ENTROPY AND DIVERGENCE RATES FOR MARKOV CHAINS: II. THE WEIGHTED CASE

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 9 Number / ENTROPY AND DIVERGENCE RATES FOR MARKOV CHAINS II THE WEIGHTED CASE Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 3 LMRS Uversté de Roue Frace; barbu@uv-rouefr Uversty of the Aegea Deartmet of Mathematcs Greece; alexkaragrgorou@aegeagr 3 Uversty of Bucharest ad ISMMA of the Romaa Academy Romaa; vaslereda0@gmalcom Corresodg author Vlad Stefa BARBU Uversté de Roue Laboratore de Mathématues Rahaël Salem UMR 6085 Aveue de l Uversté BP F7680 Sat-Étee-du-Rouvray Frace E-mal barbu@uv-rouefr Abstract I ths ork e cosder eghted versos of the geeralzatos of Alha dvergece measure (Cheroff [8] Amar ad Nagaoka [] ad Beta dvergece measures (Basu et al [4] for Markov chas ad vestgate ther lmtg behavor Ths s a cotuato of the develomets reseted [] alyg the oto of eghted dvergece measure (Belş ad Guaşu [5] Guaşu [] Kaur [] Ths ork s cotued [3] here e reset geeralzed Cresse ad Read oer dvergece class of measures ad umercally vestgate some roertes of all these geeralzed dvergece measures ad rates Key ords dvergece measures formato measures Markov chas etroy dvergece rates eghted dvergece measures PRELIMINARIES I ths ork e focus o some geeralzatos of Alha dvergece measure (Cheroff [8] Amar ad Nagaoka [] ad Beta dvergece measures (Basu et al [4] ad rates for Markov chas reseted [] For these dvergeces ad rates e cosder the assocated eghted versos follog the cocets troduced by Belş ad Guaşu [5] Guaşu [] As the results o formato measures cosder oly the robablty mass fucto or the robablty desty fucto of a radom varable thout takg to accout ts value Belş ad Guaşu [5] hghlghted the mortace of tegratg the uattatve objectve ad robablstc cocets of formato th the ualtatve subjectve ad o-stochastc cocet of utlty By cosderg the to basc cocets of objectve robablty ad subjectve utlty they troduced the cocet of eghted etroy ad thus costructed a shft-deedet formato measure th roertes smlar to those of the Shao etroy Afterard Guaşu [] characterzed axomatcally the eghted etroy measure D Crescezo ad Logobard [9] troduced the cocet of eghted resdual etroy ad eghted ast etroy May other researchers vestgated dfferet asects ad geeralzatos of eghted etroes ad dvergeces; amog them e ca cte Bhullar et al [6] Casulho [7] Dal ad Taeja [0] Kaur [] Sharma et al [3] Śmeja [4] Suhov et al [5 6] Taeja [7] Taeja ad Tuteja [8 9] The aer s orgazed as follos I the rest of ths secto e recall from [] the deftos of geeralzed Alha ad Beta dvergece measures; the Secto the corresodg eghted geeralzed dvergece measures are reseted ad the assocated rates are obtaed Let (X N be a ergodc tme-homogeeous Markov cha th fte state sace χ = { M} For ths Markov cha e cosder to dfferet robablty las Uder the frst la let = P(X = χ deote the tal dstrbuto of the cha ad j = P (X k+ = j X k = j χ the assocated trasto robabltes Let also deote the jot robablty dstrbuto of (X X X e ( = χ ere e deoted by the -tule ( χ Smlarly e defe uder the secod la j ( ad Uder ths settg of fte state sace Markov chas the Alha-

2 4 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA Gamma measure betee the to models s defed as the Alha-Gamma measure betee the to jot robablty dstrbutos ad (cf [] ad s rtte uder the ormalzed form as DAG ( = log ( ( ( χ ( here th j ad = = j j χ defed by j = = / / j ( ( ( ( j = = / / j ( ( ( ( Smlarly the Beta-Gamma measure betee the to Markov models s defed by (cf [] DBG ( = log ( ( χ here th j ad j j χ defed by = = j + + ( ( ( ( j = = / ( + / ( + j = = / ( + / ( + j + + ( ( ( ( ( (3 (4 (5 (6 WEIGHTED ALPHA AND BETA DIVERGENCE RATES FOR MARKOV CHAINS I ths secto e frst recall the otos of eghted etroes ad dvergece measures ad the troduce e cocets of eghted Alha ad Beta dvergeces The these measures ll be defed for fte Markov chas ad the corresodg rates ll be obtaed As metoed the Itroducto Belş ad Guaşu [5] troduced the cocet of eghted etroy ad Guaşu [] characterzed t axomatcally Let = ( be a fte robablty dstrbuto corresodg to ossble states or outcomes ad = ( be a vector of eghts assocated th these states 0 = Defto (cf [] The eghted Shao etroy measure s defed by ( S I ; = log( = (7

3 3 Etroy ad dvergece rates for Markov chas I The Alha-Gamma ad Beta-Gamma case 5 Whe cosderg a absolutely cotuous robablty measure μ th a desty th resect to a certa measure μ ad a eght fucto assumed to be measurable ad ostve the the eghted Shao etroy measure s defed by S I ( ; = xx ( ( log( x ( d μ( x (8 Let us cosder = ( ad = ( to fte robablty dstrbutos corresodg to ossble states ad let = ( be a vector of eghts assocated th these states 0 = th 0 for some It s clear that for real alcatos fact > 0 for all hch ll be assumed all alog ths aer Note that the eghted Shao dvergece measure (relatve etroy betee ad ca be defed as follos Defto (cf [8] The eghted Shao dvergece measure betee ad s gve by S D ( ; = log = (9 Kaur [] defed axomatcally the eghted drected dvergece cocet Let us cosder = ( ad = ( to fte robablty dstrbutos corresodg to ossble states ad let = ( be a vector of eghts assocated th these states 0 = Defto 3 (cf [] A measure D( s sad to be a arorate measure of eghted drected dvergece f the follog axoms are fulflled It s a cotuous fucto of ( ( ad ( It s ermutatoally symmetrc e t does ot chage he the trlets ( ( ( are ermuted amog themselves 3 It s alays o-egatve ad vashes he = for all = 4 It s a covex fucto of ( hch has ts mmum value zero he = for all = 5 It reduces to a ostve multle of a ordary measure of eghted drected dvergece he all the eghts are eual Note that the eghted Shao dvergece troduced Defto does ot verfy Codto 3 of the Kaur s defto of a eghted drected dvergece For ths reaso Kaur [] roosed aother eghted Shao dvergece measure hch satsfes all the above set of axoms of a eghted drected dvergece Defto 4 (cf [] The eghted Shao drected dvergece measure corresodg to the Kullback-Lebler measure s gve by D ; = + S 0 ( log = (0 Note that the eghted Shao drected dvergece measure gve the above defto verfes Codto 3 of the Kaur's defto of a eghted drected dvergece; deed oe has to factorze by the factor log ( + (0 ad the to study the sg of the fucto f ( t = log( t t+ We ll troduce o the cocets of eghted Alha ad Beta dvergece measures as ell as the

4 6 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 4 eghted Alha-Gamma ad Beta-Gamma dvergece measures Let us cosder to absolutely cotuous robablty measures μ ad μ th corresodg destes ad th resect to a certa measure μ ad a eght fucto assumed to be measurable ad ostve Defto 5 The eghted Alha ad Beta dvergece measures are gve by DA ( ; = x ( ( x ( x xx ( ( + ( xx ( ( d μ( x ; ( + + DB ( ; = x ( ( x + xx ( ( ( x + x ( ( x μ( x + d Note that here ( x = ( x ( x ad x ( = xx ( ( Note also that A ( ( D ; = D B ( ( D ; = D A B ( ( d + ( x ( x ( x + x ( ( xx ( here = ad = Trasformg the eghted Alha dvergece DA ( ; gve Defto (as doe [] for the Alha dvergece e obta the eghted Alha-Gamma dvergece measure ( x ( x ( x DAG ( log d μ ; = = D ( AG ( xd ( d μ( x xd ( d μ( x = log ( x ( xd μ( x ( d ( x here e have set ( x = ( x dμ ( x ad ( ( x x = ( x dμ ( x Smlarly trasformg the eghted Beta dvergece D ( the eghted Beta-Gamma dvergece measure gve by here e have set x ( x ( = + ( xd dμ( x D ( log ( ( ( ( BG x ; = x d μ x = DBG ( / + B (3 ; gve Defto e obta ad x ( x ( = ( / + + ( xd d μ( x Note that ad resectvely ad ca be see as the destes (or mass fuctos the dscrete case of corresodg measures μ ad μ resectvely μ ad μ th resect to a certa (4

5 5 Etroy ad dvergece rates for Markov chas I The Alha-Gamma ad Beta-Gamma case 7 measure μ Thus the Alha ad Beta dvergece measures D ( ad D ( A B that aear ( ad ( as ell as the Alha-Gamma ad Beta-Gamma dvergece measures DAG ( ad D ( BG that aear (3 ad (4 are ell defed Let us o focus o eghted dvergece measures for Markov chas We lace ourselves the same frameork as the revous secto (X N s a ergodc tme-homogeeous Markov cha th fte state sace χ = { M} ad e cosder to dfferet robablty las for ths cha j ( are the corresodg robabltes uder the frst la hle j ( are the same uattes uder the secod la Let us also cosder W( = {( ( χ } N * be a set of eghts assocated th the states χ ( > 0; as revously e deote ( by ( Let us cosder ( ( ( = ( ( ( = ( ( + ( = ( ad ( + ( = Uder these otatos takg to accout the eghted Alha-Gamma dvergece measure troduced (3 ad the eghted Beta-Gamma dvergece measure troduced (4 e defe the eghted Alha-Gamma ad Beta-Gamma dvergece measures for fte Markov models Defto 6 The eghted Alha-Gamma measure betee the to Markov models s defed as the eghted Alha-Gamma measure betee the to jot robablty dstrbutos ad that s D ( ; = D ( AG AG ( ( = log ( χ ( ( ( ( χ χ (5 The eghted Beta-Gamma measure betee the to Markov models s defed as the eghted Beta-Gamma measure betee the to jot robablty dstrbutos ad that s D ( ; = D ( BG BG ( ( log = /( + /( + + χ ( ( + χ ( χ We ll cosder to artcular cases of eghts that could be of terest ractce The frst oe s ( = (7 here = ( M s a vector of eghts assocated th the states χ = { M} > 0 = M Here s cosdered as a eght assocated to the state ad relato (7 meas that the eghts are deedet The secod case cosdered s here ( = = ( M s a vector of eghts assocated th the states χ = { M} = Mhle = ( χ s a system of eghts assocated th χ χ j j 0 j χ Here j (6 (8 0 j

6 8 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 6 s cosdered as a eght assocated to the coule of state ( j ad relato (8 meas that the eghts have a deedece of a Markov tye PROPOSITION Uder the frst artcular case of eghts gve (7 The eghted Alha-Gamma measure betee the to Markov models ca be rtte uder the ormalzed form here th here j ad DAG ( ; = log ( ( ( χ = = j j χ defed by j j = = / / j ( ( ( ( j j = = / / j ( ( ( ( The eghted Beta-Gamma measure betee the to Markov models ca be rtte uder the ormalzed form th j DBG ( ; = log ( ( χ ad j j χ defed by = = j j = ( / + = / ( + j + + ( ( ( ( j j = ( / + = / ( + j + + ( ( ( ( Uder the secod artcular case of eghts gve (8 The eghted Alha-Gamma measure betee the to Markov models ca be rtte uder the ormalzed form DAG ( ; = log ( ( ( χ (9 (0 ( ( (3 (4 (5

7 7 Etroy ad dvergece rates for Markov chas I The Alha-Gamma ad Beta-Gamma case 9 here th here j ad = = j j χ defed by j j = = / / j ( ( ( ( j j = = / / j ( ( ( ( The eghted Beta-Gamma measure betee the to Markov models ca be rtte uder the ormalzed form th j DBG ( ; = log ( ( χ ad j j χ defed by = = j j = ( / + = / ( + j + + ( ( ( ( j j = ( / + = / ( + j + + ( ( ( ( (30 The follog result gves the dvergece rates of eghted Alha-Gamma ad Beta-Gamma measures th the eghts of the tyes gve (7 ad (8 The roof s smlar as the roof of Theorems ad from [] ad ll ot be rovded here THEOREM Uder the settg of the reset secto e have lm DAG ( ; = log λ( (a here λ( = lm λ( (assumed to exst here λ ( s the largest ostve egevalue of R ( = ( ř j( j χ here ř j( = j j th j ad j defed Euatos (0 ad ( resectvely f cosderg the frst tye of eghts gve (7 or Euatos (6 ad (7 resectvely f cosderg the secod tye of eghts gve (8 lm DBG ( ; = log λ( (b here λ( = lm λ ( (assumed to exst here λ ( s the largest ostve egevalue of (6 (7 (8 (9

8 0 Vlad Stefa BARBU Alex KARAGRIGORIOU Vasle PREDA 8 R ( = ( r j( j χ here r j ( = j j th j ad j defed Euatos (3 ad (4 resectvely f cosderg the frst tye of eghts gve (7 or Euatos (9 ad (30 resectvely f cosderg the secod tye of eghts gve (8 ACKNOWLEDGEMENTS The frst to authors ould lke to exress ther arecato to the Uversty of Roue ad Uversty of the Aegea for the oortuty to exchage several vsts both sttutos The research ork of Vlad Stefa Barbu as artally suorted by the rojects XTerM Comlex Systems Terrtoral Itellgece ad Moblty (04 08 ad MOUSTIC Radom Models ad Statstcal Iformatcs ad Combatorcs Tools (06 09 th the Large Scale Research Netorks from the Rego of Normady Frace REFERENCES S AMARI H NAGAOKA Methods of Iformato Geometry Oxford Uversty Press Ne York 000 VS BARBU A KARAGRIGORIOU V PREDA Etroy ad dvergece rates for Markov chas I The Alha-Beta ad Alha-Gamma case submtted 07 3 VS BARBU A KARAGRIGORIOU V PREDA Etroy ad dvergece rates for Markov chas III The Cresse ad Read case ad alcatos submtted 07 4 A BASU IR HARRIS NL HJORT MC JONES Robust ad effcet estmato by mmsg a desty oer dvergece Bometrka M BELIŞ S GUIAŞU A uattatve-ualtatve measure of formato cyberetc systems IEEE Trasactos o Iformato Theory JS BHULLAR OP VINOCHIA M GUPTA Geeralzed measure for to utlty dstrbutos Proceedgs of the World Cogress of Egeerg Lodo UK vol III 00 7 JP CASQUILHO Combg exected utlty ad eghted G-Smso dex to a o-exected utlty devce Theoretcal Ecoomcs Letters H CHERNOFF A measure of asymtotc effcecy for tests of a hyothess based o a sum of observatos A Math Statst A D CRESCENZO M LONGOBARDI O eghted resdual ad ast etroes Scetae Math Jao G DIAL IJ TANEJA O eghted etroy of tye ( β ad ts geeralzatos Alkace matematky htt//eudmlorg/doc/5 S GUIAŞU Weghted etroy Reorts o Mathematcal Physcs JN KAPUR Measures of Iformato ad Ther Alcatos Wley Ne Delh BD SHARMA J MITTER M MOHAN O measures of useful formato Iformato ad Cotrol M ŚMIEJA Weghted aroach to geeral etroy fucto IMA Joural of Mathematcal Cotrol ad Iformato do htts//doorg/0093/mamc/dt044 5 Y SUHOV S YASAEI SEKEH I STUHL Weghted Gaussa etroy ad determat eualtes 05 8 htts//arxvorg/df/ df 6 Y SUHOV I STUHL S YASAEI SEKEH M KELBERT Basc eualtes for eghted etroes Aeuatoes mathematcae do0007/s IJ TANEJA O geeralzed eghted dvergece measures htt// /vedoc/doload?do= &re=re&tye=df 8 HC TANEJA RK TUTEJA Characterzato of a uattatve-ualtatve measure of relatve formato Iformato Sceces HC TANEJA RK TUTEJA Characterzato of a uattatve-ualtatve measure of accuracy Kyberetka htt//eudmlorg/doc/7733 Receved May 3 07

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 19, Number 3/2018, pp

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Seres A OF THE ROMANIAN ACADEMY Volume 9 Number 3/208. 43 42 MATHEMATICS ENTROPY AND DIVERGENCE RATES FOR MARKOV CHAINS. III. THE CRESSIE AND READ