Calculation of Some Expected Values for Parameterized Mean Model with Gaussian Noise

Calculation of Some Exp ected V alues for P arameterized Mean Mo del with Gaussian Noise Um ut Orguner ∗ No v em b er 21 , 2018 Consider the measurement model y = g ( x ) + v (1) where • x ∈ R is the unknown scalar we w ould like to estimate described by the prior distr ibution N ( x ; 0; σ 2 x ) where the no tation N ( · ; ¯ x, Σ x ) denotes a (real- v ariate) Gaussia n density with mean ¯ x and co v ari- ance Σ x . • y ∈ R n y is, in genera l, a c omplex mea suremen t vector; • g ( · ) : R → C n y is, in genera l, a c omplex-v alued obser v ation function; • v ∈ C n y is cir cular symmetric complex Gaus s ian measurement noise with zero -mean and cov ariance σ 2 v I n y where I n y denotes an iden tity matrix o f size n y × n y . The noise v is assumed indep enden t of x . In this do cumen t we ar e going to derive analytical formulae for the following functions. µ y ,x ( s 1 , s 2 , h 1 , h 2 ) , E h min( L s 1 ( y , x + h 1 , x ) , 1) min( L s 2 ( y , x + h 2 , x ) , 1) i , (2) µ y | x ( s 1 , s 2 , h 1 , h 2 , x ) , E y h min( L s 1 1 ( y , x + h 1 , x ) , 1) min( L s 2 1 ( y , x + h 2 , x ) , 1)    x i , (3) µ x ( s 1 , s 2 , h 1 , h 2 ) , E x h min( L s 1 2 ( x + h 1 , x ) , 1) min( L s 2 2 ( x + h 2 , x ) , 1) i . (4) where s 1 , s 2 ∈ Z , h 1 , h 2 ∈ R n x and L ( y , x, ξ ) , p ˜ y , ˜ x ( y , x ) p ˜ y, ˜ x ( y , ξ ) , L 1 ( y , x, ξ ) , p ˜ y | ˜ x ( y | x ) p ˜ y | ˜ x ( y | ξ ) , L 2 ( x, ξ ) , p ˜ x ( x ) p ˜ x ( ξ ) . (5) 1 Calculation of µ y ,x ( s 1 , s 2 , h 1 , h 2 ) W e first calculate L ( y , x + h, x ) a s L ( y , x + h, x ) = C N ( y ; g ( x + h ) , σ 2 v I n y ) N ( x + h ; 0; σ 2 x ) C N ( y ; g ( x ) , σ 2 v I n y ) N ( x ; 0; σ 2 x ) (6) = exp  2 σ 2 v R  v H d ( x, h )  − b ( x, h )  (7) where the no tation C N ( y ; ¯ y , Σ y ) denotes a circular symmetr ic complex Gaus s ian density with mea n ¯ y and cov ar iance Σ y and d ( x, h ) , g ( x + h ) − g ( x ) , (8) b ( x, h ) , 1 σ 2 v k d ( x, h ) k 2 + 1 σ 2 x x T h + 1 2 σ 2 x k h k 2 . (9) ∗ Departmen t of Electrical & Electronics Engineering, Middle East T ec hnical Unive rs it y , Ank ar a, T urkey 1 Then we hav e µ y ,x ( s 1 , s 2 , h 1 , h 2 ) = Z Z min  exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b ( x, h 1 )  , 1  × min  exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b ( x, h 2 )  , 1  p ( v ) d v p ( x ) d x. (10) W e a re first going to handle the inner integral on the right hand side of (10 ) which we call as I 1 as follows. I 1 , Z min  exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b ( x, h 1 )  , 1  × min  exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b ( x, h 2 )  , 1  p ( v ) d v = Z V 1 p ( v ) d v + Z V 2 exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b ( x, h 2 )  p ( v ) d v + Z V 3 exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b ( x, h 1 )  p ( v ) d v + Z V 4 exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b ( x, h 1 )  exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b ( x, h 2 )  p ( v ) d v (11) where the sets V 1 , V 2 , V 3 and V 4 are defined as follows. V 1 , n v ∈ C n y    R  v H d ( x,h 1 )  ≥ σ 2 v b ( x,h 1 ) 2 & R  v H d ( x,h 2 )  ≥ σ 2 v b ( x,h 2 ) 2 o , (12a) V 2 , n v ∈ C n y    R  v H d ( x,h 1 )  ≥ σ 2 v b ( x,h 1 ) 2 & R  v H d ( x,h 2 )  < σ 2 v b ( x,h 2 ) 2 o , (12b) V 3 , n v ∈ C n y    R  v H d ( x,h 1 )  < σ 2 v b ( x,h 1 ) 2 & R  v H d ( x,h 2 )  ≥ σ 2 v b ( x,h 2 ) 2 o , (12c) V 4 , n v ∈ C n y    R  v H d ( x,h 1 )  < σ 2 v b ( x,h 1 ) 2 & R  v H d ( x,h 2 )  < σ 2 v b ( x,h 2 ) 2 o . (12d) Substituting (9) and the identit y p ( v ) = C N ( v ; 0 , σ 2 v I n y ) into (11), we get I 1 = P  v ∈ V 1   v ∼ C N ( v ; 0 , σ 2 v )  + exp  s 2 2 σ 2 v k d ( x, h 2 ) k 2 − s 2 b ( x, h 2 )  P  v ∈ V 2   v ∼ C N ( v ; s 2 d ( x, h 2 ) , σ 2 v )  + exp  s 2 1 σ 2 v k d ( x, h 1 ) k 2 − s 1 b ( x, h 1 )  P  v ∈ V 3   v ∼ C N ( v ; s 1 d ( x, h 1 ) , σ 2 v )  + exp  1 σ 2 v k s 1 d ( x, h 1 ) + s 2 d ( x, h 2 ) k 2 − s 1 b ( x, h 1 ) − s 2 b ( x, h 2 )  × P  v ∈ V 4   v ∼ C N ( v ; s 1 d ( x, h 1 ) + s 2 d ( x, h 2 ) , σ 2 v )  . (13) W e now define the real scalars a 1 and a 2 as a 1 , R  v H d ( x, h 1 )  and a 2 , R  v H d ( x, h 2 )  . Since each of the pro babilities on the right hand side of (13) ar e conditioned on v being distributed with a cir cular symmetric complex Ga ussian density and s ince a 1 , a 2 are linearly dep enden t on v , w e have the vector a , [ a 1 , a 2 ] T distributed with a Gaussian dens it y which gives I 1 = P  a 1 ≥ σ 2 v b ( x,h 1 ) 2 a 2 ≥ σ 2 v b ( x,h 2 ) 2     a ∼ N  a ; ¯ a ′ 1 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 )   2 + exp  s 2 2 σ 2 v k d ( x, h 2 ) k 2 − s 2 b ( x, h 2 )  P  a 1 ≥ σ 2 v b ( x,h 1 ) 2 a 2 < σ 2 v b ( x,h 2 ) 2     a ∼ N  a ; ¯ a ′ 2 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 )   + exp  s 2 1 σ 2 v k d ( x, h 1 ) k 2 − s 1 b ( x, h 1 )  P  a 1 < σ 2 v b ( x,h 1 ) 2 a 2 ≥ σ 2 v b ( x,h 2 ) 2     a ∼ N  a ; ¯ a ′ 3 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 )   + exp  1 σ 2 v k s 1 d ( x, h 1 ) + s 2 d ( x, h 2 ) k 2 − s 1 b ( x, h 1 ) − s 2 b ( x, h 2 )  × P  a 1 < σ 2 v b ( x,h 1 ) 2 a 2 < σ 2 v b ( x,h 2 ) 2     a ∼ N  a ; ¯ a ′ 4 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 )   (14) where ¯ a ′ 1 ( x, h 1 , h 2 ) , [0 , 0] T , (15a) ¯ a ′ 2 ( x, h 1 , h 2 ) , s 2  R  d H ( x, h 1 ) d ( x, h 2 )  , k d ( x, h 2 ) k 2  T , (15b) ¯ a ′ 3 ( x, h 1 , h 2 ) , s 1  k d ( x, h 1 ) k 2 , R  d H ( x, h 1 ) d ( x, h 2 )  T , (15c) ¯ a ′ 4 ( x, h 1 , h 2 ) ,  s 1 k d ( x, h 1 ) k 2 + s 2 R  d H ( x, h 1 ) d ( x, h 2 )  , s 2 k d ( x, h 2 ) k 2 + s 1 R  d H ( x, h 1 ) d ( x, h 2 )  T , (15d) Γ( x, h 1 , h 2 ) ,  k d ( x, h 1 ) k 2 R  d H ( x, h 1 ) d ( x, h 2 )  R  d H ( x, h 1 ) d ( x, h 2 )  k d ( x, h 2 ) k 2  . (16) Each of the pr obabilities on the right hand side of (14) ca n b e wr itten using the cumulative distribution function N cdf 2 ( · , · , · ) o f a biv ariate Gauss ian random v ariable to give I 1 = N cdf 2  − σ 2 v b ( x, h 1 ) 2 , − σ 2 v b ( x, h 2 ) 2  T ; ¯ a 1 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 2 σ 2 v k d ( x, h 2 ) k 2 − s 2 b ( x, h 2 )  × N cdf 2  − σ 2 v b ( x, h 1 ) 2 , σ 2 v b ( x, h 2 ) 2  T ; ¯ a 2 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 1 σ 2 v k d ( x, h 1 ) k 2 − s 1 b ( x, h 1 )  × N cdf 2  σ 2 v b ( x, h 1 ) 2 , − σ 2 v b ( x, h 2 ) 2  T ; ¯ a 3 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  1 σ 2 v k s 1 d ( x, h 1 ) + s 2 d ( x, h 2 ) k 2 − s 1 b ( x, h 1 ) − s 2 b ( x, h 2 )  × N cdf 2  σ 2 v b ( x, h 1 ) 2 , σ 2 v b ( x, h 2 ) 2  T ; ¯ a 4 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! (17) where ¯ a 1 ( x, h 1 , h 2 ) , − ¯ a ′ 1 ( x, h 1 , h 2 ) , (18a) ¯ a 2 ( x, h 1 , h 2 ) , h − 1 0 0 1 i ¯ a ′ 2 ( x, h 1 , h 2 ) , (18b) ¯ a 3 ( x, h 1 , h 2 ) , h 1 0 0 − 1 i ¯ a ′ 3 ( x, h 1 , h 2 ) , (18c) ¯ a 4 ( x, h 1 , h 2 ) , ¯ a ′ 4 ( x, h 1 , h 2 ) , (18d) Γ( x, h 1 , h 2 ) ,  k d ( x, h 1 ) k 2 −R  d H ( x, h 1 ) d ( x, h 2 )  −R  d H ( x, h 1 ) d ( x, h 2 )  k d ( x, h 2 ) k 2  . (19) 3 Using (9) in (17), we get I 1 = N cdf 2  − σ 2 v b ( x, h 1 ) 2 , − σ 2 v b ( x, h 2 ) 2  T ; ¯ a 1 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 − s 2 σ 2 x x T h 2 − s 2 2 σ 2 x k h 2 k 2  × N cdf 2  − σ 2 v b ( x, h 1 ) 2 , σ 2 v b ( x, h 2 ) 2  T ; ¯ a 2 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 − s 1 σ 2 x x T h 1 − s 1 2 σ 2 x k h 1 k 2  × N cdf 2  σ 2 v b ( x, h 1 ) 2 , − σ 2 v b ( x, h 2 ) 2  T ; ¯ a 3 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + 2 s 1 s 2 σ 2 v R  d H ( x, h 1 ) d ( x, h 2 )  − s 1 h 1 + s 2 h 2 σ 2 x x − s 1 2 σ 2 x k h 1 k 2 − s 2 2 σ 2 x k h 2 k 2  × N cdf 2  σ 2 v b ( x, h 1 ) 2 , σ 2 v b ( x, h 2 ) 2  T ; ¯ a 4 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! (20) = N cdf 2  − σ 2 v b ( x, h 1 ) 2 , − σ 2 v b ( x, h 2 ) 2  T ; ¯ a 1 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2  exp  − s 2 σ 2 x x T h 2 − s 2 2 2 σ 2 x k h 2 k 2  × N cdf 2  − σ 2 v b ( x, h 1 ) 2 , σ 2 v b ( x, h 2 ) 2  T ; ¯ a 2 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 1 − s 1 2 σ 2 x k h 1 k 2  exp  − s 1 σ 2 x x T h 1 − s 2 1 2 σ 2 x k h 1 k 2  × N cdf 2  σ 2 v b ( x, h 1 ) 2 , − σ 2 v b ( x, h 2 ) 2  T ; ¯ a 3 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + 2 s 1 s 2 σ 2 v R  d H ( x, h 1 ) d ( x, h 2 )   × exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2 + s 1 s 2 σ 2 x h T 1 h 2  exp  − s 1 h 1 + s 2 h 2 σ 2 x x − 1 2 σ 2 x k s 1 h 1 + s 2 h 2 k 2  × N cdf 2  σ 2 v b ( x, h 1 ) 2 , σ 2 v b ( x, h 2 ) 2  T ; ¯ a 4 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! . (21) Substituting the r esult (21) into (10) and carrying out straig h tforward algebra, we obtain µ y ,x ( s 1 , s 2 , h 1 , h 2 ) = E N ( x ;0 , σ 2 x ) " N cdf 2 " − σ 2 v b ( x,h 1 ) 2 − σ 2 v b ( x,h 2 ) 2 # ; ¯ a 1 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! # + E N ( x ; − s 2 h 2 ,σ 2 x ) " exp  s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2  × N cdf 2 " − σ 2 v b ( x,h 1 ) 2 σ 2 v b ( x,h 2 ) 2 # ; ¯ a 2 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! # 4 + E N ( x ; − s 1 h 1 ,σ 2 x ) " exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 1 − s 1 2 σ 2 x k h 1 k 2  × N cdf 2 " σ 2 v b ( x,h 1 ) 2 − σ 2 v b ( x,h 2 ) 2 # ; ¯ a 3 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! # + E N ( x ; − ( s 1 h 1 + s 2 h 2 ) ,σ 2 x ) " exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + 2 s 1 s 2 σ 2 v R  d H ( x, h 1 ) d ( x, h 2 )   exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2 + s 1 s 2 σ 2 x h T 1 h 2  × N cdf 2 " σ 2 v b ( x,h 1 ) 2 σ 2 v b ( x,h 2 ) 2 # ; ¯ a 4 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! # . (22) 2 Calculation of µ y | x ( s 1 , s 2 , h 1 , h 2 , x ) and µ x ( s 1 , s 2 , h 1 , h 2 ) W e first calculate L 1 ( · , · , · ) and L 2 ( · , · ) as follows. L 1 ( y , x + h, x ) = exp h − 1 σ 2 v k y − g ( x + h ) k 2 i exp h − 1 σ 2 v k y − g ( x ) k 2 i (23) = exp  2 σ 2 v R  y H d ( x, h )  − 1 σ 2 v k g ( x + h ) k 2 + 1 σ 2 v k g ( x ) k 2  (24) = exp  2 σ 2 v R  v H d ( x, h )  − 1 σ 2 v k d ( x, h ) k 2  (25) = exp  2 σ 2 v R  v H d ( x, h )  − b 1 ( x, h )  , (26) L 2 ( x + h, x ) = exp h − 1 2 σ 2 x k x + h k 2 i exp h − 1 2 σ 2 x k x k 2 i (27) = exp  − 1 σ 2 x x T h − 1 2 σ 2 x k h k 2  = e xp [ − b 2 ( x, h )] , (28) where b 1 ( x, h ) , 1 σ 2 v k d ( x, h ) k 2 , (29) b 2 ( x, h ) , 1 σ 2 x x T h + 1 2 σ 2 x k h k 2 . (30) Then the function µ y | x ( · , · , · , · , · ) is g iv en as µ y | x ( s 1 , s 2 , h 1 , h 2 , x ) = Z min  exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b 1 ( x, h 1 )  , 1  × min  exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b 1 ( x, h 2 )  , 1  p ( v ) d v (31) = Z V 1 p ( v ) d v + Z V 2 exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b 1 ( x, h 2 )  p ( v ) d v 5 + Z V 3 exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b 1 ( x, h 1 )  p ( v ) d v + Z V 4 exp  2 s 1 σ 2 v R  v H d ( x, h 1 )  − s 1 b 1 ( x, h 1 )  exp  2 s 2 σ 2 v R  v H d ( x, h 2 )  − s 2 b 1 ( x, h 2 )  p ( v ) d v (32) where the sets V 1 , V 2 , V 3 and V 4 are defined as follows. V 1 , n v ∈ C n y    R  v H d ( x,h 1 )  ≥ σ 2 v b 1 ( x,h 1 ) 2 & R  v H d ( x,h 2 )  ≥ σ 2 v b 1 ( x,h 2 ) 2 o , (33a) V 2 , n v ∈ C n y    R  v H d ( x,h 1 )  ≥ σ 2 v b 1 ( x,h 1 ) 2 & R  v H d ( x,h 2 )  < σ 2 v b 1 ( x,h 2 ) 2 o , (33b) V 3 , n v ∈ C n y    R  v H d ( x,h 1 )  < σ 2 v b 1 ( x,h 1 ) 2 & R  v H d ( x,h 2 )  ≥ σ 2 v b 1 ( x,h 2 ) 2 o , (33c) V 4 , n v ∈ C n y    R  v H d ( x,h 1 )  < σ 2 v b 1 ( x,h 1 ) 2 & R  v H d ( x,h 2 )  < σ 2 v b 1 ( x,h 2 ) 2 o . (33d) Substituting (29) and the ident ity p ( v ) = C N ( v ; 0 , σ 2 v I n y ) into (32), we get µ y | x ( s 1 , s 2 , h 1 , h 2 , x ) = P R { v H d ( x,h 1 ) } ≥ σ 2 v b 1 ( x,h 1 ) 2 R { v H d ( x,h 2 ) } ≥ σ 2 v b 1 ( x,h 2 ) 2      v ∼ C N ( v ; 0 , σ 2 v I n y ) ! (34) + exp  s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2  P R { v H d ( x,h 1 ) } ≥ σ 2 v b 1 ( x,h 1 ) 2 R { v H d ( x,h 2 ) } < σ 2 v b 1 ( x,h 2 ) 2      v ∼ C N ( v ; s 2 d ( x, h 2 ) , σ 2 v I n y ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2  P R { v H d ( x,h 1 ) } < σ 2 v b 1 ( x,h 1 ) 2 R { v H d ( x,h 2 ) } ≥ σ 2 v b 1 ( x,h 2 ) 2      v ∼ C N ( v ; s 1 d ( x, h 1 ) , σ 2 v I n y ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + 2 s 1 s 2 σ 2 v R{ d H ( x, h 1 ) d ( x, h 2 ) }  × P R { v H d ( x,h 1 ) } < σ 2 v b 1 ( x,h 1 ) 2 R { v H d ( x,h 2 ) } < σ 2 v b 1 ( x,h 2 ) 2      v ∼ C N ( v ; s 1 d ( x, h 1 ) + s 2 d ( x, h 2 ) , σ 2 v I n y ) ! . (35) Contin uing in the same wa y as in Section 1, w e get µ y | x ( s 1 , s 2 , h 1 , h 2 , x ) = N cdf 2 " − σ 2 v b 1 ( x,h 1 ) 2 − σ 2 v b 1 ( x,h 2 ) 2 # ; ¯ a 1 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2  × N cdf 2 " − σ 2 v b 1 ( x,h 1 ) 2 σ 2 v b 1 ( x,h 2 ) 2 # ; ¯ a 2 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 × N cdf 2 " σ 2 v b 1 ( x,h 1 ) 2 − σ 2 v b 1 ( x,h 2 ) 2 # ; ¯ a 3 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) !  + exp  s 2 1 − s 1 σ 2 v k d ( x, h 1 ) k 2 + s 2 2 − s 2 σ 2 v k d ( x, h 2 ) k 2 + 2 s 1 s 2 σ 2 v R  d H ( x, h 1 ) d ( x, h 2 )   × N cdf 2 " σ 2 v b 1 ( x,h 1 ) 2 σ 2 v b 1 ( x,h 2 ) 2 # ; ¯ a 4 ( x, h 1 , h 2 ) , σ 2 v 2 Γ( x, h 1 , h 2 ) ! . (36) 6 Similarly the function µ x ( · , · , · , · ) is g iv en a s µ x ( s 1 , s 2 , h 1 , h 2 ) = Z min (exp [ − s 1 b 2 ( x, h 1 )] , 1) min (exp [ − s 2 b 2 ( x, h 2 )] , 1) p ( x ) d x (37) = Z b 2 ( x,h 1 ) < 0 b 2 ( x,h 2 ) < 0 p ( x ) d x + Z b 2 ( x,h 1 ) < 0 b 2 ( x,h 2 ) ≥ 0 exp [ − s 2 b 2 ( x, h 2 )] p ( x ) d x + Z b 2 ( x,h 1 ) ≥ 0 b 2 ( x,h 2 ) < 0 exp [ − s 1 b 2 ( x, h 1 )] p ( x ) d x + Z b 2 ( x,h 1 ) ≥ 0 b 2 ( x,h 2 ) ≥ 0 exp [ − s 1 b 2 ( x, h 1 )] exp [ − s 2 b 2 ( x, h 2 )] p ( x ) d x (38) = P  b 2 ( x,h 1 ) < 0 b 2 ( x,h 2 ) < 0    x ∼ N ( x ; 0; σ 2 x )  + exp  s 2 2 − s 2 2 σ 2 x k h 2 k 2  P  b 2 ( x,h 1 ) < 0 b 2 ( x,h 2 ) ≥ 0    x ∼ N ( x ; − s 2 h 2 , σ 2 x )  + exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2  P  b 2 ( x,h 1 ) ≥ 0 b 2 ( x,h 2 ) < 0    x ∼ N ( x ; − s 1 h 1 , σ 2 x )  + exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2 + s 1 s 2 σ 2 x h T 1 h 2  P  b 2 ( x,h 1 ) ≥ 0 b 2 ( x,h 2 ) ≥ 0    x ∼ N ( x ; − s 1 h 1 − s 2 h 2 , σ 2 x )  (39) = P a 1 < 0 a 2 < 0      a ∼ N  a ; ˜ a ′ 1 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 )  ! + exp  s 2 2 − s 2 2 σ 2 x k h 2 k 2  P a 1 < 0 a 2 ≥ 0      a ∼ N  a ; ˜ a ′ 2 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 )  ! + exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2  P a 1 ≥ 0 a 2 < 0      a ∼ N  a ; ˜ a ′ 3 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 )  ! + exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2 + s 1 s 2 σ 2 x h T 1 h 2  × P a 1 ≥ 0 a 2 ≥ 0      a ∼ N  a ; ˜ a ′ 4 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 )  ! (40) where ˜ a ′ 1 ( h 1 , h 2 ) , 1 2 σ 2 x  k h 1 k 2 , k h 2 k 2  T , (41a) ˜ a ′ 2 ( h 1 , h 2 ) , 1 2 σ 2 x  − 2 s 2 h T 1 h 2 + k h 1 k 2 , (1 − 2 s 2 ) k h 2 k 2  T , (41b) ˜ a ′ 3 ( h 1 , h 2 ) , 1 2 σ 2 x  (1 − 2 s 1 ) k h 1 k 2 , − 2 s 1 h T 1 h 2 + k h 2 k 2  T , (41c) ˜ a ′ 4 ( h 1 , h 2 ) , 1 2 σ 2 x  (1 − 2 s 1 ) k h 1 k 2 − 2 s 2 h T 1 h 2 − 2 s 1 h T 1 h 2 + (1 − 2 s 2 ) k h 2 k 2  , (41d) Λ( h 1 , h 2 ) ,  k h 1 k 2 h T 1 h 2 h T 1 h 2 k h 2 k 2  . (42) Each of the pr obabilities on the right hand side of (40) ca n b e wr itten using the cumulative distribution function N cdf 2 ( · , · , · ) a s µ x ( s 1 , s 2 , h 1 , h 2 ) = N cdf 2  0 0  ; ˜ a 1 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 ) ! 7 + exp  s 2 2 − s 2 2 σ 2 x k h 2 k 2  N cdf 2  0 0  ; ˜ a 2 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 ) ! + exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2  N cdf 2  0 0  ; ˜ a 3 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 ) ! + exp  s 2 1 − s 1 2 σ 2 x k h 1 k 2 + s 2 2 − s 2 2 σ 2 x k h 2 k 2 + s 1 s 2 σ 2 x h T 1 h 2  N cdf 2  0 0  ; ˜ a 4 ( h 1 , h 2 ) , 1 σ 2 x Λ( h 1 , h 2 ) ! (43) where ˜ a 1 ( h 1 , h 2 ) , ˜ a ′ 1 ( h 1 , h 2 ) , (44a) ˜ a 2 ( h 1 , h 2 ) , h 1 0 0 − 1 i ˜ a ′ 2 ( h 1 , h 2 ) , (44b) ˜ a 3 ( h 1 , h 2 ) , h − 1 0 0 1 i ˜ a ′ 3 ( h 1 , h 2 ) , (44c) ˜ a 4 ( h 1 , h 2 ) , − ˜ a ′ 4 ( h 1 , h 2 ) , (44d) Λ( h 1 , h 2 ) ,  k h 1 k 2 − h T 1 h 2 − h T 1 h 2 k h 2 k 2  . (45) 3 Sp ecial Cases In this section, we are going to find the ex pressions for the following quan tities one by one: µ y ,x (1 , 1 , h, h ), µ y ,x (1 , 0 , h, h ), µ y | x (1 , 1 , h, h , x ), µ y | x (1 , 0 , h, h , x ), µ x (1 , 1 , h, h ), µ x (1 , 0 , h, h ). • µ y ,x (1 , 1 , h, h ): Substituting s 1 = s 2 = 1 and h 1 = h 2 = h in (22), w e get µ y ,x (1 , 1 , h, h ) = E N ( x ;0 , σ 2 x ) " N cdf 2 " − σ 2 v b ( x,h ) 2 − σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 1 1 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # + E N ( x ; − h,σ 2 x ) " N cdf 2 " − σ 2 v b ( x,h ) 2 σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 1 2 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # + E N ( x ; − h,σ 2 x ) " N cdf 2 " σ 2 v b ( x,h ) 2 − σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 1 3 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # + E N ( x ; − 2 h,σ 2 x ) " exp  2 σ 2 v k d ( x, h ) k 2  exp  1 σ 2 x k h k 2  × N cdf 2 " σ 2 v b ( x,h ) 2 σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 1 4 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # (46) where ¯ a 1 , 1 1 ( x, h, h ) , [0 , 0] T , (47a) ¯ a 1 , 1 2 ( x, h, h ) ,  −k d ( x, h ) k 2 , k d ( x, h ) k 2  T , (47b) ¯ a 1 , 1 3 ( x, h, h ) ,  k d ( x, h ) k 2 , −k d ( x, h ) k 2  T , (47c) ¯ a 1 , 1 4 ( x, h, h ) , 2  k d ( x, h ) k 2 , k d ( x, h ) k 2  T , (47d) and Γ( x, h, h ) ,  k d ( x, h ) k 2 k d ( x, h ) k 2 k d ( x, h ) k 2 k d ( x, h ) k 2  , (48a) Γ( x, h, h ) ,  k d ( x, h ) k 2 −k d ( x, h ) k 2 −k d ( x, h ) k 2 k d ( x, h ) k 2  . (48b) 8 Noting that Γ( x, h, h ) and Γ( x, h, h ) ar e singular , the quantities r elated to the biv ariate cumulativ e distribution function N cdf 2 ( · ) can b e written as N cdf 2 " − σ 2 v b ( x,h ) 2 − σ 2 v b ( x,h ) 2 # ; ¯ a 1 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! = N cdf 1  − σ 2 v b ( x, h ) 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  , (49a) N cdf 2 " − σ 2 v b ( x,h ) 2 σ 2 v b ( x,h ) 2 # ; ¯ a 2 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! =0 , (49b) N cdf 2 " σ 2 v b ( x,h ) 2 − σ 2 v b ( x,h ) 2 # ; ¯ a 3 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! =0 , (49c) N cdf 2 " σ 2 v b ( x,h ) 2 σ 2 v b ( x,h ) 2 # ; ¯ a 4 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! = N cdf 1  σ 2 v b ( x, h ) 2 ; 2 k d ( x, h ) k 2 , σ 2 v 2 k d ( x, h ) k 2  , (49d) where N cdf 1 ( · ) is the cumulativ e distribution function for the univ ar iate Gaussian density , which gives µ y ,x (1 , 1 , h, h ) = E N ( x ;0 , σ 2 x ) " N cdf 1  − σ 2 v b ( x, h ) 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  # + E N ( x ; − 2 h,σ 2 x ) " exp  2 σ 2 v k d ( x, h ) k 2  exp  1 σ 2 x k h k 2  × N cdf 1  σ 2 v b ( x, h ) 2 ; 2 k d ( x, h ) k 2 , σ 2 v 2 k d ( x, h ) k 2  # . (50) • µ y ,x (1 , 0 , h, h ): Substituting s 1 = 1, s 2 = 0 and h 1 = h 2 = h in (22), w e get µ y ,x (1 , 0 , h, h ) = E N ( x ;0 , σ 2 x ) " N cdf 2 " − σ 2 v b ( x,h ) 2 − σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 0 1 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # + E N ( x ;0 , σ 2 x ) " N cdf 2 " − σ 2 v b ( x,h ) 2 σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 0 2 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # + E N ( x ; − h,σ 2 x ) " N cdf 2 " σ 2 v b ( x,h ) 2 − σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 0 3 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # + E N ( x ; − h,σ 2 x ) " N cdf 2 " σ 2 v b ( x,h ) 2 σ 2 v b ( x,h ) 2 # ; ¯ a 1 , 0 4 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! # (51) where ¯ a 1 , 0 1 ( x, h, h ) , [0 , 0] T , (52a) ¯ a 1 , 0 2 ( x, h, h ) , [0 , 0] T , (52b) ¯ a 1 , 0 3 ( x, h, h ) ,  k d ( x, h ) k 2 , −k d ( x, h ) k 2  T , (52c) ¯ a 1 , 0 4 ( x, h, h ) ,  k d ( x, h ) k 2 , k d ( x, h ) k 2  T . (52d) Again due to the singula rit y o f Γ( x, h, h ) and Γ( x, h, h ), we hav e µ y ,x (1 , 0 , h, h ) = E N ( x ;0 , σ 2 x ) " N cdf 1  − σ 2 v b ( x, h ) 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  # + E N ( x ; − h,σ 2 x ) " N cdf 1  σ 2 v b ( x, h ) 2 ; k d ( x, h ) k 2 , σ 2 v 2 k d ( x, h ) k 2  # . (53) 9 • µ y | x (1 , 1 , h, h , x ): Substituting s 1 = s 2 = 1 and h 1 = h 2 = h in (36), w e get µ y | x (1 , 1 , h, h , x ) = N cdf 2 " − σ 2 v b 1 ( x,h ) 2 − σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 1 1 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! + N cdf 2 " − σ 2 v b 1 ( x,h ) 2 σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 1 2 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! + N cdf 2 " σ 2 v b 1 ( x,h ) 2 − σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 1 3 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) !  + exp  2 σ 2 v k d ( x, h ) k 2  N cdf 2 " σ 2 v b 1 ( x,h ) 2 σ 2 v b 1 ( x,h ) 2 # ; ¯ a 4 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! . (54) Using the singular it y of Γ ( x, h, h ) and Γ( x, h, h ), we obtain µ y | x (1 , 1 , h, h , x ) = N cdf 1  − σ 2 v b 1 ( x, h ) 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  + exp  2 σ 2 v k d ( x, h ) k 2  N cdf 1  σ 2 v b 1 ( x, h ) 2 ; 2 k d ( x, h ) k 2 , σ 2 v 2 k d ( x, h ) k 2  . (55) • µ y | x (1 , 0 , h, h , x ): Substituting s 1 = 1 , s 2 = 0 and h 1 = h 2 = h in (36), we get µ y | x (1 , 0 , h, h , x ) = N cdf 2 " − σ 2 v b 1 ( x,h ) 2 − σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 0 1 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! + N cdf 2 " − σ 2 v b 1 ( x,h ) 2 σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 0 2 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! + N cdf 2 " σ 2 v b 1 ( x,h ) 2 − σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 0 3 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) !  + N cdf 2 " σ 2 v b 1 ( x,h ) 2 σ 2 v b 1 ( x,h ) 2 # ; ¯ a 1 , 0 4 ( x, h, h ) , σ 2 v 2 Γ( x, h, h ) ! . (56) Using the singular it y of Γ ( x, h, h ) and Γ( x, h, h ), we obtain µ y | x (1 , 0 , h, h , x ) = N cdf 1  − σ 2 v b 1 ( x, h ) 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  + N cdf 1  σ 2 v b 1 ( x, h ) 2 ; k d ( x, h ) k 2 , σ 2 v 2 k d ( x, h ) k 2  . (57) If we now substitute b 1 ( · , · ) fr om (29) into (57), we get µ y | x (1 , 0 , h, h , x ) = N cdf 1  − k d ( x, h ) k 2 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  + N cdf 1  k d ( x, h ) k 2 2 ; k d ( x, h ) k 2 , σ 2 v 2 k d ( x, h ) k 2  (58) =2 N cdf 1  − k d ( x, h ) k 2 2 ; 0 , σ 2 v 2 k d ( x, h ) k 2  (59) =1 − erf  k d ( x, h ) k 2 σ v  (60) where we used the iden tity N cdf 1  ξ ; ¯ ξ , σ 2 ξ  = 1 2 1 + er f ξ − ¯ ξ √ 2 σ ξ !! . (61) 10 • µ x (1 , 1 , h, h ): Substituting s 1 = s 2 = 1 and h 1 = h 2 = h in (43), we get µ x (1 , 1 , h, h ) = N cdf 2  0 0  ; ˜ a 1 , 1 1 ( h, h ) , 1 σ 2 x Λ( h 1 , h 2 ) ! + N cdf 2  0 0  ; ˜ a 1 , 1 2 ( h, h ) , 1 σ 2 x Λ( h, h ) ! + N cdf 2  0 0  ; ˜ a 1 , 1 3 ( h, h ) , 1 σ 2 x Λ( h, h ) ! + exp  1 σ 2 x k h k 2  N cdf 2  0 0  ; ˜ a 1 , 1 4 ( h, h ) , 1 σ 2 x Λ( h, h ) ! (62) where ˜ a 1 , 1 1 ( h, h ) , 1 2 σ 2 x  k h k 2 , k h k 2  T , (63a) ˜ a 1 , 1 2 ( h, h ) , 1 2 σ 2 x  −k h k 2 , k h k 2  T , (63b) ˜ a 1 , 1 3 ( h, h ) , − 1 2 σ 2 x  k h k 2 , −k h k 2  T , (63c) ˜ a 1 , 1 4 ( h, h ) , 3 2 σ 2 x  k h k 2 , k h k 2  T , (63d) and Λ( h, h ) ,  k h k 2 k h k 2 k h k 2 k h k 2  , (64a) Λ( h, h ) ,  k h k 2 −k h k 2 −k h k 2 k h k 2  . (64b) Using the singular it y of Λ ( h, h ) and Λ( x, h, h ), we o bta in µ x (1 , 1 , h, h ) = N cdf 1 0; k h k 2 2 σ 2 x , k h k 2 σ 2 x ! + exp  1 σ 2 x k h k 2  N cdf 1 0; 3 k h k 2 2 σ 2 x , k h k 2 σ 2 x ! . (65) • µ x (1 , 0 , h, h ): Substituting s 1 = 1 , s 2 = 0 and h 1 = h 2 = h in (4 3), we get µ x (1 , 0 , h, h ) = N cdf 2  0 0  ; ˜ a 1 , 0 1 ( h, h ) , 1 σ 2 x Λ( h, h ) ! + N cdf 2  0 0  ; ˜ a 1 , 0 2 ( h, h ) , 1 σ 2 x Λ( h, h ) ! + N cdf 2  0 0  ; ˜ a 1 , 0 3 ( h, h ) , 1 σ 2 x Λ( h, h ) ! + N cdf 2  0 0  ; ˜ a 1 , 0 4 ( h, h ) , 1 σ 2 x Λ( h, h ) ! (66) where ˜ a 1 , 0 1 ( h, h ) , 1 2 σ 2 x  k h k 2 , k h k 2  T , (67a) ˜ a 1 , 0 2 ( h, h ) , 1 2 σ 2 x  −k h k 2 , k h k 2  T , (67b) ˜ a 1 , 0 3 ( h, h ) , − 1 2 σ 2 x  k h k 2 , −k h k 2  T , (67c) ˜ a 1 , 0 4 ( h, h ) , 1 2 σ 2 x  k h k 2 , k h k 2  T . (67d) 11 Using the singular it y of Λ ( h, h ) and Λ( x, h, h ), we obtain µ x (1 , 0 , h, h ) = N cdf 1 0; k h k 2 2 σ 2 x , k h k 2 σ 2 x ! + N cdf 1 0; k h k 2 2 σ 2 x , k h k 2 σ 2 x ! (68) =2 N cdf 1 0; k h k 2 2 σ 2 x , k h k 2 σ 2 x ! (69) =1 − erf  k h k 2 √ 2 σ x  . (70) 12