the smoothness and the fidelity (regression error) can be defined

݂۰|۳ୀ܍ሺ܊ሻൌ

݂۳|۰ୀ܊ሺ܍ሻ݂۰ሺ܊ሻ

݂۳ሺ܍ሻ

(5.21)

ሺ܍ሻൌ׬

݂۳|۰ୀఉሺ܍ሻ݂۰ሺߚሻ݀ߚ

ஶ

ିஶ

is called the evidence, which is a

Therefore, the posterior of a BWH baseline model is then

nal to the product of the fidelity density and the smoothness prior,

݂۰|۳ୀ܍ሺ܊ሻ∝݂۳|۰ୀ܊ሺ܍ሻ݂۰ሺ܊ሻ

(5.22)

above equation, ݂۰ሺ܊ሻ is a prior of the baseline and its smoothing

݂۲ୀ܌|ఙೄ

మሺ܌ሻ. Two hierarchical inverse Gamma priors for two

(ߪி

ଶ and ߪௌ

ଶ) are introduced for BWH. The inverse Gamma prior

defined as below, where ߙி and ߚி are two hyper-parameters for

ty prior,

݂ఙಷ

మ|ఈಷ,ఉಷሺߪி

ଶሻൌ

ߚி

ఈಷ

ߪி

ଶఈಷାଶΓሺߙிሻ

exp ቆെ^ߚி

ߪி

ଶ^ቇ

(5.23)

nverse Gamma prior for ߪௌ

ଶ is defined as below, where ߙௌ and ߚௌ

yper-parameters for the smoothness prior

݂ఙೄ

మ|ఈೄ,ఉೄሺߪௌ

ଶሻൌ

ߚௌ

ఈೄ

ߪௌ

ଶఈೄାଶΓሺߙௌሻ

exp ቆെ^ߚௌ

ߪௌ

ଶ^ቇ

(5.24)

osterior of a baseline under estimation is defined as below, where

ߪௌ

ଶ, ߙி, ߙௌ, ߚி, ߚௌሻ,

ୀ܊|܍,ణ^ሺ܊ሻ

݂۳|۰ୀ܊ሺ܍ሻ݂۲ୀ܌|ఙೄ

మሺ܌ሻ݂ఙಷ

మ|ఈಷ,ఉಷሺߪி

ଶሻ݂ఙೄ

మ|ఈೄ,ఉೄሺߪௌ

ଶሻ

(5.25)

cing every term of this posterior, the following posterior

where ߱ிൌߙி൅1 ൅ܭ/2 and ߱ௌൌߙௌ൅ሺܰ൅1ሻ/2 , is

s,