Equivariant HPD credible sets and MAP estimators

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https://hal.inria.fr/inria-00077906v2

Submitted on 6 Jun 2006

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Equivariant HPD credible sets and MAP estimators

Pierre Druilhet, Jean-Michel Marin

To cite this version:

Pierre Druilhet, Jean-Michel Marin. Equivariant HPD credible sets and MAP estimators. Bayesian Analysis, International Society for Bayesian Analysis, 2007, 2 (4), pp.681-692. �inria-00077906v2�

a p p o r t

d e r e c h e r c h e

ISSN0249-6399ISRNINRIA/RR--5923--FR+ENG

Thème COG

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Equivariant HPD credible sets and MAP estimators

Pierre Druilhet — Jean-Michel Marin

N° 5921

Juin 2006

Unité de recherche INRIA Futurs Parc Club Orsay Université, ZAC des Vignes, 4, rue Jacques Monod, 91893 ORSAY Cedex (France)

Téléphone : +33 1 72 92 59 00 — Télécopie : +33 1 60 19 66 08

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θ∈Θ

πν(θ|x) = Argmax

θ∈Θ

πλ(θ|x) g(θ)

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MAP = MAPλ

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0<γ<1

HPD^γ(θ). ^{# .%}

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θ:θ^x(1−θ)^1−x≥kγ .

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α= log(θ/(1−θ)) = logit(θ)^® HPD(α) =

α: e^(x+1)α (1 +e^α)³ ≥k_γ⁰

.

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HPDν(θ)^®lž9_

HPDν(θ) ={θ:πν(θ|x)≥kγ}. ^{# %}

º±k˜bdR bdRlkxa°jUgibnzObnk˜gCj«®

HPD(θ) = HPDλ(θ)^ª fgibdWbdRlztb

HPDν

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–=—

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θ ^ztjl} α^¯RUW:j x= 0

jlgtbdkxjUhbdRlztbSbdRUWeRUgCkxe!Wgio/z°}`giVkxjlzObnk˜jUhVW:zCacˆlrdWkxaSjUgibf}`k˜rnW:ebdwx_ŸegCjUjUW:ebdW;}¯k˜bdRbdRUWeRUgCkxe!Wgio

z{Urnk˜gCr~}Ukxacbdrnk˜žlˆ`bdkxgij£¯RUk¦eR»egCrdrnW:ad{|gijl}Uavbdg²z{Urdkxgir2¨HjUgO¯wxW:}`hCW(gCj=bdRlW°{JztrztVWbnW!r;ª¬­j›]HW;ebnk˜gCj

„`®U¯³W}`k¦aneˆlana³bdRUWkxV{Uw˜k¦e!ztbdkxgijgto acˆJeRzeRUgik¦eW~oµgCrSbdRUW}`gCV°kxjlztbdkxjUhV°W;ziadˆUrnWiª¬­j²]`W:ebnk˜gCj l®`¯³W

zC}Uzt{UbSgCˆUrfzt{l{Urdg9zieR™bngbdRUW}UW!wxkxe:zObdWe:ziadW~gio&Vg`}`W!w¦a¯k˜bdR²jHˆUk¦adzijleW2{lzirnziV°W!bdW:rna:ª

),+- *

¬­j bdRlkxa™adW:ebdkxgij«®³¯³W‰zianadˆUVWbnRUW²ˆlacˆJztwrnW!hiˆlwxzirdk˜bu_¡egijJ}`kmbnk˜gCjla°gij bdRUW²Vg`}`W!whCk˜§CW!j ž9_

f(x|θ)

adg‰bnRlzObbnRUW//k¦acRlW!r(kxj`oµgCrdV™zObnk˜gCj

I(θ) k¦a(¯%W!wxw³}UW'ljUW;}.ª=º£Wziwxadg=zianacˆlV°WŸbdRlztb

I(θ) k¦a{|gCadk˜bdkx§iW }UW'ljUk˜bdWoµgirW:zieR

θ ∈ Θª‰QSRUW†iW!Å|rnW!_`aVW:zCacˆlrdW™oµgCr

θ®F}`W:jUgtbnW:}›žH_

Jθ

®/k¦abnRUWVW;ziadˆUrdW™¯k˜bdR

}UW!jladkmbu_

|I(θ)|^{12 ¯ªr;ªb;ª} λ^ª º£W}`W:jUgtbnWvžH_

JMAP(θ) = MAPJθ(θ)^{bdRUW –=— q} giž`bztkxjUW:}™žH_°bnzi¨Hk˜jUh bnRUWv†CWÅÀrdW:_`a VW:ziadˆUrnWziaF}`gCVk˜jlztbdkxjUhVW:zCacˆlrdWCª/]HkxVk˜w¦ztrnw˜_C®i¯%Wv}`W:jUgtbnWž9_

JHPD(θ) = HPDJθ(θ)

bnRUWšfq³œ€rnW!hikxgij¯kmbnRbdRUW†CWÅÀrdW:_`a³VW;ziadˆUrdWzCa}`giVkxjlzObnk˜jUhVW:zCacˆlrdWCª/º£W2Rlz¥§iW.-

JMAP(θ) = Argmax

θ∈Θ

f(x|θ)|I(θ)|⁻¹² πλ(θ), ^{#@‚ %}

zijl}

JHPD(θ) ={θ : f(x|θ)|I(θ)|⁻¹² πλ(θ)≥kγ}. ^#p‹.%

QSRUW 'lracb%Vgtbnk˜§OztbdkxgijŸoµgCrSˆlackxjUh†Cšvqœ~afztjl}†

–¸—

qaSk¦a³bdRlztb%bnRUW!_™wxW:zC}ŸbngW;©9ˆUk˜§Ozirdk¦ztj9b³kxj`oµW:rc·

W:jleW2ˆUjJ}`W!r}`k˜Å|W:rdW:j9bdk¦ztžUwxW2rdW:{lztrztVWbnrdkx¢:ztbdkxgij«ª/ygiˆJadadW:ziˆztjJ}yfgiž|W!rdb #@„tŠiŠ9‚%³žUrnk˜W l_egCjladkx}`W:r%z

adkxV°kxw¦ztr2kx}UW:z²kxj£z‰}`k¦aneˆlanackxgij»gij›z{lzt{|W!rgio

Ä

W!rnjlzirn}`g #7„iŠiŠ9‚%ªX³gijJack¦}`W!r2jUgO¯ z²}`kmÅÀW!rnW!j9bnkxzižUw˜W

rnW!{JztrztVWbnrdkx¢:ztbdkxgij

α=h(θ)®lbdRUW({Jg9aubnW!rnk˜gCrS}UW!jladkmbu_gto

α^¯ªr:ªb:ª bngbdRUW†CWÅÀrnW!_`aSVW:ziadˆUrnW~oµgir

α^®

}UW!jUgibdW:}žH_

Jα

®Uk¦a(-

πJα(α|x) = πλ h⁻¹(α)|x _dα^d h⁻¹(α)

|I(h⁻¹(α))|⁻¹²

_dα^d h⁻¹(α)

=πJθ h⁻¹(α)|x

. ^{# %}

Í&ÍÏ67891:

/lrdgCV©|ª5# %®i¯%WSgiž`bztkxj°bdRUWSoµˆUjJebdkxgijJztwUW;©Cˆlk˜§Oztrnk¦ztjle!W%{lrdgC{JW:rcbnk˜W;agtolbnRUWv† –=— q¡zijl}bdRUWf†9šfqœ -

JMAP(h(θ)) = Argmax

α∈h(Θ)

πJα(α|x)

= Argmax

α∈h(Θ)

πJθ h⁻¹(α)|x

= h(JMAP(θ)). ^{# %}

zijl}

JHPD(α) = {α:πJα(α|x)≥kγ}

=

α:πJθ h⁻¹(α)|x

≥kγ

= h(JHPD(θ)). ^{#pƒ %}

¹ WbˆlaegCjlack¦}`W:r2bdRUWŸjlgirnVziw W¤UztV{UwxW™gtofadW:ebdkxgij …t®

X|θ ∼ N(θ, σ²) ^zijl} θ ∼ N(µ, τ²)^ªº»W

RJz¥§iW

I(θ)∝1 ^ztjl} JMAP(θ) = MAPλ(θ)^{ª /Ugir} α= exp(θ)®$¯%W™e!zij£}`W:rdkx§iW

JMAP(α) ^oµrngiV JMAP(θ)^žH_

JMAP(α) =e^JMAP(θ).

X³gCjladkx}`W:r2jUgO¯€bnRUW Ä W!rnjUgiˆUwxwxk W¤UztV{UwxWgtof]HW:ebdkxgij …i®

X|θ ∼ B(1, θ) ^zijl} θ ∼ U^]0,1[ª™º£W™Rlz¥§CW

I(θ) = 1/(θ(1−θ)) ^zijl}

JHPD(θ) =n

θ:θ^x+1/2(1−θ)^1−x+1/2≥kγ

o.

/lgir

α=h(θ) = log(θ/(1−θ))®U¯%WRlz¥§CW

JHPD(α) =

α: exp((x+ 1/2)α) (1 + exp(α))² ≥kγ

= h(JHPD(θ)).

//kxhiˆUrnW:a „v{UrnW:adW!j9bna$bdRUW{|gCacbdW:rdkxgir}`W:jlack˜bdkxW:a/gio

θ^ztjl} α^¯ªr;ªb;ª«bnRUW†iW!Å|rnW!_`a}`giVkxjlzObnk˜jlhVW:zCacˆlrdW;a!ª X³gCj9bdrztrn_bngbdRUW

¹

W!ž|W:adhiˆUW}`gCVk˜jlztbdkxjUhVW:ziadˆUrnW:aSe!zCacWC®`¯³W2gCž`bnzik˜jbu¯³gi·Ìadk¦}`W:}rdW:hikxgijla:ª

QSRUWSgibdRUW:r/Vgibdkx§¥ztbdkxgij(oµgir/bdRUWeRUgCkxe!Wgio

Jθ

zia}`gCVk˜jlztbdkxjUh~V°W;ziadˆUrnWk¦a/bdRlztb&bnRUW†iW!Å|rnW!_`a/VW:zO·

adˆUrnW(k¦a2ze!wxzCadadkxe:ztw/jUgijU·@kxj`oµgirnV™zObnk˜§CW({Urnk˜gCrvoµgCr

θ #p†CWÅÀrnW!_`a® …;ƒi‹U…zCadaSztjJ}º›zianadW!rnVzijÀ® …:ƒiƒC‹%ª yfW:e!ziw˜w$bdRlztb:®À{UrngO§Hkx}`kxjUhbnRUW!rnWzirdWjlgŸjHˆUkxanztjJeW{lzirnziVWbdW:rna:® Ä W:rdjlzirn}Ug #c…:ƒ Oƒ %vacRUgO¯%W:}bnRlzObbdRUW

†CWÅÀrnW!_`aF{UrnkxgirS}`kxacbdrnkxžUˆ`bdkxgijV°kxjUkxVk˜¢:W:a³bdRUW2ziad_HV{`bdgibdk¦efW¤`{|W:ebdW:}2ˆUw˜wxžlzCe¨C· ¹ W:k˜žlw˜W:r³}`k¦aubztjle!Wvž|W·

bu¯%W!W:j²bdRUW({Urdkxgirztjl}bnRUW({|gCacbdW:rdkxgirf}`k¦acbdrnk˜žUˆUbdkxgijla:ª+vadk˜jlhŸgiˆUrzt{U{UrngCzCeR«®Uk˜o jUgŸ{Urnk˜gCrf¨HjUgO¯wxW:}`hCW

k¦a(z¥§Ozik˜w¦ztžUwxW™gij

θ ztjl}¡k˜of¯%Wzie:eW!{UbbdRUW†iW!Å|rnW!_`aVW;ziadˆUrdWziajUgCjUkxj`oµgirnV™zObdkx§iW{Urdkxgir;®&bdRUW:j¡bdRUW

† –=— q k¦aW;©CˆJztw.bngbdRUW2oµrnW:©9ˆUW:jCbnkxacb –=¹ ¯RlztbdW:§iW!rSbnRUW{lzirnziVWbdrnkx¢:zObnk˜gCjk¦a!®l{UrdgO§Hk¦}`W:}bdRlW3/&k¦adRUW!r

kxj`oµgCrdV™zObnk˜gCjk¦av}UW'ljUW;}.ª – girnW!gO§iW:r:®Hkxj²bdRUk¦ave:ziadWi®l†9šfq³œae!ztjbnRUW!j²ž|WbdRUgCˆUhiR9bvzCafernW:}`kxžUwxWacW!bna

ztrngiˆUjJ} bdRUW –=¹ ®J¯RUW:rdW(bdRUWbdW:rdV zirdgCˆUjl}²V™z¥_²žJW°ztžUˆladkx§iW(kxj¸e:ziadW(gtoFVˆUw˜bdkxV°g`}Uziw&{|gCacbdW:rdkxgir

}Ukxacbdrnk˜žlˆ`bdkxgij #µVˆUwmbnk˜Vg`}Uziw«kxav}`W'ljlW:}¯ªr;ªb:ª/bdRUW†CWÅÀrnW!_`aSVW:ziadˆUrnW1%ª

ʵÇ/Í«Ê É

0.0 0.2 0.4 0.6 0.8 1.0

0.000.050.100.150.200.250.30

θ

−10 −5 0 5 10

0.000.050.100.150.200.250.30

α

/&kxhiˆlrdW„- qgCacbdW!rnkxgir}`W!jJack˜bdkxW:aSgio

θ ^ztjJ} α^¯ªr:ªb:ªbnRUW†CWÅÀrdW:_HaSVW;ziadˆUrdW~¯RUW:j

x= 0

),3™ !"' °"& % / (01%/S

¬­j»bdRlkxaadW:ebdkxgij«®$¯³W™}`k¦ade!ˆlana~adW!§CW!rztwVWbnRUg`}Ua~bdg‰}`W:rdkx§iW™ztj£W:©9ˆUkx§OztwxW!j9bgtoS† –=— qaztjJ}»†9šfqœ~a

¯RlW!j™jHˆUk¦adzijleWf{JztrztVWbnW!rFztrnWf{UrnW:adW!j9bFkxj™bdRlWvVg`}`W!w@ªº»WvzCadadˆUVWbnRlzObFbnRUWv{JztrztVWbnW!r

θk¦a³ad{Uwxkmb kxj9bdgbu¯³g{lztrdbna -

θ= (θ1, θ2)∈Θ1⊗Θ2

¯RUW!rnW

θ1

kxabdRlW{lzirnziVWbdW:rvgiok˜j9bnW!rnW:acb~ztjJ}

θ2

k¦abdRUW

jHˆUk¦anztjle!Wiª²º»W}`W!jlgtbdWž9_

πν(θ1|x) bdRUW}`W!jJack˜bu_»giobdRUWVzirdhCk˜jJztwF{|gCacbdW:rdkxgir(}Ukxacbdrnk˜žlˆ`bdkxgij¡gio

θ1

¯ªr;ªb;ªbdRUWVW:zCacˆlrdW

νªFQSRUWe!girnrdW;ac{|gijl}Uk˜jUh –=— q

ν

ztjl}²šfq³œ

ν

zirdW

MAPν(θ1) = Argmax

θ1∈Θ1

πν(θ1|x) ^#c…:Š.%

ztjJ}

HPDν(θ1) ={θ1:πν(θ1|x)≥kγ} ^#c…i…1%

Ä W;e!ztˆJacW~bdRlW†iW!Å|rnW!_`a{Urnkxgirf}`gHW:afjUgtbv}`k¦acbdkxjUhiˆUk¦adRž|Wbu¯%W!W:j²{lztrztVWbnW!rgio/kxj9bdW!rnW:acbvztjJ}jHˆUk¦anztjle!W

{JztrztVWbnW!r;® Ä W:rdjlzirn}Ug #u…;ƒ Oƒ%({Urngi{|gCadWz»jUW!¯'zi{U{UrngCzieR¶e:ztwxw˜W;}¡rnWoµW!rnW!jJeW{lrdkxgirzt{U{lrdg9zieR«ª»¬­j

bnRUk¦aS]`W:ebnk˜gCj«®H¯³W2adRUgO¯±RUgO¯±¯%W~e:ztjŸˆladW:}ŸbdRUk¦aSrnWoµW!rnW!jJeWv{lrdkxgir%ziaS}`giVkxjlzObnk˜jlh(VW:zCacˆUrnWfbng°}UW'ljUW

W;©9ˆUk˜§Ozirdk¦ztj9b –=— qa2ztjl}=švqœ~a!®.e!ziw˜wxW:}=ž9_²ztjJztwxgihi_¯kmbnR£]HW;ebnk˜gCj¸„`®À† –=— qa2ztjl}‰†9šfq³œa:ª~Q&¯%g

e:ziadW:aŸzirdW»egijJack¦}`W!rnW:} -=bdRlW»e!zCacW¸¯RUW!rnW‰bdRlW!rnW=kxaegCjl}`k˜bdkxgijlziwadˆUž`suW;ebdkx§iW¸k˜j`oµgCrdV™ztbdkxgij oµgirbdRUW

jHˆUk¦anztjle!W~{JztrztVWbnW!rztjl}bdRlWe!ziadW2¯RUW!rnW~bdRUW:rdW2k¦aSjUgijlWiª

! "$#%&' "$(*),+%-. /-01(23(*)(4%)

]`ˆU{U{|gCadW=bdRJzOb‰z adˆUž`suW:ebdkx§iW¸e!gijl}Ukmbnk˜gCjlztw~{Urnk˜gCrk¦az¥§Oztkxw¦ztžUwxW‰oµgir

θ2

hCk˜§CW!j

θ1

ª º£W£}`W:jUgtbnW¸žH_

πλ(θ2|θ1) kmba°}UW!jladkmbu_›¯ªr:ªb:ª

몣¬­j¶bdRUk¦ae!ziadWi®F]Hˆlj²zijl} Ä W!rnhiW!r#u…:ƒCƒ %({Urngi{|gCadW™bu¯³g£}`k˜Å|W:rdW:jCb zi{U{UrngCzCeRUW:avbdg=}`W:rdkx§iW°rnWoµW!rnW!jJeW{Urnk˜gCrnaoµgir

θ1

ª™º»W™VkxVkxe°bnRUW!kxr(zt{l{Urdg9zieRUW;azijl}»bdRUW:rdWk¦a2bu¯%g

rnW:zCacgCjlztžlw˜Wgi{`bnk˜gCjla%oµgir 'ljJ}`k˜jlhŸz°}UgiVk˜jJzObdkxjUhVW:zCacˆUrnW2gij

Θ1

ª

56 87:9;egCjladkx}`W:rfbnRUWV™zirdhCk˜jlziw$Vg`}`W!w

f(x|θ1) =R

f(x|θ1, θ2)πλ(θ2|θ1)dθ2

ªœvW:jUgtbnW(žH_

Im(θ1)bnRUW//kxadRUW!r$k˜j`oµgCrdV™ztbdkxgij2V™zObdrnk˜¤voµgir

θ1

gCž`bnzik˜jlW:}oµrdgCVbnRUWFV™ztrnhikxjlztwOVg`}`W:w7ª — }`giVkxjlzObnk˜jUh

Í&ÍÏ67891:

VW;ziadˆUrdWSe!zijž|WSbdRlWS†CWÅÀrnW!_`aVW:ziadˆUrnW³gCj

Θ1

¯k˜bdR™}`W:jlack˜bu_¯ªr:ªb:ª

λ{Urngi{|girdbdkxgijJztw9bdg

|Im(θ1)|^1/2ª

¬­jbnRUkxae:ziadWi®HbnRUW† –¸— q±ztjl}bnRUW†Cšvqœ€zirdW}`W('ljUW:}žH_

JMAP1(θ1) = Argmax

θ1∈Θ1

πλ(θ1|x)

|Im(θ1)|^1/2

,

JHPD1(θ1) =

θ1: πλ(θ1|x)

|Im(θ1)|^1/2 ≥kγ

. JMAP1

zijl}

JHPD1

zirdWgižH§Hk˜gCˆlacwx_vW;©9ˆUk˜§Ozirdk¦ztj9b«oµgCr&z

1−1acVgHgtbnRrdW:{lztrztVW!bdrnk˜¢;zObdkxgij~gij2bnRUW%{lzO·

rztVW!bdW!r.gio9kxj9bdW:rdW;aub;ª +fjUoµgirdbdˆUjlztbdW:w˜_C®!bnRUW //kxadRUW:r.k˜j`oµgCrdV™ztbdkxgij2V™zObdrnk˜¤voµgir

Rf(x|θ1, θ2)πλ(θ2|θ1)dθ2

k¦aSgto bnW!j‰}`kŸeˆUw˜bbdgŸe!giV{Uˆ`bnWiªFQSRUk¦a}`k™e!ˆUw˜bu_Vgtbnk˜§OzObnW:a%bnRUWk˜j9bnrdg`}`ˆlebdkxgijgto ztjUgibdRUW:rgi{`bnk˜gCj«ª

56 87 ; /lgiwxw˜gO¯kxjUh Ä W!rnjlztr}`g #u…;ƒ Oƒ.%®(]HˆUj‰zijl} Ä W:rdhCW!r)#c…:ƒiƒ%²{Urngi{|gCadW£bdg±V™zO¤`kxV°kx¢!W

zCac_HV{`bngtbdk¦e!ziw˜wx_¶bnRUW¸W!¤`{JW;ebdW;} 2ˆUwxw˜žlzCe¨9·¹ W!kxžUw˜W:r}`kx§iW:rdhCW!jle!W‰žJW!bu¯³W:W!j bdRlW¸V™ztrnhikxjlztw{|gCacbdW:rdkxgir

gio θ1

zijl}bdRUW(VzirdhCk˜jJztw.{Urnk˜gCrgto

θ1

ªSQSRUkxafw˜W;zi}UaSbngbdRUW}`kxacbdrnkxžUˆ`bdkxgij²gCj

Θ1

¯kmbnR=}`W!jladk˜bu_¯ªr:ªb:ª

λ{UrdgC{JgCrcbnk˜gCjlztw|bng

exp 1

2 Z

πλ(θ2|θ1) log

|I(θ1, θ2)|

|Ic(θ2|θ1)|

dθ2

¯RlW!rnW

I(θ1, θ2) ^k¦abnRUW /&k¦adRUW!rk˜j`oµgCrdV™ztbdkxgij¶V™zObnrdk˜¤›žlziadW:}›gCj

f(x|θ1, θ2) ^ztjl} Ic(θ2|θ1) ^kxa(bdRUW

//k¦acRlW!rSk˜jUoµgirnVztbdkxgijVztbdrnkm¤ŸžJziadW:}ŸgCjbnRUW2VgH}UW!w

f(x|θ1, θ2)^¯RUW:rdW θ1

k¦aS¨HjUgO¯j«ª QSRUk¦a%kxaSW;adadW!j`·

bnkxziw˜wx_²bdRlWadgiwxˆ`bnk˜gCj¸ˆladW:}¸k˜j Ä W!rnhiW!rSztjJ} Ä W:rdjlzirn}Ug$#u…;ƒ iƒH® …;ƒiƒ9„%®«žUˆUb2RlW!rnW°zacˆlž`suW:ebdkx§iWegijJ}`km·

bnk˜gCjlztwÀ{Urnkxgir%oµgir%bdRlW~jHˆUk¦anztjle!W{lzirnziVWbdW:r³hCk˜§CW!jbdRUW2{lzirnziV°W!bdW:r%gio$kxjCbnW!rnW:acbkxa%ˆladW:}.ª º£W2{Urngi{|gCadW

bngˆJacW~bdRlW~{lrdW:§9kxgiˆJa%}Ukxacbdrnk˜žlˆ`bdkxgij²ziaS}`gCV°kxjlztbdkxjUhVW;ziadˆUrdW~gij

Θ1

ª¬­jbdRUk¦ae!zCacWC®HbdRUW†

–=—

q ztjJ}

bnRUW†9šfqœ€zirdW}`W('ljUW;}žH_

JMAP2(θ1) = Argmax

θ1∈Θ1





πλ(θ1|x) expn

1 2

Rπλ(θ2|θ1) log_|I(θ

1,θ2)|

|Ic(θ2|θ1)|

dθ2

o



,

JHPD2(θ1) =







θ1: πλ(θ1|x) expn

1 2

R πλ(θ2|θ1) log

|I(θ1,θ2)|

|Ic(θ2|θ1)|

dθ2

o ≥kγ





 . JMAP2

ztjJ}

JHPD2

ztrnW~gCžH§9kxgiˆJacwx_ŸW:©9ˆUkx§Oztrnkxzij9b%oµgCrvz

1−1 acVgHgtbnRrnW!{JztrztVWbnrdkx¢:ztbdkxgijgCjbdRUW {JztrztVWbnW!rSgto/k˜j9bdW:rdW;aub;ª

¹ Wbˆla³egCjlack¦}`W:rFzadziV°{lw˜W

X1, . . . , Xn

oµrngiV z2jlgirnVziwl}`k¦aubnrdkxžUˆ`bnk˜gCj¯kmbnRŸW¤`{|W:ebzObnk˜gCj

θ2=µ

zijl}¶acbnzijl}Uztr}¶}`W:§Hkxztbdkxgij

θ1 = σ^® bdRlW{lzirnziVWbdW:rgiofk˜j9bnW!rnW:acb:ª¶]HˆU{U{|gCadWbdRlztbbnRUWe!gijl}Ukmbnk˜gCjlztw {lrdkxgir}`k¦acbdrnk˜žUˆUbdkxgij£oµgir

µ ^hikx§iW:j σ k¦ajUgirnV™ztw ¯k˜bdR¡W!¤H{|W:ebnztbdkxgij

m ztjJ}»§Ozirdk¦ztjle!W

τ^{2ª —} {U{Uwx_Hk˜jUh {lrdgC{Jg9ack˜bdkxgij „¸gto]Hˆlj²zijl} Ä W!rnhiW!r #u…;ƒiƒ %®%Y{`bnk˜gCj…}UgiVk˜jJzObdkxjUh»V°W;ziadˆUrnWŸRlzia°}`W:jlack˜bu_¡¯ªr:ªb:ª

λ {Urngi{|girdbdkxgijJztw bng

1

σ² + σ²

(n−1)(σ²+nτ²)² 1/2

®Fzijl} Y{`bnk˜gCj „=}UgiVk˜jJzObdkxjUh=VW;ziadˆUrdWRlzia

}UW!jladkmbu_~¯ªr:ªb:ª

λ{lrdgC{JgCrcbnk˜gCjlztwObng

1/σª&QSRUW;acW bu¯%gv}`gCVk˜jlztbdkxjUhvVW;ziadˆUrdW;a«ztrnW}`k˜ÅÀW!rnW!j9b:ª/šfgO¯%W!§iW:r:®

¯RlW!j

n→ ∞®`bdRlWacW;egijJ}}`W:jladkmbu_e!gijH§iW:rdhCWˆljUkmoµgCrdVwx_™bdg°bnRUW 'lraubfgCjUWiª

ʵÇ/Í«Ê É

¹ WbˆJa(jUgO¯ e!gijladkx}UW!rbnRUWžUkx§OztrnkxztbdWŸžlk˜jUgCVkxziw³Vg`}`W!w³{Urngi{|gCadW:}›žH_¡X³rngO¯f}`W:rSzijl}²]H¯³W:Wbnk˜jUh

#c…:ƒiƒ%SztjJ}rnW!§Hk¦ack˜bdW;}žH_qgCwxadgijztjl}º›zianacW:rdV™ztj #u…;ƒiƒiŠ%-

f(x1, x2|θ1, θ2) = m

x1

θ^x₁¹(1−θ1)^m−x1 x1

x2

θ^x₂²(1−θ2)^x1−x2 I1,...,m(x1)I1,...,x1(x2).

¯RlW!rnW

IA

k¦aSbnRUWkxjl}`k¦e!zObngirSoµˆUjJebdkxgij‰gCj

A^ztjl} mk¦afacˆl{U{Jg9acW;}bdg™ž|W¨HjUgO¯j«ª³]Hˆl{U{Jg9acW2bnRlzObvbdRUW e!gijl}Ukmbnk˜gCjlztw }`k¦aubnrdkxžUˆ`bnk˜gCj¸gio

θ2

hikx§iW!j

θ1

k¦az Ä W!bnz}`kxacbdrnkxžUˆ`bdkxgij£¯k˜bdR£{lztrztVWbnW!r

a ^ztjl} b^{ª /lgir}

adˆleR²z°V°g`}`W:w7®

|I(θ1, θ2)|= (1−θ1)⁻¹θ₂⁻¹(1−θ2)^{−1 ztjJ}} |Ic(θ2|θ1)|=θ1(θ2(1−θ2))^{−1ª ¬Ìbfkxa}

§CW!rn_W:ziad_bdgadRUgO¯±bdRJzObY{`bdkxgij»…~zijl}‰Y{`bdkxgij²„(}`gCVk˜jlztbdkxjUh°V°W;ziadˆUrnW:a%ztrnWvbnRUWadziV°W2ztjJ}ŸRlz¥§CW

}UW!jladkmbu_¯ªr;ªb;ª

λ{Urngi{|girdbdkxgijlziw|bdg

θ^−1/2₁ (1−θ1)^−1/2ª

]Hˆlj²zijl} Ä W!rnhiW!r #c…:ƒiƒ%2egCjlack¦}`W:rdW;}=bnRUWŸe!zCacW°¯RUW!rnW

θ1

ztjl}

θ2

ztrnWk˜jl}UW!{|W!jl}`W:j9b:ª /lgir2bdRlkxa

gibdRUW:rv{Urnkxgirfkxj`oµgirnV™zObnk˜gCj«®l¯%W(e!zij=ztw¦acg™VkxVkxebdRUW:k˜r~zt{U{UrngCzCeRbng}`W('ljUW(z}`giVkxjlzObnk˜jUhV°W;ziadˆUrnW

gCj

Θ1

ª $ ' ! "$#$' "$(4) +$- /- ( 3(* )(4%)

¬Ìojlg adˆUž`suW:ebdkx§iW£egijJ}`kmbnk˜gCjlztw{Urnk˜gCrŸoµgir

θ2

hCk˜§CW!j

θ1

k¦az¥§¥zik˜w¦ztžlw˜WC®¯³W»{UrdgC{Jg9acW=bdg VkxVkxe¸bdRUW

rnWoµW:rdW:jleWS{Urnkxgir zt{U{lrdg9zieR(gio Ä W:rdhCW!rztjl} Ä W!rnjlztr}`g#u…;ƒ iƒU®U…:ƒCƒC„%ªQSRUk¦a wxW:zi}la/bdg2bdRlWf}`giVkxjlzObnk˜jUh

VW;ziadˆUrdW~gij

Θ1

¯k˜bdR‰}`W!jJack˜bu_Ÿ¯ªr:ªb:ª

λ{lrdgC{JgCrcbnk˜gCjlztw|bdg

exp 1

2 Z

|Ic(θ2|θ1)|^1/2log

|I(θ1, θ2)|

|Ic(θ2|θ1)|

dθ2

.

Yvo bnW!j«®|bnRUWkxj9bdW:hirztw

R|Ic(θ2|θ1)|^1/2log_|I(θ

1,θ2)|

|Ic(θ2|θ1)|

dθ2

kxajUgib~}`W('ljUW:}«ª~QSRUWegCV{lziebacˆl{U{JgCrcb

zirdhCˆUVW!j9bbdRlztbFkxa bu_9{lkxe:ztwxw˜_ˆJacW;}kxj°bdRUWrnWoµW:rdW:jleWS{UrnkxgirFzt{l{Urdg9zieR #Ä W!rnhiW:rSzijl} Ä W!rnjlztr}`gH®l…;ƒiƒC„%

V™z¥_¸bdRUW:j¡žJWzi{U{Uwxk˜W;}›RUW!rnWiª‰X³RUgHgCadWŸz²jUW:acbdW;}¡acW;©9ˆUW!jle!W

Θ1 ⊂ Θ2 ⊂ . . . gtoegCV{lzieb(acˆUžJacW!bna gio³bnRUWŸ{lzirnziVWbdW:rac{lzCeW

Θ acˆJeR»bnRlzOb

∪ⁱΘi = Θ ^ztjJ} |Ic(θ2|θ1)|^{1/2 RJzia} 'ljlkmbnWŸV™ziana2gij

Ωi = {θ2; (θ1, θ2)∈Θi}oµgirfztwxw

θ1

ª ¹ Wb

Ki(θ1) =R

Ωi|Ic(θ2|θ1)|^1/2dθ2

ztjl}

πi(θ1) = exp 1

2 Z

|Ic(θ2|θ1)|^1/2log

|I(θ1, θ2)|

|Ic(θ2|θ1)|

dθ2

.

QSRlW2}UgiVk˜jJzObdkxjUhVW:zCacˆUrnW2gij

Θ1

Rlzia%bnRUW!j‰}`W:jladkmbu_Ÿ¯ªr;ªb;ª

λ{Urngi{|girdbdkxgijlziwJbng

i→∞lim

Ki(θ1)πi(θ1) Ki(θ⁽⁰⁾₁ )πi(θ₁⁽⁰⁾)

¯RlW!rnW

θ⁽⁰⁾₁ kxazijH_'U¤`W:}{JgCk˜j9bkxj

Θ1

ªFœztbcbnzzijl}‰Z~RUgCadR #c…:ƒCƒi‹%%RlzCa³W;aubztžUwxk¦acRUW;}ŸbnRUW2k˜jH§Oztrnk¦ztjle!W

gioÀbdRUk¦aF{Urng`eW;}`ˆUrnWˆUjl}`W:r

1−1acVgHgtbnR™rdW:{lztrztVW!bdrnk˜¢;zObdkxgijgij

θ1

ªQSRlW!rnWoµgirnWi®ibdRUWegirnrnW:ad{JgCjl}`kxjUh

†

–=—

q\zijl}›†Cšvqœ zirdWW;©Cˆlk˜§Oztrnk¦ztj9b~ˆljl}`W!rz‰acVgHgtbnR»rnW!{lzirnziVWbdrnkx¢:zObnk˜gCj¸gCj¸bnRUWŸ{lzirnziV°W!bdW:r~gio

kxj9bdW:rdW;aub;ª

Í&ÍÏ67891:

¹ Wbfˆlae!gijladkx}UW!rzth9ztkxjzadziV{Uw˜W

X1, . . . , Xn

oµrngiVzjlgirnVziw.}`kxacbdrnkxžUˆ`bdkxgij²¯k˜bdRW¤`{|W:ebzObnk˜gCj

µ ztjJ}£acbnzijl}Uzirn}»}`W!§Hk¦zObdkxgij

σ®&¯RUW!rnW

σ k¦a2bnRUW™{lztrztVW!bdW!rgio³kxj9bdW!rnW:acb:ª — {U{Uwx_Hk˜jlh²bdRUW™{UrnW!§Hkxgiˆla {lrdg`eW;}`ˆUrnWi®HbnRUW}`giVkxjlzObnk˜jlh°VW;ziadˆUrdW2gCj

σ Rlzia}UW!jladkmbu_¯ªr;ªb:ª

λ{lrdgC{JgCrcbnk˜gCjlztwJbdg

1/σ®`¯RUk¦eRk¦a bnRUW%k˜jH§Oztrnkxzij9b$VW:ziadˆUrnWFoµgCr/ane!ziw˜WFVg`}`W:wxa:ª&º£WSzianacˆUVW³jUgO¯£bdRlztb/jUg~{Urnk˜gCr$kxj`oµgirnV™zObnk˜gCjkxa/z¥§OztkxwxzižUwxW

gCj σªv]Hgl®lbdRUW(jUgij`·Ìkxj`oµgirnV™zObdkx§iWrnWoµW:rdW:jleW{lrdkxgirfk¦afhCk˜§CW!j²žH_

πλ(σ) = 1/σ^ª ©9ˆUkx§OztwxW!j9bdwx_i®|kmo bnRUW {JztrztVWbnW!rSgtok˜j9bnW!rnW:acbfkxa

σ²®UbnRUWrdW!oµW!rnW!jle!W2{Urdkxgirfzijl}²}`giVkxjlzObnk˜jlh°VW;ziadˆUrdW2Rlz¥§CW}`W!jladk˜bu_¯ªr:ªb:ª

¹

W!ž|W:adhiˆUW™{lrdgC{JgCrcbnk˜gCjlztw/bdg

1 σ²

®/¯RUkxeR¡k¦azihCztkxj»bdRUWe!girnrdW;ac{|gijJ}`k˜jlhbnRUWŸkxj9§Ozirdk¦ztj9bVW:ziadˆUrnWiª¬­j

bnRlzObŸe:ziadW

JMAP(σ²) kxaW;©CˆJztwSbdg£bnRUWoµrdW;©CˆlW!j9bdk¦aubŸy

–=¹

#py ³ack¦}`ˆlziw

–

zO¤`kxV(ˆUV

¹

k˜¨CW!wxk˜Rlg9g`}%

W;aubnk˜V™ztbdgir;ªQSRUk¦ak¦a~zjUW:¯€kxjCbnW!rn{UrnWbnztbdkxgij=gioFbdRlW°y

–=¹

W:acbdkxVztbdgCrv¯RUk¦eR»ziwxadge!girnrdW;ac{|gijl}labdg

bnRUW –=— qˆUjl}`W:rbdRUW ¹ zt{lwxzCeW{Urnk˜gCroµgCr

σ² #7švztrn§Hk˜wxwxWt®…;ƒ %ª Ä _¸W;©Cˆlk˜§Oztrnk¦ztjle!W°gio³bnRUWŸ† –=— q®

¯%W(RJz¥§iWi®

JMAP(σ) =√

REMLªv¬Ìo³¯³W°eRlztjUhCW

σ ^kxjCbng log(σ)®ÀbdRlW!j=bnRUW(rnWoµW:rdW:jleW{Urnk˜gCrvk¦afbnRUW

¹ W!ž|W:adhiˆUWvV°W;ziadˆUrnWiª/¬­j™bdRlztb%e:ziadWi®

JMAP(log(σ)) = MAP(log(σ)) = ¹₂log(REML)^ª vgtbnWfbdRJzOb bnRUW:adW2rdW;acˆlwmbae!ztj²ž|W2W¤HbdW:jl}`W:}bng™V°gCrdW~hiW:jUW!rztw.§Ozirdk¦ztjle!W~e!giV{|gijUW:jCbaSVgH}UW!w¦a!ª

3³ "$!3

QSRlW†

–=—

qaztjl}£†9šfqœ~a{Urngi{|gCadW:}¸k˜j»bdRUk¦a{lzi{JW:r2hikx§iWzadkxV°{lw˜W™ztjJ}»e!giRUW:rdW:jCbztw˜bdW:rdjJzObdkx§iWbdg

bnRUWfˆladˆlziw

–=—

q³aztjJ}°švqœ~a!®Hz¥§igCkx}`kxjUh2{|W:eˆlw˜k¦ztrFž|W!RJz¥§9kxgiˆlr ˆUjl}`W:rFrdW:{lztrztVW!bdrnk˜¢;zObdkxgij$ª$šfgO¯%W!§CW!r;®

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+ ™3' (01&"

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Ä

z¥_CW:adkxzij=zt{U{lrdg9zieR

gio&§Oztrnk¦ztjle!W2e!giV{|gijUW:jCbVg`}`W!w¦a:ª

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W:}Ukmbnk˜gCj«ª

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