Statistical modeling with stochastic processes. Alexandre Bouchard-Côté Lecture 11, Monday April 4
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1 Sttistil mdeling with sthsti presses Alexndre Buhrd-Côté Leture 11, Mndy April 4 1
2 Prgrm fr tdy Bet, Pissn nd Gmm presses DDP nd sequene memizer 2
3 Pitmn-Yr press Pitmn-Yr press: Strt with the CRP, nd bst the prbbility f tble retin while preserving exhngebility This hs the sme nrmliztin s the DP: α0 + n New ustmer... n1 - d nt - d α0 + t d Jin tble #1, with lredy n1 peple sitting there Jin tble # t, with lredy nt peple sitting there Crete new tble Disunt: d [0, 1] 3
4 PY: stik breking nstrutin Dirihlet press: defined G = f(β, θ) fr n iid sequene f θi ~ G0 nd: βi ~ Bet(1, α0), Pitmn-Yr: Sme but nw bet s re nt identilly dist.: βi ~ Bet(1 - d, α0 + i d) 4
5 The infinite HMM Infinite HMMs: π Trnsitin prmeters π x x1 x2 x3 Emissin prmeters θ x x=1.. y1 y2 y3 5
6 Feture bsed representtins Stte-split Feture xi 2 F sttes (1) xi (2) xi 2 sttes 2 sttes... yi (F) xi 2 sttes yi Hw mny fetures? Will see sn slutin: Bet press 6
7 Ltent Feture Mdels Bet press DP: Ltent Feture Mdels DP: Mixture inditr prirs: Dtpint index z1 z2 z1 zz3 2 zz34 zz45 z56 φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 Cluster index φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 Dirihlet press; Pitmn-Yr press z6 Desired: Feture index φ φ φ φ φ φ φ Desired: 1 Feture inditr prirs: Dtpint index z1 zz12 z2 z3 z3 zz4 4 zz5 5 zz φ8 φ9 φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 Bet press 7
8 Preditive distributin: resturnt metphr Insted f sit-dwn resturnt, think f buffet with n infinite sequene f dishes θi smpled by ustmers Custmer #1 Custmer #2... z 1 z 2 z 3 z 4 z 5 z 6 θ1... θ2θ3θ4 Obvius: rder f the lumns nt imprtnt/exhngeble (beuse the θi s will be generted iid) Less bvius: hw t mke the rder f the rws exhngeble 8
9 Preditive distributin: resturnt metphr First ustmer:... Smple Pissn(α) number f dishes. Furth ustmer:... Smple Pissn(α/i) number f new dishes. Smple previusly tried dishes in prprtin t the number f peple wh hve previusly tried them. (Exmple n the brd) Slide frm Kurt Miller 9
10 Bet press: stik breking representtin Interprettin f the sequene f stiks (πj)j=1.. πj is the prir prbbility f piking rw j Cnsequene: the stiks n lnger sum t ne! Cnstrutin (will me bk t it lter): Bet press: β k Bet(1, α) k π k = (1 β l ) l=1 Cf.: Dirihlet press β k Bet(1, α) k 1 π k = β k l=1 (1 β l ) 10
11 Pissn presses 11
12 Pissn presses Anther rndm disrete mesure, but unnrmlized: Let P0 be distributin n smple spe Ω (the bse distributin) nd (A1,..., Ak) be prtitin f Ω. We sy P PP(P 0 ) i.e., P is Pissn Press, if fr ll prtitins nd ll k. ind. P (A 1 ) Pi(P 0 (A 1 )) A1 A2 12
13 Cf: Dirihlet Press Let G0 be distributin n smple spe Ω (the bse distributin) α0 be psitive rel number (the nentrtin), nd (A1,..., Ak) be prtitin f Ω. We sy i.e., G is Dirihlet Press, if fr ll prtitins nd ll k. G DP(α 0,G 0 ) (G(A 1 ),..., G(A k )) Dir(α 0 G 0 (A 1 ),..., α 0 G 0 (A k )) 13
14 Cnsisteny/existene Let P0 be distributin n smple spe Ω (the bse distributin) nd (A1,..., Ak) be prtitin f Ω. We sy i.e., P is Pissn Press, if fr ll prtitins nd ll k. P PP(P 0 ) ind. P (A 1 ) Pi(P 0 (A 1 )) A1 A2 B1 B2 B3 14
15 Cmpbell s therem Assume P0 is prbbility mesure, f is bunded, nd P ~ PP(P0). Let ls: Σ = X P f(x) { } Then: E [ e itσ] = exp (e itf(x) 1)P 0 (dx) Ω 15
16 Sequene memizer 16
17 Bk t hierrhil mdels Hyper-prir ver wrds---nt speifi t prefix Distributin ver wrds in text dtset Distributin ver wht fllws fter the prefix Distributin ver wht fllws fter the prefix Fix... Distributin ver wht fllws fter the prefix fix
18 Mre elbrte exmple Trining: BEG END H G[BEG] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [ END () Prefix trie fr. 18
19 Mrginliztin Trining: BEG END H G[BEG] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [ END () Prefix trie fr. 19
20 Anlyti mrginliztin H Trining: BEG END ] G[BEG] G [] G [] G [] G [] G [] G [] G [] G d G d G [] END Anlytilly pssible when: G s G σ(s) PY(α σ(s) d s,d s ) 20
21 Cnditin fr nlyti mrginliztin H G[BEG] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [] G [ END () Prefix trie fr. 21
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