Knowledge structures (Doignon & Falmagne, 1985, 1999) Parameter estimation in probabilistic knowledge structures with the pks package

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1 Knowlg struturs (Doignon & Flmgn, 1985, 1999) Prmtr stimtion in probbilisti knowlg struturs with th pks pkg Florin Wiklmir n Jürgn Hllr Psyhoo 2012, Innsbruk, Fbrury 9 Gols Chrtrizing th strngths n wknsss in ll prts of knowlg omin Pris, non-numril hrtriztion of th stt of knowlg tht is omputtionlly trtbl Builing upon rsults from isrt mthmtis n xploiting th powr of urrnt omputrs Aptiv knowlg ssssmnt Effiintly intifying th urrnt stt of knowlg bs on sking miniml numbr of qustions Apting to th lry givn rsponss s xprin thrs o in n orl xmintion Prsonliztion in thnology-nhn lrning Automtilly slt ontnt tht prson is ry to lrn 2 A subomin of physis: Consrvtion of mttr (1) (Tgpr t l., 1997) ) Whn i mlts n prous wtr: (i) Th wtr wighs mor thn th i. (ii) Th i wighs mor thn th wtr. (iii) Th wtr n i wigh th sm. (iv) Th wight pns on th tmprtur. b) Aftr th nil rusts, its mss: (i) is grtr thn bfor. (ii) is lss thn bfor. (iii) is th sm s bfor. (iv) nnot b prit. ) Whn 10 grms of iron n 10 grms of oxygn ombin, th totl mount of mtril ftr iron oxi (rust) is form must wigh: (i) 10 grms. (ii) 19 grms. (iii) 20 grms. (iv) 21 grms. 3 A subomin of physis: Consrvtion of mttr (2) (Tgpr t l., 1997) ) Aftr 3 mtl nuts n 3 mtl bolts r join togthr: (i) Th totl mount of mtl is th sm. (ii) Thr is lss mtl thn bfor. (iii) Thr is mor mtl thn bfor. (iv) Th mount of mtl nnot b trmin. ) Photosynthsis n b srib s: WATER + CARBON DIOXIDE hlorophyll GLUCOSE sunlight Whih of th following sttmnts bout this rtion is NOT tru? (i) As mor wtr n mor rbon ioxi rt, mor gluos is prou. (ii) Th sm mount of gluos is prou no mttr how muh wtr n rbon ioxi is vilbl. (iii) Chlorophyll n sunlight r n for th rtion. (iv) Th sm toms mk up th GLUCOSE molul s wr prsnt in WATER n CARBON DIOXIDE. 4

2 Rspons pttrns (Tgpr t l., 1997) Stunts from grs four through twlv N = Dtrministi thory Dfinitions A knowlg omin is intifi with st of (ihotomous) itms. Th knowlg stt of prson is intifi with th subst K of problms in th omin th prson is pbl of solving. A knowlg strutur on th omin is olltion K of substs of tht ontins t lst th mpty st n th st. Th substs K K r th knowlg stts Frquny 5 6 Consrvtion of mttr: Knowlg strutur (Tgpr t l., 1997) Probbilisti knowlg struturs Rtionl If thr r rspons rrors thn knowlg stts K n rspons pttrns R hv to b issoit. Dfinition A probbilisti knowlg strutur is fin by spifying knowlg strutur K on knowlg omin (i.., olltion K 2 with, K) mrginl istribution P K (K) on th knowlg stts K K th onitionl probbilitis P(R K) to obsrv rspons pttrn R givn knowlg stt K Th probbility of th rspons pttrn R R = 2 is prit by P R (R) = K K P(R K)P K (K) 7 8

3 Th bsi lol inpnn mol (BLIM) (Doignon & Flmgn, 1999) Assumption: Lol stohsti inpnn Givn th knowlg stt K of prson th rsponss r stohstilly inpnnt ovr problms th rspons to h problm q only pns on th probbilitis β q of rlss rror η q of luky guss Th probbility of th rspons pttrn R givn th knowlg stt K rs P(R K) = (1 β q ) (1 η q ). q K\R β q q K R q R\K η q q \(R K) Th pks pkg Provis funtionlity for prmtr stimtion in probbilisti knowlg struturs. Min funtions blim print, loglik plot, rsiuls simult s.pttrn,s.binmt Fitting n tsting bsi lol inpnn mols (BLIMs) Extrtor funtions gnrt rspons pttrns from givn BLIM onvrsion funtions Mximum liklihoo stimtion 9 Exmpl: Mximum liklihoo stimtion 10 EM lgorithm Formult th liklihoo s if w hv vilbl th bsolut frqunis M RK of subjts who r in stt K n prou pttrn R (omplt t) inst of th bsolut frqunis N R of th rspons pttrns R R (inomplt t). E(M RK ) = Expttion Comput N R P(K R, ˆβ (t), ˆη (t), ˆπ (t) ) Mximiztion Estimt ˆβ (t+1), ˆη (t+1), ˆπ (t+1) bs on m RK = E(M RK ) β β β β η η η η 11 12

4 Exmpl: Mximum liklihoo stimtion Mximum liklihoo stimtion blim(mttr97$k, mttr97$n.r, mtho="ml") Numbr of itrtions: 9474 Goonss of fit (2 log liklihoo rtio): G2(7) = , p = Minimum isrpny istribution (mn = ) Mn numbr of rrors (totl = ) rlss rror luky guss Problms Goo fit (w.r.t. liklihoo rtio sttisti) not suffiint for mpiril vliity of knowlg strutur Fit my b obtin by inflting rlss rror rts β q n luky guss rts η q Wht w wnt: Goo fit with smll vlus of β q n η q How to pply onstrints on th rror probbilitis tht r motivt by th knowlg strutur? (inst of brut-for onstrints, Stfnutti & Robusto, 2009) How muh of th fit is u to inflting th rror probbilitis in ML stimtion? Minimum isrpny mtho Rtionl For rspons pttrn R n knowlg stt K onsir th istn (R, K) = (R \ K) (K \ R), whih is bs on th symmtri st-iffrn. It is th numbr of itms tht r lmnts of ithr, but not both sts R n K (numbr of rspons rrors). Exmpl (10001, 10100) = Minimum isrpny mtho Rtionl For givn rspons pttrn R, onsir th minimum of th symmtri istns btwn R n ll th knowlg stts K K (R, K) = min (R, K). K K Th bsi i is tht ny rspons pttrn is ssum to b gnrt by los knowlg stt ls to xpliit (i.., non-itrtiv) stimtors of th rror probbilitis minimizs th numbr of rspons rrors n thus ountrts n infltion of rlss rror n luky guss probbilitis A prviously suggst implmnttion of this i by Shrpp (1999, 2001) os not llow for itm spifi stimts

5 Minimum isrpny mtho Exmpl: Minimum isrpny stimtion Assumptions A knowlg stt K K is ssign to rspons pttrn R R only if th istn (R, K) is miniml Eh of th miniml isrpnt knowlg stts is ssign with th sm probbility with i RK = ˆP(K R) = i RK K K i RK { 1 (R, K) = (R, K) 0 othrwis β β β β η η η η Exmpl: Minimum isrpny stimtion 17 Minimum isrpny ML stimtion 18 blim(mttr97$k, mttr97$n.r, mtho="md") Numbr of itrtions: 1 Goonss of fit (2 log liklihoo rtio): G2(7) = , p = 0 Minimum isrpny istribution (mn = ) Mn numbr of rrors (totl = ) rlss rror luky guss Moifi EM lgorithm Moify th E-stp in th EM lgorithm to implmnt th rstrition m RK = E(M RK N R, ˆβ (t), ˆη (t), ˆπ (t) ) = 0 whnvr (R, K) > (R, K). This ls to i RK P(K R, ˆβ (t), ˆη (t), ˆπ (t) ) m RK = N R K K i RK P(K R, ˆβ (t), ˆη (t), ˆπ (t) ) Th M-stp pros s usul

6 Exmpl: Minimum isrpny ML stimtion Exmpl: Minimum isrpny ML stimtion β blim(mttr97$k, mttr97$n.r, mtho="mdml") β β β η η η η Numbr of itrtions: 133 Goonss of fit (2 log liklihoo rtio): G2(7) = , p = 0 Minimum isrpny istribution (mn = ) Mn numbr of rrors (totl = ) rlss rror luky guss Outlook Th pks pkg fturs Fitting n tsting bsi lol inpnn mols (BLIMs) Rspons gnrtion from givn BLIM objt Mximum liklihoo, minimum isrpny, n MDML stimtion Work in progrss Smpling istributions for goonss of fit tsts Gnrliz MDML ritrion: troff btwn liklihoo mximiztion n rror minimiztion... Thnk you for your ttntion florin.wiklmir@uni-tubingn. Florin Wiklmir [r, ut] Jürgn Hllr [ut] Psqul Anslmi [tb] 23 24

7 Rfrns Aitionl slis Rfrns Aitionl slis Rfrns Doignon, J.-P. & Flmgn, J.-C. (1985). Sps for th ssssmnt of knowlg. Intrntionl Journl of Mn-Mhin Stuis, 23, Doignon, J.-P. & Flmgn, J.-C. (1999). Knowlg sps. Brlin: Springr. Shrpp, M. (1999). Extrting knowlg struturs from obsrv t. British Journl of Mthmtil n Sttistil Psyhology, 52, Shrpp, M. (2001). A mtho for ompring knowlg struturs onrning thir quy. Journl of Mthmtil Psyhology, 45, Stfnutti, L. & Robusto, E. (2009). Rovring probbilisti knowlg strutur by onstrining its prmtr sp. Psyhomtrik, 74, Tgpr, M., Pottr, F., Millr, G. E., & Lkshminryn, K. (1997). Mpping stunts thinking pttrns by th us of th knowlg sp thory. Intrntionl Journl of Sin Eution, 19, Exmpl: Gnrliz MDML stimtion blim(mttr97$k, mttr97$n.r, mtho="mdml", inrius=1) Numbr of knowlg stts: 15 Numbr of rspons pttrns: 32 Numbr of rsponnts: 1620 Mtho: Minimum isrpny mximum liklihoo Numbr of itrtions: 1679 Goonss of fit (2 log liklihoo rtio): G2(7) = 47.11, p = Minimum isrpny istribution (mn = ) Mn numbr of rrors (totl = ) rlss rror luky guss Rfrns Exmpl: Gnrliz MDML stimtion β 25 Aitionl slis Error n gussing prmtrs bt t b β β β η η η η 27