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Table 1 Synthetic data classification settings and prior models

From: On optimal Bayesian classification and risk estimation under multiple classes

  D M ν 0,…,ν M−1 m 0,…,m M−1 κ y (k y ) \(\frac {\mathbf {S}_{y}}{k_{y} - 2}\) Prior (cov.) λ
Model 1 2 2 12, 2 \(\left [\begin {array}{l} 0 \\ 0 \end {array}\right ], \left [\begin {array}{l} 0.5 \\ 0.5 \end {array}\right ]\) 6 (5) 0.3 I 2 Indep. arbit. \(\left [\begin {array}{ll} 0 & 2 \\ 1 & 0 \end {array}\right ]\)
Model 2 2 2 12, 2 \(\left [\begin {array}{l} 0 \\ 0 \end {array}\right ], \left [\begin {array}{l} 0.5 \\ 0.5 \end {array}\right ]\) 6 (5) 0.3 I 2 Homo. arbit. \(\left [\begin {array}{ll} 0 & 2 \\ 1 & 0 \end {array}\right ]\)
Model 3 2 5 12, 2, 2, 2, 2 \(\left [\begin {array}{l} 0 \\ 0 \end {array}\right ], \left [\begin {array}{l} 1 \\ 1 \end {array}\right ], \left [\begin {array}{l} -1 \\ -1 \end {array}\right ], \left [\begin {array}{l} 1 \\ -1 \end {array}\right ], \left [\begin {array}{l} -1 \\ 1 \end {array}\right ]\) 6 (5) 0.3 I 2 Indep. arbit. 0–1 loss
Model 4 2 5 12, 2, 2, 2, 2 \(\left [\begin {array}{l} 0 \\ 0 \end {array}\right ], \left [\begin {array}{l} 1 \\ 1 \end {array}\right ], \left [\begin {array}{l} -1 \\ -1 \end {array}\right ], \left [\begin {array}{l} 1 \\ -1 \end {array}\right ], \left [\begin {array}{l} -1 \\ 1 \end {array}\right ]\) 6 (5) 0.3 I 2 Homo. arbit. 0–1 loss
Model 5 20 2 12, 2 020,(0.05)20 −20.65 (5) 0.3 I 2 Indep. iden. \(\left [\begin {array}{ll} 0 & 2 \\ 1 & 0 \end {array}\right ]\)
Model 6 20 2 20, 20 020,020 −20.65 (5) 0.3 I 20 Indep. iden. \(\left [\begin {array}{ll} 0 & 2 \\ 1 & 0 \end {array}\right ]\)
Model 7 20 5 12, 2, 2, 2, 2 020,(0.1)20,(−0.1)20, −20.65 (5) 0.3 I 20 Indep. iden. 0–1 loss
     \(\left [\begin {array}{l} {(0.1)}_{10} \\ {(-0.1)}_{10} \end {array}\right ], \left [\begin {array}{l} {(-0.1)}_{10} \\ {(0.1)}_{10} \end {array}\right ]\)     
Model 8 20 5 20, 20, 20, 20, 20 020,020,020,020,020 −20.65 (5) 0.3 I 20 Indep. iden. 0–1 loss
  1. 0 k and (a) k represent all-zero and all-a column vectors of length k, respectively