Behavior of the Gibbs Sampler When Conditional Distributions Are Potentially Incompatible

Transcription

1 Behvio of the Gibbs Smple When Conditionl Distibutions Ae Potentilly Incomptible Shyh-Huei Chen Deptment of Biosttisticl Sciences, Wke Foest Univesity School of Medicine, Winston-Slem, NC 757 Edwd H. Ip Deptment of Biosttisticl Sciences, Deptment of Socil Sciences nd Helth Policy, Wke Foest Univesity School of Medicine, Winston-Slem, NC 757 Shyh-Huei Chen (E-mil: is ssistnt pofesso t the Deptment of Biosttisticl Sciences, Division of Public Helth Sciences, Wke Foest School of Medicine, Wells Fgo Cente 3d floo, Medicl Cente Blvd, Winston-Slem, NC 757. Edwd H. Ip (E-mil: eip@wkehelth.edu) is Pofesso t the Deptment of Biosttisticl Sciences nd the Deptment of Socil Sciences nd Helth Policy, Division of Public Helth Sciences, Wke Foest School of Medicine, Wells Fgo Cente 3d floo, Medicl Cente Blvd, Winston-Slem, NC 757. This wok ws ptilly suppoted by NIH gnt RAG (PI: Ip).

2 ABSTRACT The Gibbs smple hs been used extensively in the sttistics litetue. It elies on itetively smpling fom set comptible conditionl distibutions nd the smple is known to convege to unique invint joint distibution. Howeve, the Gibbs smple behves the diffeently when the conditionl distibutions e not comptible. Such pplictions hve seen incesing use in es such s multiple imputtion. In this ppe, we demonstte tht wht Gibbs smple conveges to is function of the ode of the smpling scheme. Besides poviding empiicl exmples to illustte this behvio, we lso explin how tht hppens though thoough nlysis of the exmples. KEY WORDS: Gibbs chin; Gibbs smple; Potentilly incomptible conditionlspecified distibution.

3 . INTRDUCTION The Gibbs smple is, if not singully, one of the most pominent Mkov chin Monte Clo (MCMC)-bsed methods. Ptly becuse of its conceptul simplicity nd elegnce in implementtion, the Gibbs smple hs been incesingly used coss vey bod nge of subject es including bioinfomtics nd sptil nlysis. While its oot dtes bck to elie wok (e.g., Hsting 970), the populity of the Gibbs smpling is commonly cedited to Gemn nd Gemn (984), in which the lgoithm ws used s tool fo imge pocessing. Its use in sttistics, especilly Byesin nlysis, hs since gown vey pidly (Gelfnd nd Smith 990; Smith nd Robets 993; Gilks, Richdson, nd Spiegelhlte 996). Fo quick intoduction of the lgoithm, see Csell nd Geoge (99). One of the ecent developments of the Gibbs smple is in its ppliction to potentilly incomptible conditionl-specified distibutions (PICSD). When sttisticl models involve high-dimensionl dt, it is often esie to specify conditionl distibutions insted of the entie joint distibutions. Howeve, the ppoch of specifying conditionl distibution hs the isk of not foming comptible joint model. Conside system of d discete ndom vibles, { } c is specified by F = f, f,, fd, whee fk f c = f ( x ) nd k xk whose fully conditionl model c xk is the eltive complement of x with espect to X. If the conditionl models e individully specified, then thee my not exist joint distibution tht will give ise to the specified set of conditionl distibutions. In such cse, we cll F incomptible. The study of PICSD is closely elted to the Gibbs smple becuse the ltte elies on itetively dwing smples fom the Mkov chin conveges to the desied joint distibution, if F to fom Mkov chin. Unde mild conditions, F is comptible. Howeve, if F is not comptible, then the Gibbs smple could exhibit etic behvio (e.g., Hobet nd Csell 998). k X = x x x d {,,, } In this ppe, ou gol is to demonstte the behvio of the Gibbs smple (o the pseudo Gibbs smple s it is not tue Gibbs smple in the tditionl sense of xk xk 3

4 pesumed comptible conditionl distibutions) fo PICSD. By using sevel simple exmples, we show mthemticlly tht wht Gibbs smple conveges to is function of the ode of the smpling scheme in the Gibbs smple. Futhemoe, we show tht if we follow ndom ode in smpling conditionl distibutions t ech itetion i.e., using ndom-scn Gibbs smple (Liu, Wong, nd Kong 995) then the Gibbs smpling will led to mixtue of the joint distibutions fomed by ech combintion of fixed-ode (o moe fomlly, fixed-scn) when d = but the esult is not tue when d >. This esult is efinement of conjectue put fowd in Liu (996). The demonsttion in this ppe is intended to povide edes not fmili with incomptible conditionl distibutions some bsic bckgound egding the mechnism diving the behvio of the Gibbs smple fo PICSD. Two ecent developments in the sttisticl nd mchine-lening litetue undescoe the impotnce of the cuent wok. The fist is in the ppliction of the Gibbs smple to dependency netwok, which is type of genelized gphicl model specified by conditionl pobbility distibutions (Heckemn et l. 000). One ppoch to lening dependency netwok is to fist specify individul conditionl models nd then pply (pseudo) Gibbs smple to estimte the joint model. The uthos cknowledged the possibility of incomptible conditionl models but gued tht when the smple size is lge, the degee of incomptibility will not be substntil nd the Gibbs smple is still pplicble. Yet nothe exmple is the use of the fully conditionl specifiction fo multiple imputtion of missing dt (vn Buen et l. 999, 006). The method, which is lso clled multiple imputtion by chined equtions (MICE), mkes use of Gibbs smple o othe MCMCbsed methods tht opete on set of conditionlly specified models. Fo ech vible with missing vlue, n imputed vlue is ceted unde n individul conditionlegession model. This kind of pocedue ws viewed s combining the best fetues of mny cuently vilble multiple imputtion ppoches (Rubin 003). Due to its flexibility ove comptible multivite-imputtion models (Schfe 997) nd bility to hndle diffeent vible types (continuous, biny, nd ctegoicl) the MICE hs gined cceptnce fo its pcticl tetment of missing dt, especilly in high-dimensionl dt sets (Rssle, Rubin, nd Zell 008). Popul s it is, the MICE hs the limittion of potentilly encounteing incomptible conditionl-egession models nd it hs been 4

5 shown tht n incomptible imputtion model cn led to bised estimtes fom imputed dt (Desche nd Rssle 008). So f, vey little theoy hs been developed in suppoting the use of MICE (White, Royston, nd Wood 0). A bette undestnding of the theoeticl popeties of pplying the Gibbs smple to PICSD could led to impotnt efinements of these imputtion methods in pctice. The ticle is ognized s follows: Fist, we povide bsic bckgound to the Gibbs chin nd Gibbs smple nd define the scn ode of Gibbs smple. Section 3 descibes simple exmple to demonstte the convegence behvio of Gibbs smple s function of scn ode, both by pplying mtix lgeb to the tnsition kenel s well s using MCMC-bsed computtion. In Section 4, we offe sevel nlytic esults concening the sttiony distibutions of the Gibbs smple unde diffeent scn pttens nd counte-exmple to sumise bout the Gibbs smple unde ndom ode of scn ptten. Finlly in Section 5 we povide bief discussion.. GIBBS CHAIN AND GIBBS SAMPLER Continuing the nottion in the pevious section, let = (,,, ) d denote T pemuttion of {,,, d}, x = (x,x,,x ) denote eliztion of X with { } x,,, C, whee C is the numbe of ctegoies of the k th k vible. Thus, k k x (x,x,,x ) is eliztion of X defined in the ode of. Fo specified F, the ssocited fixed (systemic)-scn Gibbs chin govened by scn ptten = ( cn be implemented s follows:,,, d ) d (0) (0) (0) (0). Pick n bity stting vecto x = (x,x,,x ). d. On the t th cycle, successively dw fom the full conditionl distibutions ccoding to scn ptten s follows: 5

6 . The seies (0) () () ( s) x, x, x,, x, obtined by single dw (itetion) is clled eliztion of Gibbs chin defined by F with scn ptten ; nd the seies x, x, x,, x, obtined by single cycle is eliztion of the ssocited (0) ( d ) ( d ) ( td ) Gibbs smple. Fo exmple, X = ( x, x, x, x ), = (,4,,3) 3 4, nd given initil vlue x = (x,x,x,x ), the Gibbs chin in cycle pefoms the following dws nd (0) (0) (0) (0) (0) 4 3 poduces the coesponding sttes: x f ( x x = x, x = x, x = x ), () (0) (0) (0) x f ( x x = x, x = x, x = x ), () () (0) (0) x = (x,x,x,x ) ; () () (0) (0) (0) 4 3 x = (x,x,x,x ) ; () () () (0) (0) 4 3 x f ( x x = x, x = x, x = x ), (3) () () (0) (3) () () (3) (0) x = (x,x 4,x,x 3 ) ; nd x f ( x x = x, x = x, x = x ), (4) () () (3) x = (x,x,x, x ). (4) () () (3) (4) 4 3 In this exmple, the seies F with scn ptten. (0) (4) (8) x, x, x,, is the eliztion of Gibbs smple defined by We cn lso expess Gibbs smple of ndom scn ode s Gibbs chin. Let = (,,, d ) be the set of selection pobbilities, whee k > 0 is the pobbility of visiting conditionl, nd =. The ndom-scn Gibbs smple (Levine nd Csell 006) cn be stted s follows: f k. Pick n bity stting vecto x. At the s th itetion, s =,, d k k= = (x,x,,x d ). (0) (0) (0) (0) 6

7 . Rndomly choose k {,,, d} with pobbility k ; ( s) ( s ) b. x k f ( xk x c ) k. 3. Repet step until convegence citeion is eched. 3. ILLUSTRATIVE EXAMPLES Exmple. (Comptible conditionl distibutions). Conside the following bivite joint distibution nd fo ( X, X ) defined on the domin {, }, with its π coesponding conditionl distibutions f nd (Anold, Cstillo, nd ( x x ) f( x x) Sbi 00, p. 4): π = 3 4, nd f =, f 3 = Thee e 4 possible sttes, (, ), (, ), (, ), nd (, ) fo the Gibbs chin. The tnsition fom one stte to nothe is ppently govened by the conditionl mtices f nd. As shothnd, we denote n enty in the mtix s f (.,.) ; e.g., f (,) = / 3. In f ( t) ode to keep tck of the scn ode, we denote the stte in the Gibbs chin s ( x ), ( t) if the cuent stte x t time t is the esult of dwing fom the conditionl. To fix ides, we use fixed-scn Gibbs smple with = (,) nd the conditionl distibutions f k f k ( f, f ). The tnsition kenel fo the Gibbs chin is digmmticlly epesented in Figue, whee nd P indicte locl tnsition pobbilities. Fo exmple, the locl P ( t) (t+ ) tnsition pobbility fom ( x = (,) f) to ( x = (,) f) is f (,) = 4, nd = ( t) ( x (,) f) to = (t+ ) ( x (,) f) is 0 (indicted by disconnectedness). 7

8 Figue. Tnsition pobbilities of the Gibbs chin in Exmple. By nging the stte in lexicogphicl ode such tht the fist index chnges the fst nd the lst index the slowest, the tnsition pobbility mtices T nd T tht coespond espectively to P, nd P e: T nd = T = Moe genelly, the locl tnsition pobbility (Mds 00, p. 77) fo two ( s ) ( s) successive sttes of Gibbs chin, ( x ) nd ( x ), cn be defined by f k P k f k = P ( x, x ) = k 0, othewise. ( s) ( s) ( s ) f ( ), if c c ; ( s ) ( s) x x x k k k The mtices nd T in Exmple hve two pis of identicl ows nd e idempotent but not ieducible. As this exmple illusttes, genelly Gibbs chin is not homogeneous, but if one defines suogte tnsition pobbility mtix T = T T T T, then homogeneous chin with tnsition mtix T cn be fomed fo 3 d T 8

9 the scn ptten = (,, ). In othe wods, fo collection of full conditionl distibutions F nd scn ptten, the fixed-scn Gibbs smple is homogeneous Mkov chin with tnsition mtix d T. Anlogously, ndom-scn Gibbs smple with selection pobbility = (,,, ) d cn be lso tnsfeed to homogeneous Mkov chin by defining T d k = ktk s the suogte tnsition pobbility mtix. The coesponding sttiony distibution nd cn be diectly computed by evluting π π lim T m = m T C vecto of s. π nd lim T m m T d = Cπ, whee = C k = k C C, nd is C-dimensionl T In Exmple, the tnsition mtices fo fixed- nd ndom-scns e espectively = T T fo nd fo nd Tble = (, ) T = T T = (,), T = ( T + T ) /. diectly compes the joint distibutions obtined fom the following computtions: () diect MCMC Gibbs smple fo the only two possible fixed-scn pttens = (,) ; () diect MCMC Gibbs smple fo ndom-scn pttens with the = (, ) following selection pobbilities:, nd, (3) mtix 0 = (, ) = ( 3, 3) = ( 3, 3) nd multipliction using multipliction using m T m T with low powe (m = 4) nd high powe (m = 3) nd (4) mtix lso with low nd high powes. Fo both () nd (), we used the fist 5,000 cycles s bun-in nd the subsequent,000,000 cycles fo smpling. As expected, both the fixed-scn, egdless of scn ode, nd the ndom-scn Gibbs smples numeiclly convege to the sme joint distibution (convegence is defined hee s ll cell-wise diffeences between estimtes fom two consecutive itetions to be less thn ). Tble lso demonsttes tht diect mtix multipliction of the tnsition pobbilities poduces pid convegence even fo smll m nd diffeent vlues of. Howeve, we lso obseved tht if ws hevily imblnced, it took mny moe itetions to chieve numeicl convegence (not shown) Fo exmple, if = (, ), it took m > 0 to chieve the sme numeicl convegence (up to 4 deciml plces). 9

10 Tble. Joint distibutiuons poduced by vious Gibbs smples fo Exmple. π = (, ) = (,) 0 = (, ) π ( m = 4) π ( m = 4) π ( m = 3) 0 π ( m = 3) π ( m = 3) (,) (,) (,) (,) Exmple. (Incomptible conditionl distibutions). Conside pi of conditionl distibutions f nd defined on the domin {, } s follows (Anold ( x x ) f( x x) Cstillo, nd Sbi, p. 4): = nd f 3 = f () These two conditionl distibutions e not comptible. It is esy to show tht the locl tnsition pobbility mtices e espectively: T nd = T = Tble shows the esults fo the joint distibutions deived fom the simulted Gibbs smples nd mtix-multipliction of Exmple fo conditions tht e identicl to those pesented in Tble. Sevel impotnt obsevtions cn be mde hee: () The Gibbs smples tht use the fixed-scn ptten nd espectively convege to two distinct 0

11 joint distibutions; () ech individul fixed-scn Gibbs smple conveges to the coesponding solution computed fom the mtix-multipliction method; nd (3) the ndom-scn Gibbs smple conveges to the mixtue distibution of the individul fixedscn distibutions i.e., π = ( ) π + π. Howeve, the lst obsevtion, s we shll π π π π π 0 see lte, only holds tue fo d =. Tble 3 shows the conditionl distibutions of Exmple deived fom the mtixmultipliction method (m = 3) fo., nd. Inteestingly, one of the given conditionl distibutions is lwys identicl to the conditionl distibution deived fom the joint distibution of π, k =,. Fo exmple, the given conditionl k distibution in Eq. () is numeiclly identicl to the conditionl distibution f f x x π ( ) diectly deived fom the fixed-scn Gibbs smple. On the othe hnd, f in Eq. () is identicl to the conditionl distibution f ( x x) deived fom. Indeed, s we shll π see lte, fo given set of full conditionls c { f = f ( x x ), k =,, d}, fo scn k k k ptten = (,,, d ), the fixed-scn Gibbs smple π lwys hs t lest one f s its conditionl distibutions. To illustte the eo of the joint distibution to which Gibbs smple conveges when conditionl distibutions e incomptible, we pescibe cell-wise nom-bsed metic to quntify the distnce between two distibutions. The metic computes the Eucliden distnce between the given conditionlly specified distibutions nd the deived conditionl distibutions of the joint density. Thus, when the conditionl distibutions e comptible, the distnce metic, o eo tem, is identicl to zeo. Tble 4 shows the eo tems obtined fo the vious schemes fo Exmple. Bsed on the summy sttistics, the ndom scn ppes to hve the lest eo. Howeve, we emk hee tht such esult could depend on how the distnce metic is defined. Fo exmple, when cell-wise l -nom ws used to mesue distnce, distibution tht contined the smllest eo (Tble 4). π k l - is the

12 Tble. Joint distibutiuons poduced by vious Gibbs smples fo Exmple. = (, ) = (,) 0 = (, ) π ( m = 4) π ( m = 4) π ( m = 3) 0 π ( m = 3) π ( m = 3) (,) (,) (,) (,) Tble 3. Conditionl distibutiuons deived fom the computed joint distibutions (by using mtix multipliction) in Tble. (,) (,) (,) (,) (,) (,) (,) (,) Eq.() π π π 0 π π f f

13 Tble 4. l -nom nd l -nom eos between the computed joint distibutions nd given conditionl models. π π π 0 π π l f f Totl f Totl l f 4. SOME ANALYTIC RESULTS In this section, we offe sevel genel esults egding the behvios of the fixedscn nd the ndom-scn Gibbs smple fo discete vibles in which the tnsition mtices e finite. Fo most of these esults, it is not necessy to ssume comptibility. Besides poviding some theoeticl undepinning to the pevious illusttive exmples, the esults hee llow close look t the mechnisms though which incomptibility impcts the behvios of the diffeent Gibbs smpling schemes. Note tht these esults e specil cses tht cn be deived fom moe genel theoies fo Mkov chins, but fo ou pupose focusing on the specil cse of discete vibles nd scn pttens mkes it esie to exmine the dynmics of convegence. Genel esults egding convegence of Mkov chins cn be found elsewhee (e.g., see Tieney 994; Gilks et l. 996, nd the efeences theein). All of the poofs of the following esults e included in the Appendix. Theoem : If Fis positive then the Gibbs smple, eithe fixed-scn with scn ptten o ndom scn with selection pobbility > 0, conveges to unique sttiony distibution nd espectively. π π Note tht Theoem does not equie F to be comptible. The esult ssues tht when F is positive stonge condition thn F being non-negtive ny scn ptten 3

14 cn hve one nd only one sttiony distibution. Futhemoe, the tnsition fo ny fixed-scn ptten is govened by the following theoem: Theoem : If is positive then fo ech stte set, (x,x,,x f ), k =,, d, of the F d k Gibbs chin with scn ptten = ( hs exctly one sttiony,,, d ) T T T T distibution. In pticul, π = π T nd π = T, k =,, d, nd π π. d = π k d π k k k A diect consequence of Theoem is tht fo ny fixed-scn ptten, one of the specified conditionl distibutions in F cn lwys be deived fom its sttiony distibution. This is summized in the following coolly: Coolly : If Fis positive then the sttiony distibution of the Gibbs smple hs π f d s one of its conditionl distibutions fo the scn ptten = (, i.e.,.,,, d ) π( x x,,, ) d x x = f d d When F is comptible, ll scn pttens convege to the sme joint distibution. The following theoem povides foml sttement. Theoem 3: Given F is positive. F is comptible if nd only if thee exists joint distibution π with eithe π = π,, o. Futhemoe, is π = π, π the joint distibution chcteized by F. An inteesting obsevtion bout the ndom scn is tht it foms mixtue of the fixed-scn pttens only fo counte-exmple fo d = 3. d =. We stte the Coolly fo the cse d = nd give Coolly : If F > 0 nd d = then π, = (, ), is fomed by the convex combintion of π = ( ) π + π π = nd π (,) = ; i.e., fo ll [0,], (,). 4

15 A thee-dimensionl counte exmple to Coolly fo the cse d = 3 is pesented in Tble 5. In this exmple, is positive but not comptible. Thee e totl of six scn pttens nd fo ech scn ptten, the solution to which the individul Gibbs smple conveges is shown s ow in Tble 5. The vege of ll six fixed-scn 6 Gibbs smple π = π is povided, s well s efeence. In ode to solve fo 6 i= i F = { f, f, f } 3 non-negtive line combintion (mixtue) of the fixed-scn distibutions, π 0 6 = i = c iπ i () whee 0 = ( 3, 3, 3), we teted eqution () s system of line equtions nd solved fo c = ( c ), i =,,6. As it tuned out, ou esult indicted tht thee ws no solution tht stisfied i c = ( c, c,, c ) 0. This obsevtion led us to believe tht the sumise 6 (Liu 996) tht the sttiony distibution fo ndom-scn Gibbs smple is mixtue of the sttiony distibutions fo ll systemtic scn Gibbs smples is not tue in genel. It only holds fo d =. Tble 5. A thee-dimensionl counte exmple fo Coolly. (,,) (,,) (,,) (,,) (,,) (,,) (,,) (,,) f f f π = (,3,) π = (3,,) π 3 = (,,3) π 4 = (3,,) π 5 = (,3,) π 6 = (,,3) π π 0 =( 3, 3, 3 )

16 5. DISUSSION This ppe povides some simple exmples to illustte the behvios of the Gibbs smple fo full set of conditionlly specified distibutions tht my not be comptible. We show tht fo given scn ptten, homogeneous Mkov chin is fomed by the Gibbs smpling pocedue nd unde mild conditions, the Gibbs smple conveges to unique sttiony distibution. Unlike comptible distibutions, diffeent scn pttens led to diffeent sttiony distibutions fo PICSD. The ndom-scn Gibbs smple genelly conveges to something in between but the exct weighted eqution only holds fo simple cses i.e., when the dimension is two. Ou findings hve sevel implictions fo the pcticl ppliction of the Gibbs smple, especilly when they opete on PICSD. Fo exmple, the MICE often elies on single fixed-scn ptten. This implies tht the imputed missing vlues could chnge beyond expected sttisticl bounds when seemingly innocuous chnge in the ode of the vible is being mde. Although in this ppe we hve not studied the issue of which fixed-scn ptten poduces the best joint distibution, some ecent wok hs been done in tht diection. Fo exmple, Chen, Ip, nd Wng (0) poposed using n ensemble ppoch to deive n optiml joint density. The uthos lso showed tht the ndomscn pocedue genelly poduces pomising joint distibutions. It is possible tht in some cses the gin fom using multiple Gibbs chins, s in the cse of ndom-scn, is mginl. As gued by Heckemn et l. (000), the single-chin fixed-scn (pseudo) Gibbs smple symptoticlly woks well when the extent to which the specified conditionl distibutions e incomptible is miniml. This my be tue fo models tht e pplied to one single dt set with lge smple size. Howeve, the extent of incomptibility could be much highe when multiple dt sets e used nd when multiple sets of conditionl models e specified. While it is likely tht even in moe complex pplictions bute-foce implementtion of the (pseudo) Gibbs smple will still povide some kinds of solutions, the qulities nd behvios of such solutions will need to be igoously evluted. 6

17 APPENDIX: PROOFS OF ANALYTIC RESULTS Poof of Theoem. We need lemm to pove Theoem bout ieducibility (bility to ech ll inteesting points of the stte-spce) nd peiodicity (etuning to given stte-spce t iegul times). Lemm : If F is positive then T nd T e ieducible nd peiodic w..t. ny given nd > 0. Poof. Let x = (x,x,x,,x ) nd x = (x,x,x,,x ) be two sttes fo the 3 k 3 k chin induced by T o T. Without loss genelity, we lso let = (,,3,, d) nd = (,,, d d d ), T = TT T3 Td nd T = ( T + T + + Td ). d To pove tht T nd T e peiodic, we must hve ( T ) ii > 0 nd ( T ) ii > 0, i. By the definition of locl tnsition pobbility, we hve ( T ) > 0, k, if F is positive. Consequently, d d ( T ) ii ( T ) 0 k = k ii > nd ( T ) ii d ( T ) = 0 k = k ii >, i. k ii To pove tht T nd T e ieducible is equivlent to pove tht x nd x commute, i.e., to show the tnsition pobbility scn ptten we hve P( x x ) > 0 nd P( x x ) > 0. Given the P( x x ) = f (x,x,,x,x ) f (x,x,x,,x ) f (x,x,x,,x ) > 0, d d 3 d d 3 d nd P( x x ) = f (x,x,,x,x ) f (x,x,x,,x ) f (x,x,x,, x ) > 0. d d 3 d d 3 d Similly, fo the ndom-scn cse we hve d P( x x ) ( ) d f(x, x,,x d,x d ) f(x,x,x 3,,x d ) fd (x, x,x 3,,x d ) > 0, nd 7

18 d P( x x ) ( ) d f(x,x,, x d,x d ) f(x,x,x 3,,x d ) fd (x,x,x 3,,x d ) > 0. It is well known tht if Mkov chin is ieducible nd peiodic, then it conveges to unique sttiony distibution (Nois 997). Consequently, we hve the uniqueness nd existence theoem (Theoem ) fo the Gibbs smple nd Gibbs chin. Poof of Theoem. We need lemm to pove Theoem. Lemm : If F is positive then the sttiony distibution π of the Gibbs smple hs f d s one of its conditionl distibution fo the scn ptten = (,,, ), i.e., π ( x x, x,, x ) = f. d d d d Poof: Since x f ( x x, x,, x ), it follows ( td ) ( td ) ( td ) ( td ) d d d d c c π f ( x x, x,, x ). Consequently, π ( x x ) = f ( x x ). d d d d d d d d Theoem esily follows fom Lemm. Poof of Theoem 3. If pt. Since F is positive nd comptible, thee exists positive joint distibution π > 0 chcteized by F. Unde the positive ssumption of π, it is well known tht the Gibbs smple govened by F detemines π (Besg, 994). Only if pt. Let i = (,,, d ) be scn ptten with d = i, i =,, d. Assuming tht thee exists π such tht π = π,. Fom Theoem nd Lemm, it c c c c c follows tht π ( x x ) = f ( x x ),. Thus, π ( xi xi ) = π ( xi xi ) = fi( xi xi ), i d d d d d Hence F is comptible nd π is the joint distibution of F.. i 8

19 Assuming tht thee exists π such tht π = π,. We only need to pove tht π = π,. Fom Theoem, we hve π T = π. By the definition of ndom-scn Gibbs smple, we hve π Tk = π = π, k,. It follows tht π = πt = ( πt ) T = = π ( T T T T ) = πt. d d d d d Fom Theoem, π is uniquely detemined by T. As esult, π = π = π,,. Poof of Coolly. The poof follows diectly fom Theoem nd 3. Poof of Coolly. Since F is positive, π, π nd π e sttiony distibutions uniquely detemined by, nd, espectively. Theefoe, [ ] T T π ( ) π + T + ( ) T T T T T T T T T T T = π T + ( ) π T + ( ) π T + ( ) π T = π + ( ) π + ( ) π + ( ) π = π + ( ) π Becuse T = T + ( ) T is the tnsition kenel fo the ndom-scn Gibbs chin with selection pobbility, we hve the uniquely detemined π which equls π + ( ) π. 9

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