The optimal rate for resolving a near-polytomy in a phylogeny

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Giles Foster
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1 Publsh n "Journal of Thortcal Bology : 7 79, 7" whch shoul b ct to rfr to ths work. Th optmal rat for rsolvng a nar-polytomy n a phylogny Mk Stl a,, Chrstoph Lunbrgr b a Bomathmatcs Rsarch Cntr, Unvrsty of Cantrbury, 8, Chrstchurch, Nw Zalan b Départmnt mathématqus, Unvrsté Frbourg, Chmn u Musé, 7 Frbourg, Swtzrlan Introucton Th rconstructon of phylogntc trs from scrt charactr ata typcally rls on mols that assum th charactrs volv unr a contnuous-tm Markov procss opratng at som ovrall rat λ. Whn λ s too hgh or too low, t bcoms ffcult to stngush a short ntror g from a polytomy (th tr that rsults from collapsng th g). In ths not, w nvstgat th rat that maxmzs th xpct log-lklhoo rato (.. th Kullback Lblr sparaton) btwn th four-laf unrsolv (star) tr an a four-laf bnary tr wth ntror g lngth. For a smpl two-stat mol, w show that as convrgs to th optmal rat also convrgs to zro whn th four pnant gs hav qual lngth. Howvr, whn th four pnant branchs hav unqual lngth, two local optma can ars, an t s possbl for th globally optmal rat to convrg to a non-zro constant as. Morovr, n th sttng whr th four pnant branchs hav qual lngths an thr () w rplac th two-stat mol by an nfnt-stat mol or () w rtan th two-stat mol an rplac th Kullback Lblr sparaton by Euclan stanc as th maxmzaton goal, thn th optmal rat also convrgs to a non-zro constant. Whn scrt charactrs volv on a phylogntc tr unr a contnuous-tm Markov procss, th stats at th lavs prov nformaton about th ntty of th unrlyng tr. It s known that whn th ovrall substtuton rats bcoms too hgh or too low, t bcoms ncrasngly mpossbl to stngush th tr from a lss rsolv tr (or n from any othr tr) usng any gvn numbr of charactrs. In partcular, suppos w tak a tr T wth an ntror g of lngth an w sarch for an ovrall substtuton rat λ that optmally scrmnats (unr som mtrc or crtron) btwn T an T (.. th tr that has th sam topology an branch lngths as T xcpt that has bn collaps (.. has lngth )). Ths optmal rat pns n an ntrstng way on th tr's branch lngths (an th mtrc or crtron us), as rval by svral stus ovr th last two cas (s, for xampl, Fschr an Stl, 9; Lws t al., 6; Townsn, 7; Townsn an Lunbrgr, ; Yang, 998), an appl to th stuy of ata sts (s, for nstanc, Klopfstn t al., ; Townsn t al., ). In ths short not, w consr a mor lcat quston that las to som curous subtlts n ts answr. Namly, how os λ bhav as tns to zro? For smplcty, w consr th four-laf tr an two smpl substtuton mols. W fn that th answr to ths quston pns rathr crucally on thr thngs: whthr th stat spac s fnt or nfnt, th mtrc mploy, an th gr of mbalanc n th branch lngths. Our rsults prov som analytc nsght nto smulaton-bas fnngs rport by Klopfstn t al. () (n th scon part of thr scton nttl Optmum Rats of Evoluton ); spcfcally, th optmal rat n th fnt-stat sttng can bhav ffrntly from th optmal rat for gnratng charactrs that ar parsmony-nformatv an homoplasy-fr.. Optmal rat rsults Consr a bnary phylogntc tr T wth four pnant gs of lngth L an an ntror g of lngth, as shown n Fg. (). Now consr a Markovan procss that gnrats stats at th lavs of T. W consr two mols n ths papr: (a) th two-stat symmtrc mol (somtms rfrr to as th Nyman two-stat mol or th Cavnr-Farrs-Nyman mol), an (b) an nfnt-alll mol (n whch a chang of stat always las to a nw stat, a mol oftn rfrr to as th nfnt allls mol of Crow an Kmura (Kmura an Crow, 96), or th ranom clustr mol (Stl, 6)). For both mols th nuc partton of th laf st (n whch th blocks ar th substs of lavs n th sam stat) wll b rfrr to as a charactr. Thus for th two-stat mol, thr ar xactly ght possbl charactrs that can ars on T, whl for th nfnt-alll mol, thr ar Corrsponng author. E-mal arsss: mk.stl@cantrbury.ac.nz (M. Stl), chrstoph.lunbrgr@unfr.ch (C. Lunbrgr).

2 Fg.. () A bnary four-laf tr T wth a short ntror g of lngth an four pnant gs of qual lngth L. () Th star tr obtan from T by sttng =. () A tr xhbtng two local optma for th rat λ maxmzng KL( P, P ). whn >or whn =(thr ar parttons of th st of four lavs of T, howvr whn >(rsp. =) two (rsp. thr) hav zro probablty of bng gnrat). Suppos that th branch lngths ar all multpl by a rat factor λ, an lt P b th probablty strbuton on charactrs. Lt P b th probablty strbuton on charactrs unr th corrsponng mol on th star tr T (shown n Fg. ()). Now, suppos that a ata st D of k charactrs s gnrat by an npnnt an ntcally strbut (...) procss on th (unrsolv) star tr T (unr thr Mol (a) or Mol (b)). Lt LLR not th log-lklhoo rato of th star tr T to th rsolv tr T (.. th logarthm of th rato ( DT )/ ( DT )). As k grows, LLR k convrgs n probablty to ts (constant) xpct valu, whch s prcsly th Kullback Lblr sparaton (s Covr an Thomas, 6): P () KL( P, P ) = P ( ) ln P (), whr th summaton s ovr all th possbl charactrs. Lt λ b a valu of λ that maxmzs KL( P, P ). From th prvous paragraph, ths s th rat that provs th largst xpct lklhoo rato n favour of th gnratng tr T ovr an altrnatv rsolv tr wth an ntrnal g of lngth. W ar ntrst n what happns to λ as tns to zro. In that cas, KL( P, P ) also convrgs to zro, but t s not mmatly clar whthr th optmal rat that hlps to stngush T from T by maxmzng KL( P, P ) shoul b ncrasng, crasng or convrgng to som constant valu. A larg rat mprovs th probablty of a stat-chang occurrng on th cntral g of T howvr, ths coms at th prc of gratr ranomzaton on th pnant gs, whch tns to obscur th sgnal of such a chang bas on ust th stats at th lavs. It turns out that th lmtng bhavour of λ pns crucally on whthr th stat spac s fnt or nfnt. In Part () of th followng thorm, w consr ust th two-stat symmtrc mol (s.g. Chaptr 7 of Stl (6)) but w ncat n Fg. that a smlar rsult appars to hol for th symmtrc mol on any numbr of stats. Th rsult n Part () contrasts wth that n Part () for th nfnt-alll mol, n whch homoplasy (.. substtuton to a stat that has appar lswhr n th tr) os not ars. Ths scon rsult s ffrnt from (but consstnt wth) a rlat rsult n Townsn (7). Thorm. () For th two-stat symmtrc mol, lm λ =. () By contrast, for th nfnt-alll mol, lm λ =. Proof. Part () Lt p = ( xp( λ)) b th probablty of a stat chang across th ntror g of T unr th two-stat mol. Lt p b th probablty of gnratng a charactr on T, whr on laf s n L on partton block an th othr thr lavs ar n a ffrnt partton block, an lt q b th corrsponng probablty on T. Bcaus th four pnant gs of T an T hav qual lngth, w hav: p =( pq ) + pq = q, () so qln( q / p ) =. Lt p b th probablty of gnratng thr on of th two charactrs that hav a parsmony scor of on T, an lt q b th corrsponng probablty on T. Onc agan w hav: p =( pq ) + pq = q, () an so q ln( q / p ) =. Lt p b th probablty of gnratng th charactr that has a parsmony scor of on T an a parsmony scor of on T, an lt q b th corrsponng probablty on T. Lt p b th probablty of gnratng th charactr that has parsmony scor on T an lt q b th corrsponng probablty on T. Notc that w can wrt: q = α + ( α) an q = α ( α), whr α = ( ). Morovr, p =( pq ) + pq, an p =( pq ) + pq. It follows that q ln( q / p ) = q ln( p / q ) = q ln( p + pq / q ) () an q ln( q / p ) = q ln( p / q ) = q ln( p + pq / q ). () Combnng Eqs. () () gvs: q P P q p q p q p q KL(, )=+ ln + ln + q p. If w lt θ = θ( λ) = q / q, thn: P P q θ θ p KL(, )= ln(+( ) )+ln( ( θp ) ). θ Notc that, by Eq. (), w hav: λl α α θ = ( ) ( ) = α +( α) ( + ) + ( ), an so, n Eq. (6), θ s a monoton ncrasng functon from (at λ = ) to a lmtng valu of as λ. Not also that q = q ( λ) s a monoton crasng functon from (at λ = ) to a lmtng valu of as λ. 8 Now lt us st λ = x n Eq. (6), for a fx valu of x. Thn p = ( xp( x))= x + O(), an from Eq. () w hav θ =x L + O(). Thrfor: l( x) lm ( P, P)/ = x L ln + KL + x, xl (7) an so l( x) convrgs to L as x. Nxt, suppos that λ os not convrg to zro as. Thn for som δ >an som squnc of valus whch convrgs to zro, w hav: λ > δ > (8) λ for all. Lt λ λ, θ θ( λ ) an p ( ). Notc that ( θ) p convrgs to zro as. Ths s bcaus w can wrt Bλ ( θ) p A λ ( ) for constants A, B >, an ffrn- () (6)

3 Bλ λ tal calculus shows that th maxmal valu of A ( ) as λ > vars convrgs to zro as. Snc θ s boun away from (by Inqualty (8)), t also follows that (/ θ ) p = ( θ) p/ θ convrgs to zro as. Consquntly, both ( θ ) p an ( θ) p/ θ wll both l wthn (, ) for all I for som suffcntly larg fnt valu I (pnnt on δ). W now apply th followng nqualty an xpanson whch hol for all x, y (, ): x ln( + x) < x +, an ln( y) = wth x =( θ) p/ θ an y =( θ ) p n Eq. (6). Notng that th two lnar trms n p from Eq. (6) cancl w obtan only quaratc an hghr trms n p. Thus, for all I: p P P q λ (, ) < ( ) θ θ ( )( KL ) + θ y ( θ p ). (9) Morovr, q ( λ) < q () = (by Eq. (8)) an p λ for all, sow can wrt: KL( P, P ) λ < ( θ )( θ ) + θ Lt y()= t t ( ) t t ( + ) + ( ) ( θ) λ. (). It can b vrf that t ( yt ( )) yt ( ) < for all t >. Applyng ths wth t yt () obtan th followng boun on th frst trm n Inqualty (): λ θ θ ( )( ) <. θ L =λl,w In aton, th scon trm on th rght n Inqualty () convrgs to zro as grows, snc th summaton trm s absolutly boun (not that λ( θ) as λ ) an snc th numrator trm out front,, convrgs to. In summary, for suffcntly larg valus of, w hav KL( P, P )/ < L. Thus, by slctng x suffcntly larg w can nsur that l( x) (gvn by Eq. (7), an whch s bas on a λ valu that convrgs to zro as ) taks a largr valu for KL( P, P ) than th valu λ. Ths complts th proof of Part (). For Part (), lt y = xp( λl) an lt ζ = xp( λ); ths ar th probablts of a stat chang on a pnant an th ntror g, rspctvly, n th nfnt-alll mol. Th parttons of { a, b, c, } that can b gnrat wth strctly postv probablty on T fall nto fv sont classs (an th probablts of gnratng th parttons wthn a gvn class ar th sam). W labl ths classs C,, C whr C ={ abc}, C ={ a bc, b ac, c ab, abc}, C = { acb, abc, bca, bac }, C = { abc, cab } an C ={ } abc. For =,,, lt P [ ] (rsp. P []) b th probablty that th gnrat partton ls n Class on T. W hav: P []=( y), P []=( y y), P []= y ( y), P[]= y ( y), P[]= y ( y)+ y. Lt P [ ] b th corrsponng probablts of T. W can thn wrt: P []=( ζp ) []+ ζd, () whr D s pnnt only on y. Mor prcsly, D = D = D =, D = ( ), D =( ). () Th xprsson for D arss bcaus whn thr s a stat chang across th ntror g of T thn a partton n C occurs prcsly whn thr s no stat chang btwn th two lavs on on s of th g (wth probablty λl ) an thr s a stat chang btwn th two lavs on th othr s of th g (wth probablty ); th coffcnt of out front rcognss that thr ar two ways that ths can occur. For D, a stat chang across th ntror g of T las to th partton n C prcsly f th lavs on on s of th ntror g ar n ffrnt stats, an so too ar th lavs on th othr s of th ntror gs, an ths two npnnt vnts hav probablty. Now, ach partton wthn any gvn class C has th sam probablty of bng gnrat on T (morovr, th sam statmnt appls for T n plac of T ). Ths allows us to wrt KL as a sum of fv trms (rathr than ) n th followng way: P P P P P P ζ P [] D KL(, )= []ln( []/ [])= []ln. P [] = = () Now, T has on atonal partton typ that t can gnrat but T can not, namly th partton { ab c}. Ths partton s gnrat by T wth probablty ζ an so P []= ζ =. By Eq. (), w obtan th ntty = = = = = ζ = P[]=( ζ) P []+ ζ D = P [] ζ ( P [] D), = an snc P []=w uc that: ( P [] D)=, = () whch s quvalnt to th ntty D + + D = from Eq. (). Now, ln( x) x for all valus of x < an combnng ths wth Eqs. () an () gvs ( P P P ζ P [ ] D ) KL(, ) [ ] = ζ( P [ ] D) = ζ. P [] = = () By Eq. () w can wrt KL( P, P ) = aln( ζb), = whr a an b ar functons of λ fn for =,,,, by a = P [ ], b = b = b = b an a = D a = b ( ), = ( )( + ). For < L w hav ζb <for ach valu of. Ths s clar for =,,; th cass = an = rqur a lttl car as b, b as λ. Howvr, ζ also pns on λ an for < L w hav λ ζb ( ) < λl ( ) an ζb λ <. Thus w xpan Eq. () va ts Taylor srs an wrt b ζ ( P, P) = a, an snc th trm for = s th trm KL = λl ζ apparng n th lowr boun for KL( P, P ) (s Eq. (), w hav: ( P, P) ζ a KL = b ζ. (6) Notc that th trm on th rght of ths last nqualty can b wrttn as ( b ζ) ζ ab. = For ach =,, th trm a b s boun abov by a constant tms λl (ths s clar for =,, an th abov formula for b an b nsur t also hols for =,as thr s a trm ( ) n a to cancl ths trm n th nomnator of b ). Consquntly, from (6), w can wrt λl cλ ( P, P) ζ ζ C, KL (7)

4 whr C, c ar absolut an strctly postv constants (not pnnt on λ or ). By ffrntal calculus, λ = ln( + ) maxmzs ζ λl, an L lm λ =. Now for th valu of λ L that maxmzs KL( P, P ), Eq. (7) shows that λ must tn to zro as, snc othrws thr s a squnc of valus λ whch tns to nfnty, whch las to valus for KL( P, P ) that ar smallr than thos obtan by sttng λ = λ (u to th xponntal trms n Eq. (7)) whch contracts th optmalty assumpton on λ. Thus λ as whch mpls that lm KL( P, P )/ = λ complts th proof of Part (). Dspt th contrast xhbt by Thorm btwn th nfntalll an two-stat sttng w hav a curous corrsponnc btwn th mols for th Euclan mtrc (.. th L mtrc), as th followng rsult shows., an ths s maxmz whn λ = L. Ths Thorm. For th Euclan mtrc, th substtuton rat valu λ that maxmzs ( P, P) for th two-stat symmtrc mol s gvn by λ = ln ( + ), whch convrgs to L as. L Proof. Usng th Haamar rprsntaton for th two-stat symmtrc mol (Hny, 989), an assocat nnr prouct ntty (Eq. 7.8 n Stl (6)) shows that: Lλ λ ( P, P) = ( ). (8) Ths functon of λ has a unqu local maxmum at λ = ln ( + ). L Now, lm ln ( + ) =, an at ths valu of λ w hav lm ( P, P)/ = L. L Thorms an ar llustrat n Fg. an. Hr, th g lngths of th tr ar L= (for ach of th four pnant gs) an =. an =.. Th valus wr calculat usng Eqs. (6) an () (usng Mapl), an ar consstnt wth th xprssons us n th rvaton an statmnt of Thorms an. Th Kullback Lblr sparaton s closly rlat to th Fshr nformaton (Lhmann an Caslla, 998), an th usfulnss of th Fshr nformaton n phylogntc trs has frst bn stu n Golman (998). It follows from larg sampl thory that th varanc Fg.. Th Euclan stanc btwn P an P for a quartt tr wth xtror gs of lngths L= an ntror g of lngth =.an =.as functons of λ, for th two-stat mol (calculat usng Eq. (8)). Th optmal λ valu convrgs to =. as, gvng as th asymptotc valu of ( P, P) L. of an ffcnt stmator of th g lngth, bas on th ata st D for a larg numbr k of charactrs (an wth L an λ bng known), s nvrsly proportonal to th Fshr nformaton wth rspct to th paramtr. Th Fshr nformaton s fn by: I() = P P ln () = () ln P ( ). For th two-stat symmtrc mol, w can xpan ln P ( ) n a Taylor srs aroun =to obtan: Fg.. Kullback Lblr sparaton of P an P for a quartt tr wth xtror gs of lngths L= an ntror g of lngth =.an =.as functons of λ, for th two-stat an nfnt-alll mols (calculat usng Eqs. (6) an () rspctvly). Th graph on th lft s consstnt wth a progrsson of th optmal λ valu towars zro as crass. For th nfnt-alll mol (rght), th optmal λ valu convrgs to =. as, an ( P, P) L KL convrgs to. L

5 P ( P, P) = P()ln () KL = P ( )(ln P ( ) ln P ( )) P () P = () ln P ( ) P O ln ( ) + ( ) = = = ( P ()) P P O () ln ( ) + ( ) = = I() + O( ). (9) In th two-stat mol wth L=, analyss of th coffcnt of n Eq. (6) (usng Mathmatca) provs th followng xplct scrpton of th Fshr nformaton trm I(): 8λ xp( λ)cosh ( λ) I() = ( + cosh( λ))snh ( λ). = () Lt ^ b an asymptotcally ffcnt stmator of th short g lngth,.. on whos varanc asymptotcally achvs th Cramér-Rao boun (s Ch. 6. n Lhmann an Caslla, 998). Thn ts rlatv rror, bas on a larg numbr k of... gnrat charactrs, s roughly rlatv rror(^) var(^) / ki(). From (9), w gt, for small valus of, th followng approxmaton unr th two-stat symmtrc mol: rlatv rror(^) [ k ( P, P)]. KL From ths w s that th optmal rat λ mnmzs th rlatv stmaton rror for th g lngth, agan unrlyng th usfulnss of Kullback Lblr sparaton n ths sttng.. Unqual pnant g lngths Thorm () s not gnrally val for quartts whn w rop th assumpton of qual g lngths. Fg. shows th Kullback Lblr sparaton KL( P, P ) pnnt on λ for a quartt tr wth ntror g lngth =. an unqual lngths of an on th pnant gs on on s of th ntror g, an also an on th pnant gs on othr s of (as shown n Fg. ()). Thr s stll a local maxmum whch tns to as but th global maxmum λ stays boun away from. Fg.. Kullback Lblr sparaton (sol ln) for a quartt tr wth xtror gs of lngths an on both ss of th ntror g of lngth =. as a functon of λ. Th ott ln shows th Fshr nformaton trm from Eq. (). Ths osyncratc shap of th curv n th hghly asymmtrc cas can b xplan as follows: If two gs on ach s of th cntral g ar vry long, thy can ssntally b gnor (th stat at ach of th two lavs s almost compltly ranom) an th stats at th lavs of th two shortr gs of lngths ar mor nformatv for nfrrng th total lngth of + of th path onng thm, an thrby for cng whthr or not =. Lt p() = (+xp( λ(+))) b th probablty that th two charactrs at th lavs of th untlngth gs ar n th sam stat. Thn th Fshr nformaton wth rspct to, whn w gnor th charactrs at th lavs of th two long gs, s th gvn by: I() = p() p p p ln () ( ()) ln ( ()) λ = xp( λ( + )). () Fg. shows I()/ as a functon of λ (th ott ln). Clarly, th global maxmum s xplan by th stmaton of va th two untlngth gs. As for Fg., th valus wr calculat by smulatng charactrs ovr a rang of λ valus (usng th R statstcal packag). By lssnng th mbalanc btwn th pnant g lngths t s possbl to mak th two local optma for th rats hav qual (global) optmal valus; whch provs an xampl whr th global optmal rat for maxmzng KL( P, P ) s not unqu. Whn th g lngth mbalanc crass furthr, smulatons suggst that Thorm () rmans val (.. th lmt λ as os not ust hol for th spcal sttng n whch all pnant gs hav xactly qual lngths)... Conclung commnts For th bologst, Thorm () provs a cauton: n rsolvng a nar-polytomy, t s tmptng to sarch for gntc ata that hav volv fast nough to hav unrgon substtuton vnts on th ntror g; howvr, a slowr-volvng ata st may, n fact, b mor lkly to stngush th rsolv tr from an unrsolv phylogny. For nfnt-alll mols, howvr, Thorm () nsurs thr s a postv optmal rat rgarlss of how short th ntror g s (consstnt wth a rlat rsult from Townsn (7)). Thorm appls to balanc trs, an w also saw that for suffcntly unbalanc trs, ths fnngs can chang u to th apparanc of a scon local optmal rat that vntually bcoms th global optmal rat (cf Fg. ). Morovr, as Fg Fg ncats, th rsults stablsh for th two-stat symmtrc mol appar to hol for othr fnt-stat mols such as th four-stat symmtrc mol (oftn rfrr to as th Juks-Cantor (969)' mol (JC69); for tals s Stl (6), Scton 7..). W hav assum throughout that th lngths of th four pnant gs of th tr rman fx as th ntror g shrnks to zro. If th pnnt gs ar also allow to shrnk, thn Thorm no longr appls. For xampl, consr th bnary tr T that has an ntror g of lngth an four pnant gs of lngth L. Thn as, th optmal rat λ now ncrass towars nfnty rathr than crasng to zro as. Th rason for ths s qut smpl. Consr th tr T that has an ntror gs of lngth an four pnant gs ach of lngth L. Ths tr has som optmal rat λ* that maxmzs KL. Now T s obtan from T by multplyng ach g lngth of T by λ. Thus, for T, th optmal rat s gvn by λ = λ*/ as. Fnally, w hav consr KL( P, P ) rathr than KL( P, P ), partly bcaus th formr s asr to analys mathmatcally, an s wll-fn n th nfnt-alll sttng (KL( P, P ) s not wllfn for ths mol snc th partton { ab c} has postv

6 Fg.. Th four-stat symmtrc mol (JC69) shows smlar bhavour to th two-stat symmtrc mol. Top: Kullback Lblr sparaton of P an P for a quartt tr wth xtror gs of lngths L= an ntror g of lngth =. an =. as functons of λ. Bottom: Kullback Lblr sparaton of P an P for a quartt tr wth xtror gs of lngths L= an L= as shown n Fg. (), an =.. probablty unr P for > an zro probablty unr P ). Howvr, a mor funamntal rason for our choc of KL( P, P ) s that t s mor natural to consr th unrsolv tr ( =)asth null hypothss an th rsolv tr as th altrnatv hypothss bcaus on typcally wshs to sprov th null whch n our cas mans to rct th polytomy. Acknowlgmnts W thank Jffry Townsn an Danl Wgmann for hlpful scussons. W also thank th thr anonymous rvwrs for numrous hlpful suggstons that hav mprov th papr. Rfrncs Covr, T.M., Thomas, J.A., 6. Elmnts of Informaton Thory n a. Wly- Intrscnc. Fschr, M., Stl, M., 9. Squnc lngth bouns for rsolvng a p phylogntc vrgnc. J. Thor. Bol. 6, 7. Golman, N Phylogntc nformaton an xprmntal sgn n molcular systmatcs, Procngs Roy. Soc. B. 6, pp Hny, M.D., 989. Th rlatonshp btwn smpl volutonary tr mols an obsrvabl squnc ata. Syst. Bol. 8,. Kmura, M., Crow, J., 96. Th numbr of allls that can b mantan n a fnt populaton. Gntcs 9, Klopfstn, S., Kropf, C., Quck, D.L.J.,. An valuaton of phylogntc nformatvnss profls an th molcular phylogny of Dplazontna (Hymnoptra, Ichnumona). Syst. Bol. 9 (), 6. Lhmann, E.L., Caslla, G., 998. Thory of pont stmaton n. Sprngr. Lws, P.O., Chn, M.-H., Luo, L., Lws, L.A., Fučková, K., Nupan, S., Wang, Y.-B., Sh, D., 6. Estmatng baysan phylogntc nformaton contnt. Syst. Bol. 6 (6), 9. Stl, M., 6. Phylogny: scrt an ranom procsss n voluton. CMBS-NSF Rgonal Confrnc Srs n Appl Mathmatcs No. 89. SIAM Phlalpha PA. Townsn, J.P., 7. Proflng phylogntc nformatvnss. Syst. Bol. 6 (),. Townsn, J.P., Lunbrgr, C.,. Taxon samplng an th optmal rats of voluton for phylogntc nfrnc. Syst. Bol. 6 (), 8 6. Townsn, J.P., Su, Z., Tkl, Y.L.,. Phylogntc sgnal an nos: prctng th powr of a ata st to rsolv phylogny. Syst. Bol. 6 (), Yang, Z., 998. On th bst volutonary rat for phylogntc analyss. Syst. Bol. 7,. 6

10/7/14. Mixture Models. Comp 135 Introduction to Machine Learning and Data Mining. Maximum likelihood estimation. Mixture of Normals in 1D

10/7/14. Mixture Models. Comp 135 Introduction to Machine Learning and Data Mining. Maximum likelihood estimation. Mixture of Normals in 1D Comp 35 Introducton to Machn Larnng and Data Mnng Fall 204 rofssor: Ron Khardon Mxtur Modls Motvatd by soft k-mans w dvlopd a gnratv modl for clustrng. Assum thr ar k clustrs Clustrs ar not rqurd to hav