Page 441 - Numerical Methods for Chemical Engineering

430 8 Bayesian statistics and parameter estimation

where H α > 0 is the Hessian of h α (θ α )at θ α . Substituting this quadratic expan-
M
1
sion into (8.226) and assuming I θ (θ α ) = 1 everywhere where exp{− Nh α (θ α )} is
α
2
significantly nonzero yields
9 :
1
Q α = exp − Nh α θ α T α (8.228)
2 M
where
' 1

T α = exp − 1 θ α − θ α T NH α θ α − θ α dθ α (8.229)
M
M
2
P α

Using the Gaussian integral

1 T T (2π) 1 T −1
' P/2
exp − z Az + b z dz = exp b A b (8.230)
2 |A| 1/2 2
P

we obtain
(2π) P α /2 2π P α /2 α −1/2
T α = = |H | (8.231)
|NH α 1/2 N
|
Thus, from (8.228) and (8.223), we have
P α /2
1
N 2π
p α (y|M α ) = C α exp − h α θ α (8.232)
M
2 N
where
c
σ α α
0
C α = (8.233)
1/2
)
(2π) N/2 (σ α (N+1) H α
We have then an approximation for the Bayes factor (8.214):
(
α P α /2
S α θ M 2π
C α
exp −
α 2 N
p(y|M α ) 2 σ MLE
B αβ (y) = ≈ ( (8.234)
p(y|M β ) S β θ β 2π P β /2
M
C β exp − 2
2 σ β N
MLE
where we have used (8.225). Taking the natural logarithm of (8.234), we have
α
S α θ M P α 2π
ln[B αβ (y)] ≈ ln C α + ln
α 2 2 N
−
2 σ
MLE

β
S β θ M P β 2π
− ln C β − + ln (8.235)
2
2 σ β 2 N
MLE
If we assume that C α and C β are of the same order of magnitude, and define Schwartz’s
Bayesian Information Criterion (BIC),

α
S α θ M P α N
α
BIC (y) =− + ln (8.236)
α 2 2 2π
2 σ MLE