1. 1 Finite Population Inference for the Mean from a Bayesian Perspective Edward J. Stanek III Department of Public Health University of Massachusetts Amherst, MA

2. 2 Collaborators Parimal Mukhopadhyay, Indian Statistics Institute, Kolkata, India Viviana Lencina, Facultad de Ciencias Economicas, Universidad Nacional de Tucumán, CONICET, Argentina Luz Mery Gonzalez, Departamentao de Estadística, Universidad Nacional de Colombia, Bogotá, Colombia Julio Singer, Departamento de Estatística, Universidade de São Paulo, Brazil Wenjun Li, Department of Behavioral Medicine, UMASS Medical School, Worcester, MA Rongheng Li, Shuli Yu, Guoshu Yuan, Ruitao Zhang, Faculty and Students in the Biostatistics Program, UMASS, Amherst

3. 3 Outline Guessing the Finite Population Mean using Bayesian Methods: 1.General Idea of Labels and Response 2.Notation 3.Example with Continuous Normal Priors Exchangeable Prior Distributions 1.Example with N=3 2.Notation 3.Geometric Interpretations The Data 1.Sample space as subspace of prior 2.Sample space conditional on the data Notation for Prior, Data and Posterior 1.Points in the prior and their probability 2.Partitioning prior points into the data, and remainder 3.Simplifications when data match prior 4.Posterior notation and simplification Simple Example

4. 4 Finite Population Inference for the Mean from a Bayesian Perspective Population Data What is ? ListingLatent Value Rose Lily Daisy

5. 5 Bayesian Model Population Notation Population Label Latent Value Set of Labels Parameter Population includes N subjects Vector

6. 6 Bayesian Model Prior Populations Populations Prior # Prior Parameters: Population h Prior Probability Labels Parameters Prior Prob. Usually, prior populations have the same number of subjects, N.

7. 7 Bayesian Model Prior Populations- Equality and Population h Population h* Two Prior Populations are equal if and only if The same subjects are in each population then Prior populations are equal if they include the same subjects, and a subject’s latent value in one population is equal to the same subject’s latent value in the other population.

8. 8 Bayesian Model Prior Distribution of Example Prior Populations 0 4 8 12 16 20 24 28 32

9. 9 Bayesian Model General Idea Populations # Posterior Populations: Data Prior Posterior # Prior Populations: Prior Probabilities Posterior Probabilities

10. 10 Bayesian Model General Idea Populations # Posterior Populations: Data Prior Posterior # Prior Populations: Sample Subjects

11. 11 Bayesian Model General Idea Populations # Posterior Populations: Data Prior Posterior # Prior Populations: If Sample subjects must be in the population Latent Value for the subject in the sample must equal that in the population for

12. 12 Bayesian Model General Idea Populations # Posterior Populations: Data Discrete Prior Posterior # Prior Populations: Prior Probabilities Posterior Probabilities Note: Often, the prior distribution is continuous, as if

13. 13 Bayesian Model General Idea-Example Populations # Posterior Populations: Data Continuous Prior Posterior # Prior Populations: Prior Probabilities Posterior Probabilities iid

14. 14 Bayesian Model General Idea-Example Data Continuous Prior Posterior iid Model: where Let us define

15. 15 Bayesian Model General Idea-Example Data Continuous Prior Posterior iid Model: Posterior Mean is Different Posterior Variance is smaller

16. 16 Bayesian Model Link between Prior and Data Populations # Posterior Populations: Data Prior Posterior # Prior Populations: Assume the Potential Response for Subjects in each Prior population is a vector of exchangeable random variables! Use Likelihood of Parameter, given the data to form the Posterior

17. 17 Bayesian Model Exchangeable Prior Populations- Population Labels (subjects) Let be a vector of exchangeable random variables for the population Exchangeability implies that the joint probability density, of response for associated with each permutation, p, of subjects in L is identical for all (Focus on one population, h, but drop the subscript for simplicity) where

18. 18 Bayesian Model- Exchangeable Prior Populations General Idea When N=3 Each Permutation p of subjects in L (i.e. each different listing) Joint Probability Density Must be identical Exchangeable Random Variables The common distribution General Notation Assigns (usually) equal probability to each permutation of subjects in the population.

19. 19 Bayesian Model - Exchangeable Prior Populations General Idea When N=3 Permutation Matrices (for Listings)

20. 20 Bayesian Model - Exchangeable Prior Populations N=3 Each Listing of subjects in L Random Variables For Listing Listing Labels Define the Possible Listings of subjects in L

21. 21 Bayesian Model - Exchangeable Prior Populations N=3 Latent Values for Each Listing of subjects in L ListingListings Permutation Matrices Latent Values for Listing

22. 22 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Values for Listing Latent Values for permutations of listing

23. 23 Bayesian Model- Exchangeable Prior Populations N=3 Population Point (N dimension space) Rose Daisy Lily Listing p=1 Example

24. 24 Bayesian Model-Exchangeable Prior Population Permutations Rose Daisy Lily Listing p=1

25. 25 Bayesian Model-Exchangeable Prior Population Permutations Rose Daisy Lily Listing p=1

26. 26 Bayesian Model- Exchangeable Prior Populations N=3 Permutations Rose Daisy Lily Listing p=1

27. 27 Bayesian Model- Exchangeable Prior Populations N=3 Permutations Listing p=1 for p=1 Joint Probability Density Let

28. 28 Bayesian Model- Exchangeable Prior Populations N=3 Rose Daisy Lily Listing p=2 Joint Probability Density

29. 29 Bayesian Model- Exchangeable Prior Populations N=3 Equal Probability Density Listing p=1 Joint Probability Density Listing p=2

30. 30 Bayesian Model- Exchangeable Prior Populations N=3 Permutations Listing p=1 For Equality: each point must have equal probability Listing p=2

31. 31 Bayesian Model-Exchangeable Prior Populations N=3 Equal/Unequal Probability Listing p=1Listing p=2 All equal to 1/6 Example 1 Example 2 Each equal to 0.25 Each equal to 0.125 Exchangeable does not mean equal probability for all permutations!

32. 32 Bayesian Model- Exchangeable Prior Populations N=3 The same Points have Equal Probability Listing p=1 For Equality: each point must have equal probability Listing p=2

33. 33 Bayesian Model-Exchangeable Prior Populations N=3 Permutations Rose Daisy Lily Listing p=1

34. 34 Bayesian Model- Exchangeable Prior Populations N=3 Rose Daisy Lily Listing p=2

35. 35 Bayesian Model- Exchangeable Prior Populations N=3 Rose Daisy Lily Listing p=3

36. 36 Bayesian Model- Exchangeable Prior Populations N=3 Rose Daisy Lily Listing p=4

37. 37 Bayesian Model- Exchangeable Prior Populations N=3 Rose Daisy Lily Listing p=5

38. 38 Bayesian Model- Exchangeable Prior Populations N=3 Rose Daisy Lily Listing p=6

39. 39 Bayesian Model-Exchangeable Prior Populations N=3 Permutations of Listings Listing p=1 Point for Listing Listing p=2 Listing p=3 Listing p=4 Listing p=5 Listing p=6

40. 40 Bayesian Model Exchangeable Prior Populations- Population Labels (subjects) Let be a vector of exchangeable random variables for the population Exchangeability implies that the joint probability density, of response for associated with each permutation, p, of subjects in L is identical for all When N=3, possible points for Y are given in the previous slide! Comparable points must have equal probability in each listing.

41. 41 Bayesian Model Link between Prior and Data Populations # Posterior Populations: Data Prior Posterior # Prior Populations: Assume the Potential Response for Subjects in each Prior population is a vector of exchangeable random variables! Use Likelihood of Parameter, given the data to form the Posterior N=3 Suppose n=2

42. 42 Bayesian Model Link between Prior and Data Populations Data Prior # Prior Populations: N=3 Suppose n=2 Realizations of are the Data

43. 43 Bayesian Model Link between Prior and Data Populations Data Prior # Prior Populations: N=3 Suppose n=2 Sample subjects must be in the population Latent Value for the subject in the sample must equal that in the population

44. 44 Bayesian Model Exchangeable Prior Populations N=3 10 5 5 2 2 Listing p=1 Sample Space n=2 Prior Listing p=1

45. 45 Bayesian Model Exchangeable Prior Populations N=3 10 5 5 2 2 Listing p=2 Sample Space n=2 Prior Listing p=2

46. 46 Bayesian Model Exchangeable Prior Populations N=3 10 5 5 2 2 Listing p=3 Sample Space n=2 Prior Listing p=3

47. 47 Bayesian Model Exchangeable Prior Populations N=3: Sample Point n=2 10 5 5 2 2 Listing p=3 10 5 5 2 2 Listing p=4 10 5 5 2 2 Listing p=1 Listing p=2 10 5 5 2 2 5 5 2 2 Listing p=5 10 5 5 2 2 Listing p=6

48. 48 Bayesian Model -Exchangeable Prior Populations N=3 Sample Points Rose Daisy Lily Listing p=1

49. 49 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points When 10 5 5 2 2 Listing p=1 Sample Space n=2 when Prior Listing p=1

50. 50 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points When 10 5 5 2 2 Listing p=1 Sample Space n=2 when Posterior Points Listing p=1

51. 51 Bayesian Model Exchangeable Prior Populations N=3 Sample Points Rose Daisy Lily Listing p=2

52. 52 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=2 Sample Space n=2 when Prior

53. 53 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=2 Sample Space n=2 when Posterior Points

54. 54 Bayesian Model Exchangeable Prior Populations N=3 Sample Points Rose Daisy Lily Listing p=3

55. 55 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=3 Prior Listing p=3 Sample Space n=2 when

56. 56 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=3 Sample Space n=2 when Posterior Points

57. 57 Bayesian Model Exchangeable Prior Populations N=3 Sample Points Rose Daisy Lily Listing p=4

58. 58 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when Listing p=4 10 5 5 2 2 Prior Sample Space n=2 when

59. 59 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when Listing p=4 10 5 5 2 2 Sample Space n=2 when Posterior Points

60. 60 Bayesian Model Exchangeable Prior Populations N=3 Sample Points Rose Daisy Lily Listing p=5

61. 61 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=5 Prior Sample Space n=2 when

62. 62 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=5 Sample Space n=2 when Posterior Points

63. 63 Bayesian Model Exchangeable Prior Populations N=3 Sample Points Rose Daisy Lily Listing p=6

64. 64 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=6 Prior Sample Space n=2 when

65. 65 Bayesian Model- Exchangeable Prior Populations N=3 Sample Points when 10 5 5 2 2 Listing p=6 Sample Space n=2 when Posterior Points

66. 66 Bayesian Model Exchangeable Prior Populations N=3: Sample Points n=2 10 5 5 2 2 Listing p=3 10 5 5 2 2 Listing p=4 10 5 5 2 2 Listing p=1 Listing p=2 10 5 5 2 2 5 5 2 2 Listing p=5 10 5 5 2 2 Listing p=6

67. 67 Bayesian Model Exchangeable Prior Populations N=3: Sample Points n=2 10 5 5 2 2 Listing p=3 10 5 5 2 2 Listing p=4 10 5 5 2 2 Listing p=1 Listing p=2 10 5 5 2 2 5 5 2 2 Listing p=5 10 5 5 2 2 Listing p=6 Positive Prob.

68. 68 Bayesian Model-- Exchangeable Prior Populations N=3 Prior Distribution and Data: Sample Points with Positive Probability n=2 10 5 5 2 2 Listing p=3 10 5 5 2 2 Listing p=4 10 5 5 2 2 Listing p=1 Listing p=2 10 5 5 2 2 5 5 2 2 Listing p=5 10 5 5 2 2 Listing p=6

69. 69 Bayesian Model-- Exchangeable Prior Populations N=3 Prior Distribution and Data: Sample Points with Positive Probability n=2 Conclusions : For all listings, the same sample points have positive probability The sample points correspond to a permutation of response for subjects in the data For different listings, the permutation that results in the sample points is different We need a way of representing these results in general First, we’ll define notation for points in the prior distribution. We call these points “potential response”

70. 70 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Values for Listing Latent Values for permutations of listing

71. 71 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Values for Listing Latent Values for permutations of listing

72. 72 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Values for permutations of listing Let Also, Latent Values Subject labels

73. 73 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Values for permutations of listing Let Also, Latent Values Subject labels Same Points

74. 74 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Value Vectors for permutations of listing Potential response for Random Variables For Listing p Let us define indicator random variables for permutations of subjects in a listing: Then

75. 75 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Potential response for Random Variables For Listing p Let us define a permutation matrix of indicator random variables for a listing: Then Since where

76. 76 Bayesian Model- Exchangeable Prior Populations N=3 Distribution of Listings Latent Value Vectors for permutations of listing Potential Response Joint Probability Density

77. 77 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects In each listing, the same points occur!

78. 78 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Value Vectors for permutations of listing Potential response for Random Variables For Listing p Circled points are equal and have equal probability, for different listings.

79. 79 Bayesian Model- Exchangeable Prior Populations N=3 Potential Response for Each Listing of subjects Listings Latent Value Vectors for permutations of listing Potential response for Random Variables For Listing p Same Color Circled points have equal probability, for different listings.

80. 80 Bayesian Model- Exchangeable Prior Populations N=3 The Data Prior Data Posterior Consider the Data: A set Points in the Prior are vectors

81. 81 Bayesian Model- Exchangeable Prior Populations N=3 The Data- Represent as a set of Vectors Define: and to be an permutation matrix, k=1,…,n! and Also: Finally, define A set A set of vectors

82. 82 Bayesian Model- Exchangeable Prior Populations N=3 The Data Prior Data Posterior if Sample subjects must be in the population Latent Value for the subject in the sample must equal that in the population For each Define:

83. 83 Bayesian Model- Exchangeable Prior Populations N=3 The Data Prior Data Posterior

84. 84 Bayesian Model- Exchangeable Prior Populations N=3 The Data Prior Data Posterior

85. 85 Bayesian Model- Exchangeable Prior Populations N=3 The Data Prior Data Posterior When

86. 86 Bayesian Model- Exchangeable Prior Populations The Data- Match Orders: Data and Prior Listing PriorData Subjects Must Match Use this order to determine the Prior listing corresponding to p=1 for h implying that Define a data initial order for the subjects in the data (k=1), and their values as: where and Note: This is possible since we do this only when and

87. 87 Bayesian Model- Exchangeable Prior Populations The Data- Notation PriorData Subjects Must Match Since If subjects in the data are part of population h, and hence

88. 88 Bayesian Model- Exchangeable Prior Populations The Data- Notation PriorData Subjects Must Match Initial order for the subjects in the data: Initial order for the subjects in the prior for population h: Now implying that so that Subjects in the data

89. 89 Bayesian Model- Exchangeable Prior Populations The Data- Notation PriorData Subjects Must Match Subjects in the data For Subjects to match, Must be equal Only points in the prior where this is true are in the posterior Subjects in the data

90. 90 Bayesian Model- Exchangeable Prior Populations The Posterior- Notation PriorData Subjects Must Match Requires Letcorrespond to the k where Now Points are in posterior if and

91. 91 Bayesian Model- Exchangeable Prior Populations The Posterior- Simplified Notation Points are in posterior ifand Let defined over all points This is an indicator variable for points in the posterior distribution Consider the Example with N=3

92. 92 Bayesian Model- Exchangeable Prior Populations N=3 Simpler Notation Prior Data Posterior Points are in posterior if and

93. 93 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points (given h) Listings Latent Value Vectors for permutations of listing Must Match Data Prior Data Prior Points are in posterior if and

94. 94 Bayesian Model- Exchangeable Prior Populations N=3 Simpler Notation for Posterior Points Listings Points in Posterior Distribution Data We want to define a notation that will make it easy to represent the points in the posterior. Points are in posterior if and

95. 95 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing Data For each point Define Prior Points are in posterior if and

96. 96 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing Data For each point since When Prior Points are in posterior if and

97. 97 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing For each point when

98. 98 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing For each point when

99. 99 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing For each point when

100. 100 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing For each point when since

101. 101 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing For each point when

102. 102 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing For each point when

103. 103 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for permutations of listing Prior Data Prior

104. 104 Bayesian Model- Exchangeable Prior Populations N=3 Simpler Notation for Posterior Points Listings Latent Value Vectors for permutations of listing Prior Data Prior

105. 105 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Prior Data Listings Latent Value Vectors for permutations of listing Prior

106. 106 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Prior Probability given h and p Listings Prior Probability of Latent Value Vectors for permutations of listing Prior Data Prior Probability given h When

107. 107 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Prior Probability for Points in the Posterior given h and p Listings Prior Probability of Latent Value Vectors for permutations of listing that will be in Posterior Prior Data Prior Probability given h if in Posterior When Note: Doesn’t depend on p since the probability is equal for the same point in each p. This is due to the assumption of exchangeability.

108. 108 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Prior Probability for Points in the Posterior given h and p Prior Data Prior Probability given h if in Posterior When We assume Then

109. 109 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Prior Probability for Points in the Posterior given h and p Listings Prior Data Prior Probability given h if in Posterior When Prior Probability of Latent Value Vectors for permutations of listing that will be in Posterior

110. 110 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Probability given h and p Listings Posterior Probability of Latent Value Vectors for permutations of listing given h Prior Data Posterior Probability given h When

111. 111 Bayesian Model- Exchangeable Prior Populations N=3 Notation for Posterior Points Listings Latent Value Vectors for posterior Given h Prior Data Points in the Posterior given h For Posterior Points: When

112. 112 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Points with Positive Prob Prior Data When Listings Latent Value Vectors for posterior Given h

113. 113 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Points with Positive Prob Prior Data When Listings

114. 114 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Points with Positive Prob Prior Data Any Listing where

115. 115 Bayesian Model- Exchangeable Prior Populations N=3 Prior Data Any Listing where Often, implying that the probability of a permutation of the data is independent of a permutation of the remainder where Define Doesn’t depend on h

116. 116 Bayesian Model- Exchangeable Prior Populations N=3 the Posterior Random Variables Prior Data and then Any Listing where Recall that When, Independent, random permutation matrices

117. 117 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Distribution-Accounting for Populations Posterior Points with Positive Prob Prior Data Any Listing where Define populations that include the data

118. 118 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Distribution-Accounting for Populations Posterior Points with Positive Prob Prior Data Define Any Listing where where

119. 119 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Distribution-Accounting for Populations Points in Posterior Distribution Prior Data

120. 120 Bayesian Model- Exchangeable Prior Populations N=3 Posterior Distribution-Accounting for Populations Points in Posterior Distribution Prior Data since where

121. 121 Bayesian Model- Exchangeable Prior Populations N=3 Expected Value of Posterior Random Variables Prior Data Assume permutations of subjects in listing p are equally likely:

122. 122 Bayesian Model- Exchangeable Prior Populations N=3 Variance of Posterior Random Variables Prior Data Assume permutations of subjects in listing p are equally likely: The mean of random variables for the data is the mean for the data. Random variables representing the data are independent of the remainder. The distribution of random variables for the data is a random permutation distribution

123. 123 Exchangeable Prior Bayesian Model An Example: H=3, N=3, n=2 Populations ? Data Prior Posterior # Prior Populations:

124. 124 Exchangeable Prior Bayesian Model- Example #1: H=3, N=3, n=2 Populations ? Data Prior Posterior 1053 56 1221053 56 122

125. 125 Exchangeable Prior Bayesian Model- Example #1: H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 122 1053 56 1221053 56 122 Prior

126. 126 Exchangeable Prior Bayesian Model- Example #1: H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 122 1053 56 1221053 56 122 105 Suppose the Data is Prior

127. 127 Exchangeable Prior Bayesian Model- Example #1: H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 122 1053 56 1221053 56 122 105 Prior

128. 128 Exchangeable Prior Bayesian Model-An Example: H=3, N=3, n=2 Populations Data Prior Posterior 1053 5 56 1221053 56 Posterior Prior

129. 129 Exchangeable Prior Bayesian Model- Example 2. H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 122 1053 56 1221053 56 122 1012 Suppose the Data is Prior

130. 130 Exchangeable Prior Bayesian Model: Example 2. H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 1221053 56 122 1012 PriorPosterior

131. 131 Example 2: Exchangeable Prior Bayesian Model: H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 1221053 56 122 1012 PriorPosterior

132. 132 Exchangeable Prior Bayesian Model Example 3. H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 122 1053 56 1221053 56 122 1012 Suppose the Data is Prior

133. 133 Exchangeable Prior Bayesian Model Example 3. H=3, N=3, n=2 Posterior 1056 122 1053 56 122 Prior Populations Data Prior 1053 56 122 1053

134. 134 Exchangeable Prior Bayesian Model Example 3. H=3, N=3, n=2 Populations Data Prior Posterior 1053 56 12210122 Prior Posterior