Optimal inference of sameness Supporting information

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1 Optmal nference of amene Supportng nformaton Content Decon rule of the optmal oberver.... Unequal relablte.... Equal relablte... 5 Repone probablte of the optmal oberver Equal relablte Unequal relablte Proof that A non-negatve defnte n the unequal-relablte cae Suboptmal model Repone probablte for the ngle-crteron and blocwe-crteron model Sngle-crteron model for Experment Blocwe-crteron model for Experment Maxmum-abolute-dfference model... 4 Bayean model comparon... 5 Smulaton of anmal cognton experment Method for Experment, color Supplementary Table Supplementary Fgure... 5 Decon rule of the optmal oberver Snce equal relablte (Experment ) are a pecal cae of unequal relablte (Experment ), we treat the latter frt.. Unequal relablte The generatve model of the ta gven n Fgure A. The varable are a follow: C a bnary varable that denote amene (+ for ame, for dfferent), denote the vector of orentaton preented, and x denote the correpondng vector of nternal repreentaton. Each x drawn from a Gauan dtrbuton wth mean and tandard devaton. The Gauan aumpton common, and reflect the preence of many ndependent, unbaed ource of noe (). Although orentaton pace crcular, we can treat t a the real lne, becaue the nternal repreentaton are cloe to the true tmulu and much maller than the

2 crcle crcumference. We refer to / a the relablty of the th obervaton. When the tmul are the ame, each orentaton equal to. When the tmul are dfferent, each drawn ndependently from a Gauan dtrbuton wth mean and tandard devaton. Regardle of amene, the value of drawn from a unform dtrbuton on [L,L) (one could tae L=/ here, but any L wor). Optmal (Bayean) oberver bae ther decon on the poteror probablty dtrbuton over the varable of nteret, here C, gven ngle-tral obervaton, here x=(x,,x ) T. Snce C a bnary random varable, th poteror reduce to a ngle number, whch can be expreed n many way. A partcularly convenent way to expre t a a log poteror rato: p C x p x C p C d log log log. (S) p C x p x C p C Evaluatng the lelhood n th expreon, p(x C) for C=±, requre margnalzaton over the tmulu orentaton, =(,, ) T, and ther mean, :, p x C p x p C p d d. (S) We aume that the tandard devaton,, of the noe aocated wth a tmulu nown to the oberver for each tmulu and each tral, a would be the cae n a Poon-le populaton code (). Therefore, we do not need to margnalze over, but can treat t a a nown parameter. We now evaluate Eq. (S) by ubttutng the nown dtrbuton. If denote the column vector of length wth all entre equal to, and (x) the Drac Delta dtrbuton, then px C p d d L x p d L x x exp L x exp d. L d (S3) In the ret of th document, all um and product run from to unle mentoned otherwe. We rewrte the um n the exponent a

3 x x x x V, (S4) wth V x x. If all relablte are equal, o that =, then V mplfe to Var x V x x. Here, Var x defned a Var x x x. For the general cae, n whch each tem come wth t own relablty, we ubttute Eq. (S4) nto Eq. (S3) to fnd V exp p x C exp d L x V exp L where n the econd equalty, we aumed that the doman of large compared to, and o the ntegral can be evaluated over the entre real lne. Th aumpton jutfed here, becaue n our ta the maxmum-lelhood etmate of a tmulu dtrbuted around the true value of the tmulu n a range that much narrower than the entre crcular pace. Startng from Eq. (S), we repeat th calculaton for the hypothe that the tmul are dfferent, that C = :, 3

4 x L px C exp exp d d L L x L exp exp d d L L L x exp L d L V exp L x exp d L L V exp L, where V x x. The log lelhood rato, the frt term on the rght-hand de of Eq. (S), now become p x C V V log log log p x C wx wx w w w wx log log w w w w ww jxx j ww jxx j j j w wx w w w w log log, w w 4

5 where w and w are column vector wth entre w, w. Therefore, the log poteror rato, d, a quadratc form n x. In matrx notaton, we can wrte d w w p w w p T ame xaσ x log log log, ame (S5) where p(c=)=p ame and A() a ymmetrc, relablty-dependent matrx wth entre ww j ww j Aj w w j w w. (S6) Maxmum-a-poteror etmaton amount to repondng ame whenever d potve. Th lead to Eq. () n the man text.. Equal relablte When all relablte are equal, = and therefore w =w and entre of A reduce to A w w j w w, a n Experment, the j, and the quadratc form x T Ax become w w Var x. The decon rule, whch decrbe when the optmal oberver repond C=, tae the form w p Var x log log ww w p ame ame. (S7) Thu, the decon rule amount to comparng the varance to a crteron that depend n a pecfc way on the common relablty of the tem (hdden n w and w ) and on et ze,. 5

6 Repone probablte of the optmal oberver In the prevou ecton, we derved the decon rule of the optmal oberver for both experment. However, thee decon rule cannot be teted drectly, nce a expermenter we do not have acce to the nternal obervaton x. We only now the preented tmul,, and the ubject repone on each tral. To obtan tetable model predcton, we compute the theoretcal probablty of a ame repone gven by the optmal oberver, when the obervaton x are drawn from the generatve model for gven. In the reultng model predcton, x no longer preent. The complete model thu cont of two part merged together : the generatve model, producng a theoretcal (or mulated) et of obervaton, and the nference model, whch decrbe the decon made by the optmal oberver baed on thoe obervaton. Formally, we denote the oberver etmate of C by Ĉ ; for the optmal oberver, t equal the gn of d. The optmal oberver repond accordng to th etmate. Eq. (S5) determne the optmal decon rule on a ngle tral. We denote the probablty of the optmal oberver repondng Ĉ when the expermenter preent a pecfc tmulu et by ˆ obtaned by averagng over the hypothetcal obervaton x generated by : ˆ ˆ C d x ˆ,gn pc. It p C p C x p x d x p x, σ d x, (S8) where the Kronecer delta. In word, the probablty of repondng C ˆ equal to the proporton of obervaton drawn from p(x ) for whch d(x) potve. For many pychophycal ta, an ntegral le that gven n Eq. (S8) analytcally ntractable, and need to be approxmated, for example, by Monte Carlo mulaton. In the preent ta, an analytcal treatment poble.. Equal relablte We frt conder the cae of Experment, wth the decon rule gven by Eq. (S7). We are ntereted n the probablty that th nequalty atfed for x drawn from a multvarate normal dtrbuton wth mean and covarance matrx I, denoted x~(, I), wth I the dentty matrx. We wrte Var x=x T A'x, where the component of A' are A' j = j //. Th matrx ha egenvalue equal to, and one egenvalue equal to. Therefore, for an orthogonal matrx M and dagonal matrx D=dag(,,,,), we can wrte M T A'M=D. Let Y be Y=M T x. Snce M orthogonal, y~(, I), wth =M T. Snce y T T T T YDY xmdmx xax, ' (S9) t follow that x T A'x/ follow a non-central ch-quared dtrbuton wth degree of 6

7 freedom. Th varable equal to wvar x. The noncentralty parameter of the um on the left of Eq. (S9), whch can be rewrtten a T T T T μ Dμ MDM A ' w Var. Therefore, we have w Var x~ w Var. (S) From Eq. (S7), the probablty of repondng ame equal the probablty that w w p ame w Var x le than log log. Ung Eq. (S), th probablty ww w pame obtaned from a cumulatve ch-quared dtrbuton: ˆ w w p p C Pr wvar log log ww w p ame ame. (S). Unequal relablte We next conder the cae of Experment, where relablty can dffer between tmul. Then x follow a multvarate normal dtrbuton wth mean and covarance matrx =dag(, ). We defne a random varable Q(x)=x T Bx, where B a non-negatve defnte and ymmetrc matrx. We denote mean and tandard devaton of Q by Q and Q, repectvely. Lu, Tang, and Zhang (3) have found the followng approxmaton to the dtrbuton of Q: T Pr xbx Pr l t (S) where the varable l,, t,, and are defned a follow: 7

8 Q t l 3a ra ra 3 3 a 3 a ra a, f r r r r a, f r r c r 3 3/ c r c 4 c j c Tr Q j T j r r j BΣ BΣ B umercal mulaton how that for our purpoe, th a very good approxmaton. In order to apply th approxmaton to x T Ax from Eq.(S5), A ha to be both ymmetrc and non-negatve defnte. Snce A obvouly ymmetrc, what reman to be hown that n our cae, A nonnegatve defnte..3 Proof that A non negatve defnte n the unequal relablte cae The matrx A pecfed by Eq. (S6) can be wrtten a T ww ww Adag ww T w T T w. To how that A non-negatve defnte (potve emdefnte), we prove that t egenvalue are non-negatve. In Experment, m of the are equal to low, and m are equal to hgh. Wthout lo of generalty, we arrange them a =( low,, low, hgh,, hgh ),.e., the frt m correpond to low-relablty tmul, and the lat m to hgh-relablty one. Conequently, w = =w m =w low =/ low and w m+ = =w =w hgh =/ hgh. The (,j) th term of A gven by Eq. (S6). ote that 8

9 A ww j ww j Aj w w j j w w, (S3) we ee that =(,,) T an egenvector wth egenvalue. We now defne a et of m vector v, wth =,,m, a follow: v. For example, v 3 =(,,,, ) T. Thee vector are egenvector of A wth egenvalue w w, nce low low Av Av j j Aj j j j j ww ww j j j w w j j j w w ww ww j www www w w w w j j w w j j j j j w w w low wlow wlow wlow v. Smlarly, the m vector v wth =m+,,, whoe th entre are v, m are egenvector wth egenvalue w hgh w hgh. That leave one egenvalue to fnd. Conder the vector v=(m,, m,m,,m) T. We wll how that th vector an egenvector. We frt calculate the product of any of the frt m row of A wth v, mang ue of Eq. (S3) 9

10 Av m m A m A j j jm j wlow w wlow whgh w low lowwhgh mwlow wlow m m m m w w w w w w w w w w w w m m m m w w w w w w w w m w w low hgh low hgh low hgh low hgh low hgh low hgh m m m m m m low hgh low hgh low hgh low hgh m m m m Smlarly, for any of the lat m row of A, we get. Av m. m low m hgh m low m hgh Thu, m m m m low hgh low hgh the fnal egenvalue of A. Snce all egenvalue are non-negatve, A non-negatve defnte, and therefore we may apply the approxmaton dcued n Secton..

11 3 Suboptmal model 3. Repone probablte for the ngle crteron and blocwe crteron model Calculatng a model predcton for the probablty of repondng ame gven a et of preented orentaton cont of two tep: to determne the decon rule for repondng ame, and to apply th decon rule to the et of nternal repreentaton on each ndvdual tral. In non-optmal oberver model, the frt tep (decon rule) dfferent, but the econd tep (calculatng the probablty wth whch the decon atfed) dentcal. In partcular, a long a the decon rule of the form T xbx, (S4) wth B a non-negatve defnte, ymmetrc matrx, we can tll ue Eq. (S) to approxmate the model repone probablte. The only dfference from the optmal model that dfferent expreon mut be ubttuted for B and/or. In the pecal cae that the decon rule Var x< and the relablte are equal, the model repone probablte are gven by an exact expreon analogou to Eq. (S), ˆ Pr Var p C w w. 3. Sngle crteron model for Experment In Experment, the SC oberver aume n the decon rule that all relablte are equal, = aumed for all. In other word, th oberver doe not weght the obervaton by ther correct repectve relablte. We wll how here that th model equvalent to one n whch the oberver compare the ample varance to a ngle crteron. The decon rule the ame one a n Experment, Eq. (S7), but wth aumed ntead of. Th rule can be rewrtten a p Var x log log aumed aumed ame aumed pame. (S5) Snce aumed a free parameter, the rght-hand de can aume any value on the real lne. To ee th, frt note that th expreon contnuou. Then compute the two lmt

12 aumed aumed p ame lm log log aumed pame aumed aumed p ame lm log log aumed pame lm aumed log a umed aumed aumed aumed Therefore, we can mply replace the entre rght-hand de by a ngle free parameter,.e., Var x<, thereby jutfyng the termnology ngle-crteron model. (ote that n order to obtan model predcton, we tll need to ft the parameter low and hgh.) Thu, the SC model ha only two varant: wth and wthout lape rate. 3.3 Blocwe crteron model for Experment In Experment, the BC model the model n whch aumed may vary by bloc type (LOW, MIXED, or HIGH). Then, the decon rule equvalent to Var x< bloc, where each bloc type ha t own free parameter bloc. Th model analogou to the BC model n Experment, wth relablty condton (LOW, MIXED, or HIGH) tang the place of et ze. A n the SC model, there are only two varant: wth and wthout lape rate. 3.4 Maxmum abolute dfference model For the MAD model, the repone probablte cannot be calculated analytcally. We etmated thee probablte numercally, by mulatng, et of nternal repreentaton for each of the 7 expermental tral, and applyng the model decon rule to each et; the frequency of ame repone wa an etmate of the probablty of repondng ame on the gven tral. 4 Bayean model comparon We denote by p(ĉ,m,) the probablty predcted by model M wth parameter of the ubject actual repone on the th tral, Ĉ, when the preented tmul are. We computed the log probablty of the data gven M by margnalzng over. For the pror over parameter, we aumed a unform dtrbuton, p()=/vol, where Vol the volume of parameter pace. We calculated the parameter lelhood by aumng that the data are condtonally ndependent acro tral: p(data M,)= p(ĉ,m,). For the MAD model, we approxmated the margnalzaton through a Remann um; the parameter range were to 4 for the decon crteron,. to. for, and to.5 for the lape rate, each n 35 tep. For the other model, nce analytcal expreon were avalable, we ued the Laplace approxmaton (4); the ze of the parameter range were for, for the aumed,.4 for p ame, and.4 for the lape rate.

13 5 Smulaton of anmal cognton experment To examne whether the proporton of dfferent repone from the optmal oberver correlate wth the entropy of the tmulu et, we mulated the tmulu et ued by Young et al. (5) (ee Table n ther paper) and ued thee a nput to the optmal-oberver model. Thee et alway contaned 6 tem, wth everal ubet of dentcal tem (e.g., 4 ubet of 4 dentcal tem each). In our mulaton, for each ubet, a random tmulu value wa drawn from a Gauan dtrbuton (wth σ =5) and agned to all tem n that ubet. A total of tral were mulated per tmulu et. On each tral, tmulu obervaton were mulated by addng Gauan noe to the tmulu orentaton (σ=) and a repone wa generated by applyng the decon rule from the optmal-oberver model. Entropy and caled logt of percent dfferent repone were computed a decrbed by Young et al. (5). To examne the effect of et ze on the proporton of dfferent repone n the optmal-oberver model, we mulated tral per et ze, wth σ =8 and σ=8. To examne the effect of tmulu vblty, we vared nternal noe, σ, and mulated tral per noe level (wth σ =5, =8, p ame =.6, and a gueng rate of.5). Whle model parameter were eparately choen for each experment, the oberved trend were robut under a wde range of parameter value. 6 Experment (color) Method The method for Experment, color, were dentcal to the Experment for orentaton, except for the followng dfference. Stmul conted of a et of colored dc wth a radu of.4 deg. The color of the dc were drawn ndependently from 8 color value unformly dtrbuted along a crcle of radu 5 n CIE 976 (L*,a*,b*) color pace. Th crcle had contant lumnance (L*=58) and wa centered at the pont (a*=, b*=3). The tmul were preented on a grey bacground of lumnance cd/m. Set ze wa choen randomly on each tral. In the model, blocwe crteron wa now defned a one crteron per et ze, rather than per bloc. The experment conted of 3 eon wth 6 bloc of 5 tral. Two author and fve pad, nave ubject partcpated n the experment. Reult The SC model can be ruled out but not the BC model (Fg. SA-B; log lelhood dfference between the optmal model and the SC, BC, MAD-SC, and MAD-BC model were 8.7±4.9, 3.9±.5, 7.3±6.3, and.8±.3, repectvely). Even though the BC model jut edge out the optmal model, the decon crtera and noe level from the bet BC model are very cloe to thoe predcted by the bet optmal-model varant (Fg. SC; p>. for all x t-tet). Fnally, the noe level exhbt a wea dependence on et ze (Fg. SC; power law ft yeld a power of.3±.4). 3

14 7 Supplementary Table Table S. Overvew of model parameter. The model have the ame parameter n both experment, except that n Experment, σ=(σ =, σ =4, σ =8 ) and =( =, =4, =8 ), whle n Experment, σ=(σ low, σ hgh ) and =( LOW, MIXED, HIGH ). Model Optmal Optmal Optmal Optmal Optmal Optmal Optmal Optmal Sngle-crteron Sngle-crteron Blocwe-crteron Blocwe-crteron MAD wth ngle crteron MAD wth ngle crteron MAD wth blocwe crteron MAD wth blocwe crteron Free parameter σ σ, λ σ, σ σ, σ, λ σ, p ame σ, p ame, λ σ, p ame, σ σ, p ame, σ, λ σ, σ,, λ σ, σ,, λ σ, σ,, λ σ, σ,, λ Table S. Overvew of maxmum-lelhood parameter value of the bet fttng optmal model n Experment. An empty cell ndcate that the repectve parameter wa not a free parameter n the bet-fttng model varant. p ame σ λ AK DS.4.4 HB MH. ML.4.5 RB.3 RC.6 TR.46. 4

15 Table S3. Overvew of maxmum-lelhood parameter value of the bet-fttng optmal model n Experment. An empty cell ndcate that the repectve parameter wa not a free parameter n the bet-fttng model varant. 8 Supplementary Fgure A Proporton "dfferent" repone B Relatve log lelhood Subject number p ame σ λ B.53.8 DB. DS 9.5. HB.45. KJ.45.7 MH.44. ML MV.46.4 RB RC.55 Optmal SC BC MAD-SC MAD-BC Standard devaton () Standard devaton () Standard devaton () Standard devaton () Standard devaton () SC BC MAD-SC MAD-BC Optmal model better Optmal model wore 4 8 Set ze Fg. S Comparon of model n Experment 3. Crcle and error bar repreent mean and.e.m. of ubject data. Shaded area repreent.e.m. of model ft. (A) Proporton dfferent repone a a functon of ample tandard devaton for optmal and uboptmal model. (B) Bayean model comparon. Each bar repreent the log lelhood of the optmal model mnu that of a uboptmal model. (C) The decon crtera (left) and the nternal noe level (rght) for the betfttng BC model are nearly dentcal to thoe of the bet-fttng optmal-oberver model. Th ugget that the human crtera are cloe to optmal. C Decon crteron ( ) BC Optmal Internal noe, σ () BC Optmal 4 8 Set ze 5

16 A Optmal model B BC model hgh = Probablty.. Same Dfferent 5 5 Probablty hgh = Probablty Probablty hgh =3 Probablty Probablty hgh =4 Probablty Probablty hgh =5 Probablty.. Probablty Decon varable ( ) Decon varable ( ) Fg. S. A comparon of the decon tratege of the optmal and BC model n Experment. (A) Dtrbuton of the decon varable of the optmal model. A the number of hgh-relablty tmul n the dplay, hgh, ncreae, the dtrbuton become more eparable, mang the ta eaer. The optmal oberver et h crteron (dahed lne) n uch a way that performance maxmzed. (B) Dtrbuton of the decon varable of the BC model (Var x). Th model ue the ame crteron regardle of hgh (dahed lne). Hence, t cannot maxmze performance for all value of hgh.the ame hold for the SC, MAD-SC, and MAD-BC model. The dtrbuton dffer between the model becaue the decon varable do.... Green DM & Swet JA (966) Sgnal detecton theory and pychophyc (John Wley & Son, Lo Alto, CA).. Ma WJ, Bec JM, Latham PE, & Pouget A (6) Bayean nference wth probabltc populaton code. at euroc 9(): Lu H, Tang Y, & Zhang HH (9) A new ch-quare approxmaton to the dtrbuton of non-negatve defnte quadratc form n non-central normal varable. Comp Stattc and Data Analy 53: MacKay DJ (3) Informaton theory, nference, and learnng algorthm (Cambrdge Unverty Pre, Cambrdge, UK). 5. Young ME & Waerman EA (997) Entropy detecton by pgeon: repone to mxed vual dplay after ame-dfferent dcrmnaton tranng. J Exp Pychol Anm Behav Proce 3():

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