. Then E ( Y 2 k + 1 ) = E ( Y 2 k ) . This is not the case if X is a pure image. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 22 / 52

  38. Probabilities Independence Outline Probabilities 1 Experiments and Events Probabilities Random variables Independence Probability Distributions 2 The uniform distribution Some other distributions Hypothesis tests 3 Introduction Pairs of Values Steganalysis 4 Pairs of Values Generalised χ 2 test 5 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 23 / 52

  39. Probabilities Independence Independence The definition Consider two events A and B . The probability that both happens is denoted Pr ( A , B ) . Definition We say that A and B are independent if Pr ( A , B ) = Pr ( A ) Pr ( B ) . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 24 / 52

  40. Probabilities Independence Independence The definition Consider two events A and B . The probability that both happens is denoted Pr ( A , B ) . Definition We say that A and B are independent if Pr ( A , B ) = Pr ( A ) Pr ( B ) . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 24 / 52

  41. Probabilities Independence Independence Examples Example You roll two dice, a green and a red one. Let A be the event of a 6 on the red die, and B a 6 on the green die. We have Pr ( B ) = Pr ( A ) = 1 / 6 and Pr ( A , B ) = 1 / 36. The events are independent. Example Let C be the event of rolling a sum of 6 on the two dice. Now Pr ( C ) = 5 / 36, but if A the sum is at least 7, so Pr ( A , C ) = 0. The events A and C are dependent . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 25 / 52

  42. Probabilities Independence Independence Examples Example You roll two dice, a green and a red one. Let A be the event of a 6 on the red die, and B a 6 on the green die. We have Pr ( B ) = Pr ( A ) = 1 / 6 and Pr ( A , B ) = 1 / 36. The events are independent. Example Let C be the event of rolling a sum of 6 on the two dice. Now Pr ( C ) = 5 / 36, but if A the sum is at least 7, so Pr ( A , C ) = 0. The events A and C are dependent . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 25 / 52

  43. Probabilities Independence Conditional probability Definition Consider two events A and B . The probability of A happening in the case when B is known to happen is known as the conditional probability Pr ( A | B ) . Theorem (Bayes formula) Pr ( A | B ) Pr ( B ) = Pr ( B | A ) Pr ( A ) = Pr ( A , B ) Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 26 / 52

  44. Probability Distributions The uniform distribution Outline Probabilities 1 Experiments and Events Probabilities Random variables Independence Probability Distributions 2 The uniform distribution Some other distributions Hypothesis tests 3 Introduction Pairs of Values Steganalysis 4 Pairs of Values Generalised χ 2 test 5 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 27 / 52

  45. Probability Distributions The uniform distribution The uniform distribution Let X be a random variable from a set Ω . Let p ( x ) be the probability distribution of X . or the probability density function if X is continuous If p ( x ) = p ( y ) for all x , y ∈ Ω , then we say that X is uniformly distributed on Ω Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 28 / 52

  46. Probability Distributions The uniform distribution Distributions in graphics Discrete random variable Continuous random variable Probability distribution Probability density function as histogramme as plot Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 29 / 52

  47. Probability Distributions The uniform distribution Distributions in graphics Discrete random variable Continuous random variable Probability distribution Probability density function as histogramme as plot Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 29 / 52

  48. Probability Distributions Some other distributions Outline Probabilities 1 Experiments and Events Probabilities Random variables Independence Probability Distributions 2 The uniform distribution Some other distributions Hypothesis tests 3 Introduction Pairs of Values Steganalysis 4 Pairs of Values Generalised χ 2 test 5 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 30 / 52

  49. Probability Distributions Some other distributions The χ 2 distribution The χ 2 distribution will be used later. Defined for k degrees of freedom k = 1 , 2 , . . . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 31 / 52

  50. Probability Distributions Some other distributions Probability density function The χ 2 distribution Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 32 / 52

  51. Probability Distributions Some other distributions How to use it (Usually) No need to learn the formulæ Look probabilities up in tables . . . or in Matlab chi2pdf ( x, k ) gives the PDF at x for k degrees of freedom chi2cdf ( x, k ) gives P ( X ≤ x ) for k degrees of freedom Inverse function exists too (use help functions) Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 33 / 52

  52. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  53. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  54. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  55. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  56. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  57. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  58. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  59. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  60. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  61. Probability Distributions Some other distributions Random bit strings We will refer to random bit strings X = ( X 1 , X 2 , . . . , X N ) If X is an unbiased random bit string, then X i are identically distributed with P ( X i = 0 ) = P ( X i = 1 ) = 1 2 ; and each X i is independent of all other X j . If X is an biased random bit string, then X i are identically distributed with P ( X i = 0 ) = p � = 1 2 ; and each X i is independent of all other X j . Note the significance of independence Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 34 / 52

  62. Hypothesis tests Introduction Outline Probabilities 1 Experiments and Events Probabilities Random variables Independence Probability Distributions 2 The uniform distribution Some other distributions Hypothesis tests 3 Introduction Pairs of Values Steganalysis 4 Pairs of Values Generalised χ 2 test 5 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 35 / 52

  63. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  64. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  65. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  66. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  67. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  68. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  69. Hypothesis tests Introduction Hypothesis tests Hypothesis testing is a recurring theme in statistics. Typical hypotheses Treatment A makes patients recover more quickly than no treatment. The climate in South-East Britain is as warm today as it was a 100 years ago. The image sent by Alice is a stegogramme. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 36 / 52

  70. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  71. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  72. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  73. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  74. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  75. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  76. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  77. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

  78. Hypothesis tests Introduction The test statistic Make a null hypothesis H 0 . Formulate an alternative hypothesis H 1 (negating H 0 ). Identify a random variable X with known distribution assuming H 0 Observe the random variable. Is the observed value x a likely result under H 0 ? We decide on a threshold t such that Pr ( X > t | H 0 ) is small If the observed x > t we reject H 0 . We can never prove H 0 , or reject H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2007 37 / 52

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