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Statistics and Steganalysis CSM25 Secure Information Hiding Dr Hans Georg Schaathun University of Surrey Spring 2009 Week 2 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 Week 2 1 / 54 Learning Outcomes After this

  1. Visual Steganalysis The Histogram Outline Visual Steganalysis 1 The LSB plane The Histogram Limitations Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 17 / 54

  2. Visual Steganalysis The Histogram A typical image Image histogram made by imhist in Matlab Gives number of pixels per colour-value Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 18 / 54

  3. Visual Steganalysis The Histogram And a stego-image Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 19 / 54

  4. Visual Steganalysis The Histogram And a stego-image Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 19 / 54

  5. Visual Steganalysis The Histogram What happened? Histogram of stego-image: More ragged Every other bar sticks out. Why? 50.8% 1-s in the binary message. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 20 / 54

  6. Visual Steganalysis The Histogram What happened? Histogram of stego-image: More ragged Every other bar sticks out. Why? 50.8% 1-s in the binary message. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 20 / 54

  7. Visual Steganalysis The Histogram What happened? Histogram of stego-image: More ragged Every other bar sticks out. Why? 50.8% 1-s in the binary message. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 20 / 54

  8. Visual Steganalysis The Histogram What happened? Histogram of stego-image: More ragged Every other bar sticks out. Why? 50.8% 1-s in the binary message. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 20 / 54

  9. Visual Steganalysis The Histogram What happened? Histogram of stego-image: More ragged Every other bar sticks out. Why? 50.8% 1-s in the binary message. Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 20 / 54

  10. Visual Steganalysis The Histogram What is characteristic? Pairs of values Consider colour 2 i ( i = 0 , 1 , . . . , 127) What happens under LSB embedding? 2 i → 2 i , 2 i + 1 Never 2 i → 2 i − 1. Likewise 2 i + 1 → 2 i , 2 i + 1 ( 2 i , 2 i + 1 ) is a Pair of Values A pixel in ( 2 i , 2 i + 1 ) before embedding ... is a pixel in ( 2 i , 2 i + 1 ) after embedding Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 21 / 54

  11. Visual Steganalysis The Histogram What is characteristic? Pairs of values Consider colour 2 i ( i = 0 , 1 , . . . , 127) What happens under LSB embedding? 2 i → 2 i , 2 i + 1 Never 2 i → 2 i − 1. Likewise 2 i + 1 → 2 i , 2 i + 1 ( 2 i , 2 i + 1 ) is a Pair of Values A pixel in ( 2 i , 2 i + 1 ) before embedding ... is a pixel in ( 2 i , 2 i + 1 ) after embedding Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 21 / 54

  12. Visual Steganalysis The Histogram What is characteristic? Pairs of values Consider colour 2 i ( i = 0 , 1 , . . . , 127) What happens under LSB embedding? 2 i → 2 i , 2 i + 1 Never 2 i → 2 i − 1. Likewise 2 i + 1 → 2 i , 2 i + 1 ( 2 i , 2 i + 1 ) is a Pair of Values A pixel in ( 2 i , 2 i + 1 ) before embedding ... is a pixel in ( 2 i , 2 i + 1 ) after embedding Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 21 / 54

  13. Visual Steganalysis Limitations Outline Visual Steganalysis 1 The LSB plane The Histogram Limitations Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 22 / 54

  14. Visual Steganalysis Limitations Visual methods Advantages and Limitations Human perception is very flexible can exploit the unexpected you don’t have to know what you look for Manual work the process cannot be automated or computerised How do you check a million images? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 23 / 54

  15. Statistics Outline Visual Steganalysis 1 Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 24 / 54

  16. Statistics Statistical models Outline Visual Steganalysis 1 The LSB plane The Histogram Limitations Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 25 / 54

  17. Statistics Statistical models The remit of statistics Statistics can estimate ‘normal’ behaviour and compare behaviours Advantages Automated decisions Extract detail Exact, quantifiable features Aggregate measures Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 26 / 54

  18. Statistics Statistical models The remit of statistics Statistics can estimate ‘normal’ behaviour and compare behaviours Advantages Automated decisions Extract detail Exact, quantifiable features Aggregate measures Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 26 / 54

  19. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  20. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Is the image a stegogramme? Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  21. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Is it a probable, natural image? Is it a probable stegogramme? Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  22. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Is it a probable, natural image? Is it a probable stegogramme? Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  23. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Is it a probable, natural image? Is it a probable stegogramme? Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  24. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Is it a probable, natural image? Is it a probable stegogramme? Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  25. Statistics Statistical models The fundamental question Wendy the Warden intercepts an image. Is it a probable, natural image? Is it a probable stegogramme? Depends on a model for natural images Statistical models and probability distributions With a perfect model, cipher with ciphertexts distributed as natural images If Wendy has a better model than Alice and Bob, then she can do effective steganalysis In reality, we do not know what a natural image looks like Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 27 / 54

  26. Statistics Pairs of Values Outline Visual Steganalysis 1 The LSB plane The Histogram Limitations Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 28 / 54

  27. Statistics Pairs of Values Pairs of Values The statistic Image X . Random variable Y k = # { ( x , y ) | X xy = k } The Y k -s is the Histogramme. Recall that ( 2 l , 2 l + 1 ) is a pair of values. First 7 pixel bits determined by image colour. i.e. which pair Last bit (LSB) determined by message i.e. which half of the pair Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 29 / 54

  28. Statistics Pairs of Values Pairs of Values Expected behaviour Sum Y 2 l + Y 2 l + 1 unaffected by embedding. For a random message steganogram, Expect 50-50 2 l and 2 l + 1 i.e. E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) For a natural image, what can we expect? In a given image, we can observe Y 2 l . Is the observation probable? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 30 / 54

  29. Statistics Pairs of Values Pairs of Values Expected behaviour Sum Y 2 l + Y 2 l + 1 unaffected by embedding. For a random message steganogram, Expect 50-50 2 l and 2 l + 1 i.e. E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) For a natural image, what can we expect? In a given image, we can observe Y 2 l . Is the observation probable? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 30 / 54

  30. Statistics Pairs of Values Pairs of Values Expected behaviour Sum Y 2 l + Y 2 l + 1 unaffected by embedding. For a random message steganogram, Expect 50-50 2 l and 2 l + 1 i.e. E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) For a natural image, what can we expect? In a given image, we can observe Y 2 l . Is the observation probable? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 30 / 54

  31. Statistics Pairs of Values Pairs of Values Expected behaviour Sum Y 2 l + Y 2 l + 1 unaffected by embedding. For a random message steganogram, Expect 50-50 2 l and 2 l + 1 i.e. E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) For a natural image, what can we expect? In a given image, we can observe Y 2 l . Is the observation probable? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 30 / 54

  32. Statistics Pairs of Values Hypothesis testing The principle We have two possible hypotheses H 0 The image is a steganogram with random message 1 Known distribution: E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) H 1 The image is a natural image 2 Unknown distribution Statistics allows us to answer is the observed Y 2 l -s likely under H 0 ? We cannot ask a similar question under H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 31 / 54

  33. Statistics Pairs of Values Hypothesis testing The principle We have two possible hypotheses H 0 The image is a steganogram with random message 1 Known distribution: E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) H 1 The image is a natural image 2 Unknown distribution Statistics allows us to answer is the observed Y 2 l -s likely under H 0 ? We cannot ask a similar question under H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 31 / 54

  34. Statistics Pairs of Values Hypothesis testing The principle We have two possible hypotheses H 0 The image is a steganogram with random message 1 Known distribution: E ( Y 2 l ) = E ( Y 2 l + 1 ) = 1 2 ( Y 2 l + Y 2 l + 1 ) H 1 The image is a natural image 2 Unknown distribution Statistics allows us to answer is the observed Y 2 l -s likely under H 0 ? We cannot ask a similar question under H 1 . Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 31 / 54

  35. Statistics Pairs of Values The χ 2 test Statistical hypothesis tests exist for many purposes The χ 2 test can compare different distributions i.e. the H 0 distribution and the observed distribution aggregate several numbers i.e. Y 2 l for every l Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 32 / 54

  36. Statistics Pairs of Values The χ 2 statistic Several random variable F 0 , F 1 , . . . , F m Known expectations E ( F 0 ) , E ( F 1 ) , . . . , E ( F m ) m ( F o − E ( F o )) 2 � S = E ( F o ) i = 0 Definition 127 127 ( Y 2 l − 1 2 ( Y 2 l + Y 2 l + 1 )) 2 1 2 ( Y 2 l − Y 2 l + 1 ) 2 � � S PoV = = 1 Y 2 l + Y 2 l + 1 2 ( Y 2 l + Y 2 l + 1 ) l = 0 l ∈ 0 χ 2 distributed with m degrees of freedom Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 33 / 54

  37. Statistics Pairs of Values Making Conclusions If the observed S is likely under χ 2 distribution, the assumed distribution (and thus H 0 ) is plausible If the observed S is unlikely under χ 2 distribution, H 0 is implausible Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 34 / 54

  38. Statistics Pairs of Values Making Conclusions If the observed S is likely under χ 2 distribution, the assumed distribution (and thus H 0 ) is plausible If the observed S is unlikely under χ 2 distribution, H 0 is implausible Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 34 / 54

  39. Statistics Pairs of Values The χ 2 PDF Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 35 / 54

  40. Statistics Pairs of Values The Pairs-of-Values χ 2 Distribution χ 2 PDF 127 degrees of freedom Red: 2% prob. +Green: 5% +Blue: 10% Cumulative Density Function (CDF) Area under the curve Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 36 / 54

  41. Statistics Pairs of Values The Pairs-of-Values χ 2 Distribution χ 2 PDF 127 degrees of freedom Red: 2% prob. +Green: 5% +Blue: 10% Cumulative Density Function (CDF) Area under the curve Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 36 / 54

  42. Statistics Pairs of Values The Pairs-of-Values χ 2 Distribution χ 2 PDF 127 degrees of freedom Red: 2% prob. +Green: 5% +Blue: 10% Cumulative Density Function (CDF) Area under the curve Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 36 / 54

  43. Statistics Pairs of Values The p -value Let S be a stochastic χ 2 distributed variable Let s be the observed χ 2 statistic Define p -value: p = P ( S < s ) I.e. low p -value ⇒ s is unusually small Improbable if the image is a stegogramme. Conclusion: probably natural image Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 37 / 54

  44. Statistics Pairs of Values p -value illustrated We read the statistic ( χ 2 ) on the x -axis The p -value is the area under the PDF to the right Compute it with chi2cdf Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 38 / 54

  45. Statistics Pairs of Values Corrections You may have to exclude pixel values which do not occur have at least four pixels of each pair of values used This keeps the χ 2 distribution a good approximation This reduces the degrees of freedom Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 39 / 54

  46. Statistics Pairs of Values χ 2 in Matlab Defined in the Statistics toolbox Simplified functions available on website: chi2pdf (the PDF) chi2cdf(s,v) – P ( S ≤ s ) when S ∼ χ 2 ( v ) chi2inv(p,v) – s such that P ( S ≤ s ) = p Note that the p -value is P ( S ≥ s ) = 1 − P ( S ≤ s ) use chi2cdf to calculate it Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 40 / 54

  47. Statistics Pairs of Values χ 2 in Matlab Defined in the Statistics toolbox Simplified functions available on website: chi2pdf (the PDF) chi2cdf(s,v) – P ( S ≤ s ) when S ∼ χ 2 ( v ) chi2inv(p,v) – s such that P ( S ≤ s ) = p Note that the p -value is P ( S ≥ s ) = 1 − P ( S ≤ s ) use chi2cdf to calculate it Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 40 / 54

  48. Statistics I visual approach Outline Visual Steganalysis 1 The LSB plane The Histogram Limitations Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 41 / 54

  49. Statistics I visual approach Part-image The χ 2 statistic is effective when the image is full of hidden information What happens if only a small part is used? Basic LSB embedding uses the first N pixels We calculate the χ 2 and p values for every N The result can be plotted use plot or fplot in Matlab Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 42 / 54

  50. Statistics I visual approach Plots No message χ 2 statistic p -value Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 43 / 54

  51. Statistics I visual approach Plots 30% of capacity χ 2 statistic p -value Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 44 / 54

  52. Statistics I visual approach Plots 60% of capacity χ 2 statistic p -value Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 45 / 54

  53. Statistics I visual approach Plots 100% of capacity χ 2 statistic p -value Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 46 / 54

  54. Statistics Error types Outline Visual Steganalysis 1 The LSB plane The Histogram Limitations Statistics 2 Statistical models Pairs of Values I visual approach Error types Postlogue 3 Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 47 / 54

  55. Statistics Error types Classification errors Steganalysis is a binary classification problem identify an unknown object (image) as either suspicious innocent Two error types False positive an innocent image wrongly accused False negative a «guilty» image not identified Which type is most severe? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 48 / 54

  56. Statistics Error types Classification errors Steganalysis is a binary classification problem identify an unknown object (image) as either suspicious innocent Two error types False positive an innocent image wrongly accused False negative a «guilty» image not identified Which type is most severe? Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 48 / 54

  57. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  58. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  59. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  60. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  61. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  62. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  63. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  64. Statistics Error types Hypothesis testing and errors Hypothesis testing is a recurring theme in statistics. Typical null hypotheses Treatment A makes patients recover no more quickly than no treatment. The climate in South-East Britain is as warm/cold today as it was a 100 years ago. The image sent by Alice is a natural (innocent) image. When the hypothesis has been phrased, experiments can tell us whether it is plausible or not. Wrongly accepting the null hypothesis is the least serious error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 49 / 54

  65. Statistics Error types Asymmetry of hypothesis testing Treatment A makes patients recover more quickly than no treatment. One error is more serious than another. Type I: Accepting the hypothesis when it is wrong Patients get ineffective (or unhealthy) medicine. Type II: Rejecting the hypothesis when it is right More research will be made to optimise the treatment. H 0 retained H 0 rejected H 0 true No error Error Type I H 0 false Error Type II No error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 50 / 54

  66. Statistics Error types Asymmetry of hypothesis testing Treatment A makes patients recover more quickly than no treatment. One error is more serious than another. Type I: Accepting the hypothesis when it is wrong Patients get ineffective (or unhealthy) medicine. Type II: Rejecting the hypothesis when it is right More research will be made to optimise the treatment. H 0 retained H 0 rejected H 0 true No error Error Type I H 0 false Error Type II No error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 50 / 54

  67. Statistics Error types Asymmetry of hypothesis testing Treatment A makes patients recover more quickly than no treatment. One error is more serious than another. Type I: Accepting the hypothesis when it is wrong Patients get ineffective (or unhealthy) medicine. Type II: Rejecting the hypothesis when it is right More research will be made to optimise the treatment. H 0 retained H 0 rejected H 0 true No error Error Type I H 0 false Error Type II No error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 50 / 54

  68. Statistics Error types Asymmetry of hypothesis testing Treatment A makes patients recover more quickly than no treatment. One error is more serious than another. Type I: Accepting the hypothesis when it is wrong Patients get ineffective (or unhealthy) medicine. Type II: Rejecting the hypothesis when it is right More research will be made to optimise the treatment. H 0 retained H 0 rejected H 0 true No error Error Type I H 0 false Error Type II No error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 50 / 54

  69. Statistics Error types Asymmetry of hypothesis testing Treatment A makes patients recover more quickly than no treatment. One error is more serious than another. Type I: Accepting the hypothesis when it is wrong Patients get ineffective (or unhealthy) medicine. Type II: Rejecting the hypothesis when it is right More research will be made to optimise the treatment. H 0 retained H 0 rejected H 0 true No error Error Type I H 0 false Error Type II No error Dr Hans Georg Schaathun Statistics and Steganalysis Spring 2009 – Week 2 50 / 54

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