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Submitted on 24 Jun 2010
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Bayesian variable selection for probit mixed models
Meïli Baragatti
To cite this version:
Meïli Baragatti. Bayesian variable selection for probit mixed models. 42èmes Journées de Statistique,
2010, Marseille, France, France. �inria-00494781�
❇❆❨❊❙■❆◆ ❱❆❘■❆❇▲❊ ❙❊▲❊❈❚■❖◆ ❋❖❘ P❘❖❇■❚
▼■❳❊❉ ▼❖❉❊▲❙
▼❡ï❧✐ ❇❛r❛❣❛tt✐
∗✱❛✱❜❛
■♥st✐t✉t ❞❡ ▼❛t❤é♠❛t✐q✉❡s ❞❡ ▲✉♠✐♥② ✭■▼▲✮✱ ❈◆❘❙ ▼❛rs❡✐❧❧❡✱ ❝❛s❡ ✾✵✼✱ ❈❛♠♣✉s ❞❡
▲✉♠✐♥②✱ ✶✸✷✽✽ ▼❛rs❡✐❧❧❡ ❈❡❞❡① ✾✱ ❋r❛♥❝❡
❜
■♣s♦❣❡♥ ❙❆✱ ▲✉♠✐♥② ❇✐♦t❡❝❤ ❊♥tr❡♣r✐s❡s✱ ❈❛s❡ ✾✷✸✱ ❈❛♠♣✉s ❞❡ ▲✉♠✐♥②✱ ✶✸✷✽✽ ▼❛rs❡✐❧❧❡
❈❡❞❡① ✾✱ ❋r❛♥❝❡
❆❜str❛❝t
■♥ ❝♦♠♣✉t❛t✐♦♥❛❧ ❜✐♦❧♦❣②✱ ❣❡♥❡ ❡①♣r❡ss✐♦♥ ❞❛t❛s❡ts ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ✈❡r②
❢❡✇ ✐♥❞✐✈✐❞✉❛❧ s❛♠♣❧❡s ❝♦♠♣❛r❡❞ t♦ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ♠❡❛s✉r♠❡♥ts ♣❡r s❛♠✲
♣❧❡✳ ❚❤✉s✱ ✐t ✐s ❛♣♣❡❛❧✐♥❣ t♦ ♠❡r❣❡ t❤❡s❡ ❞❛t❛s❡ts ✐♥ ♦r❞❡r t♦ ✐♥❝r❡❛s❡ t❤❡
♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ❞✐✈❡rs✐❢② t❤❡ ❞❛t❛✱ ❛❧❧♦✇✐♥❣ ❛ ♠♦r❡ r❡❧✐❛❜❧❡ s❡❧❡❝✲
t✐♦♥ ♦❢ ❣❡♥❡s r❡❧❡✈❛♥t t♦ t❤❡ ❜✐♦❧♦❣✐❝❛❧ ♣r♦❜❧❡♠✳ ❚❤✐s ♥❡❝❡ss✐t❛t❡s t❤❡ ✐♥tr♦✲
❞✉❝t✐♦♥ ♦❢ t❤❡ ❞❛t❛s❡t ❛s ❛ r❛♥❞♦♠ ❡✛❡❝t✳ ❊①t❡♥❞✐♥❣ ♣r❡✈✐♦✉s ✇♦r❦ ♦❢ ▲❡❡
❡t ❛❧✳ ✭✷✵✵✸✮✱ ❛ ♠❡t❤♦❞ ✐s ♣r♦♣♦s❡❞ t♦ s❡❧❡❝t r❡❧❡✈❛♥t ✈❛r✐❛❜❧❡s ❛♠♦♥❣ t❡♥s ♦❢
t❤♦✉s❛♥❞s ✐♥ ❛ ♣r♦❜✐t ♠✐①❡❞ r❡❣r❡ss✐♦♥ ♠♦❞❡❧✱ ❝♦♥s✐❞❡r❡❞ ❛s ♣❛rt ♦❢ ❛ ❧❛r❣❡r
❤✐❡r❛r❝❤✐❝❛❧ ❇❛②❡s✐❛♥ ♠♦❞❡❧✳ ▲❛t❡♥t ✈❛r✐❛❜❧❡s ❛r❡ ✉s❡❞ t♦ ✐❞❡♥t✐❢② s✉❜s❡ts ♦❢
s❡❧❡❝t❡❞ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❝♦❧❧❛♣s✐♥❣ t❡❝❤♥✐q✉❡ ♦❢ ▲✐✉ ✭✶✾✾✹✮ ✐s ❝♦♠❜✐♥❡❞ ✇✐t❤ ❛
▼❡tr♦♣♦❧✐s✲✇✐t❤✐♥✲●✐❜❜s ❛❧❣♦r✐t❤♠ ✭❘♦❜❡rt ❛♥❞ ❈❛s❡❧❧❛✱ ✷✵✵✹✮✳ ❚❤❡ ♠❡t❤♦❞ ✐s
❛♣♣❧✐❡❞ t♦ ❛ ♠❡r❣❡❞ ❞❛t❛s❡t ♠❛❞❡ ♦❢ t❤r❡❡ ✐♥❞✐✈✐❞✉❛❧ ❣❡♥❡ ❡①♣r❡ss✐♦♥ ❞❛t❛s❡ts✱
✐♥ ✇❤✐❝❤ t❡♥s ♦❢ t❤♦✉s❛♥❞s ♦❢ ♠❡❛s✉r❡♠❡♥ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢♦r ❡❛❝❤ ♦❢ s❡✈❡r❛❧
❤✉♥❞r❡❞ ❤✉♠❛♥ ❜r❡❛st ❝❛♥❝❡r s❛♠♣❧❡s✳ ❊✈❡♥ ❢♦r t❤✐s ❧❛r❣❡ ❞❛t❛s❡t ❝♦♠♣r✐s❡❞ ♦❢
❛r♦✉♥❞ ✷✵✵✵✵ ♣r❡❞✐❝t♦rs✱ t❤❡ ♠❡t❤♦❞ ✐s s❤♦✇♥ t♦ ❜❡ ❡✣❝✐❡♥t ❛♥❞ ❢❡❛s✐❜❧❡✳ ❆s ❛
❞❡♠♦♥str❛t✐♦♥✱ ✐t ✐s ✉s❡❞ t♦ s❡❧❡❝t t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❣❡♥❡s t❤❛t ❝❤❛r❛❝t❡r✐③❡
t❤❡ ❡str♦❣❡♥ r❡❝❡♣t♦r st❛t✉s ♦❢ t❤❡ ❝❛♥❝❡r ♣❛t✐❡♥ts✳
❑❡② ✇♦r❞s✿ ❇❛②❡s✐❛♥ s❡❧❡❝t✐♦♥✱ s❡❧❡❝t✐♦♥ ♦❢ ❝♦✈❛r✐❛❜❧❡s✱ r❛♥❞♦♠ ❡✛❡❝ts✱ ♣r♦❜✐t
♠✐①❡❞ r❡❣r❡ss✐♦♥ ♠♦❞❡❧✱ ❝♦❧❧❛♣s✐♥❣ t❡❝❤♥✐q✉❡✱ ▼❡tr♦♣♦❧✐s✲✇✐t❤✐♥✲●✐❜❜s
❛❧❣♦r✐t❤♠
❙❡❧❡❝t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ✐s ❛ ❝♦♠♠♦♥ ♣r♦❜❧❡♠ ✐♥ ♠❛♥② s❝✐❡♥t✐✜❝ ✜❡❧❞s✱ ❛♥❞
♣❛rt✐❝✉❧❛r❧② ✐♥ ❜✐♦✐♥❢♦r♠❛t✐❝s✳ ●❡♥❡ ❡①♣r❡ss✐♦♥ ♣r♦✜❧✐♥❣ ❛♥❛❧②s❡s ❛r❡ ♥♦t♦r✐♦✉s
❢♦r ❣❡♥❡r❛t✐♥❣ ❛ ✈❡r② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ♣r❡❞✐❝t♦rs ❝♦♠♣❛r❡❞ t♦ t❤❡ ♥✉♠❜❡r ♦❢
♦❜s❡r✈❛t✐♦♥s✳ ▼✐❝r♦❛rr❛② t❡❝❤♥♦❧♦❣② ✐s ✐♠♣♦rt❛♥t ❢♦r ✜♥❞✐♥❣ ❣❡♥❡s t❤❛t ❛r❡ ✐♠✲
♣❧✐❝❛t❡❞ ✐♥ ❜✐♦❧♦❣✐❝❛❧ ♣r♦❝❡ss❡s ✐♥❝❧✉❞✐♥❣ ❞❡✈❡❧♦♣♠❡♥t✱ ❞✐s❡❛s❡✱ ❛♥❞ r❡s♣♦♥s❡ t♦
∗