njugate inverse gamma distributed priors on the variances is

in the model,

IGሺߪ௠^ଶ|ܽ௠, ܾ௠ሻൌ^ܾ௠

௔೘ߪ௠

ିଶሺ௔೘ାଵሻ

߁ሺܽ௠ሻ

exp ቆെ^ܾ௠

ߪ௠^ଶቇ

(6.46)

mixing coefficients are modelled using the non-informative priors

of ߤ଴ and ߤଵ is assigned a Gaussian prior with a zero mean and

deviation ߬௠ which is a hyper-parameter (the inverse of a

,

ߤ௠~࣡ሺ૙, ߬௠ሻ

(6.47)

over |߬௠| →0 indicates the null data. ࣡ሺܢ|ߤ଴, ߪ଴

ଶሻ is estimated

a such that |ߤ଴| →0 by the consistency. The prior means are set

ased on the observation that both centres are close to zero and set

to be 0.5. Suppose ߙൌሼܽ଴, ܽଵ, ܾ଴, ܾଵ, ߬଴, ߬ଵሽ and ߚ௠ൌߪ௠^ିଶ as

௠^ൌ߬௠

ିଶ. The posterior of DSG can be written as below,

ߙሻ∝ܲሺܼ|ߠሻܲሺߠ|ߙሻ

ൌෑሼݓ଴࣡ሺܢ|ߤ଴, ߚ଴

ିଵሻ൅ݓଵ࣡ሺܢ|ߤଵ, ߚଵ

ିଵሻሽ

௡

௜ୀଵ

ෑቊ^ܾ௠

௔೘ߚ௠

௔೘ାଵ

߁ሺܽ௠ሻ

expሺെܾ௠ߚ௠ሻ࣡ሺ૙, ߭௠ሻቋ

ଵ

௠ୀ଴

(6.48)

og-posterior can be written as below,

logܲሺߠ|ܼ, ߙሻ∝෍log݂

ଵ

ሺݖ௜|ߠሻ൅෍logܤ௠

௠ୀ଴

௡

௜ୀଵ

(6.49)

ܤ௠ is defined as,