Bayesian parameter estimation. Nuno Vasconcelos UCSD

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1 Byes prmeter estmto Nuo Vscocelos UCS

2 Byes prmeter estmto the m dfferece wth respect to ML s tht the Byes cse Θ s rdom vrble bsc cocepts trg set {... } of emples drw depedetly probblty desty for observtos gve prmeter X Θ pror dstrbuto for prmeter cofgurtos Θ tht ecodes pror belefs bout them gol: to compute the posteror dstrbuto Θ X

3 ByesvsML there re umber of sgfct dffereces betwee Byes d ML estmtes : ML produces umber the best estmte to mesure ts goodess we eed to mesure bs d vrce ths c oly be doe wth repeted epermets Byes produces complete chrcterzto of the prmeter from the sgle dtset ddto to the most probble estmte we obt chrcterzto of the ucertty lower ucertty hgher ucertty 3

4 ByesvsML : optml estmte uder ML there s oe best estmte uder Byes there s o best estmte oly rdom vrble tht tkes dfferet vlues wth dfferet probbltes techclly spekg t mkes o sese to tlk bout the best estmte 3 : predctos remember tht we do ot relly cre bout the prmeters themselves they re eeded oly the sese tht they llow us to buld models tht c be used to mke predctos e.g. the BR ulke ML Byes uses ALL formto the trg set to mke predctos 4

5 ByesvsML let s cosder the BR uder the - loss d depedet smple {... } ML-BR: pck f two steps: fd * rg m where * plug to the BR * * ; ll formto ot cptured by * s lost ot used t decso tme X Y rg m X Y Y 5

6 6 Byes BR summry pck f ote: s before the bottom equto s repeted for ech clss hece we c drop the depedece o the clss d cosder the more geerl problem of estmtg d where Y Y X Y X Y Y X rg m * Θ Θ d X X Θ Θ

7 he predctve dstrbuto the dstrbuto X X Θ Θ d s kow s the predctve dstrbuto ths follows from the fct tht t llows us to predct the vlue of gve ALL the formto vlble the trg set ote tht t c lso be wrtte s E [ ] X Θ X Θ sce ech prmeter vlue defes model ths s epectto over ll possble models ech model s weghted by ts posteror probblty gve trg dt 7

8 he predctve dstrbuto suppose tht ~ N d ~ N X Θ Θ weght π π Θ X weght π weght π π the predctve dstrbuto s verge of ll these Gusss d X X Θ Θ 8

9 he predctve dstrbuto Byes vs ML ML: pck oe model Byes: verge ll models re Byes predctos very dfferet th those of ML? they c be uless the pror s rrow Θ Θ m m Byes ~ ML very dfferet 9

10 MA ppromto ths souds good why use ML t ll? the m problem wth Byes s tht the tegrl c be qute sty d X X Θ Θ prctce oe s frequetly forced to use ppromtos oe possblty s to do somethg smlr to ML.e. pck oly oe model ths c be mde to ccout for the pror by pckg the model tht hs the lrgest posteror probblty gve the trg dt MA rg m Θ

11 MA ppromto ths c usully be computed sce MA rg m rg m Θ Θ d correspods to ppromtg the pror by delt fucto cetered t ts mmum Θ Θ Θ MA MA

12 MA vs ML ML-BR pck f * where rg m * X Y rg m * ; X Y Y Byes MA-BR pck f * rg m where MA MA ; the dfferece s o-eglgble oly whe the dtset s smll there re better ltertve ppromtos X Y rg m Y Θ Y Θ Y

13 Emple let s cosder emple of why Byes s usefull emple: commuctos bt s trsmtted by source corrupted by ose d receved by decoder Y chel X Q: wht should the optml decoder do to recover Y? 3

14 Emple the optml soluto s to threshold X pck decso rule f < Y f > wht s the threshold? the mdpot betwee sgl vlues < 4

15 Emple tody we cosder slght vrto Y tmosphere X recever stll: two sttes: Y trsmt sgl s - Y trsmt sgl s sme ose model X Y ε ε ~ N 5

16 Emple the BR s stll pck f < ths s optml d everythg works woderfully oe dy we get phoe cll: the recever s geertg lot of errors! somethg must hve chged the rover there s o wy to go to Mrs d check gol: to do s best s possble wth the fo tht we hve t X d our kowledge of the system 6

17 Emple wht we kow: the receved sgl s Guss wth sme vrce but the mes hve chged there s clbrto mode: rover c sed test sequece but t s epesve c oly sed few bts f everythg s orml receved mes should be d cto: sk the system to trsmt few s d mesure X compute the ML estmte of the me of X result: the estmte s dfferet th X 7

18 Emple we eed to combe two forms of formto our pror s tht X ~ N our dt drve estmte s tht X ~ N ˆ Q: wht do we do? f ˆ for lrge f ˆ for smll f tutve combto ˆ [] 8

19 Byes soluto Guss lkelhood observtos G s kow Guss pror wht we kow G re kow hyper-prmeters we eed to compute posteror dstrbuto for 9

20 Byes soluto posteror dstrbuto ote tht ths s probblty desty we c gore costrts terms tht do ot deped o d ormlze whe we re doe we oly eed to work wth X

21 Byes soluto pluggg the Gusss - ep - ep - ep X G G

22 Byes soluto ths s Guss we just eed to put t the stdrd qudrtc form to kow ts me d vrce use the completg the squres trck - ep b c b c b b b c b c b

23 3 Byes soluto ths cse we hve - ep b b c b c b - ep

24 4 Byes soluto d usg we hve d - ep - ep / G

25 5 Byes soluto ths c be rewrtte s we c compre wth our tutve soluto G ML

26 6 Byes soluto we hd the Byes soluto s ote tht t s ectly the sme s our heurstc ML [] ˆ []

27 Byes soluto for free Byes lso gves us the weghtg costts mesure of the ucertty of our estmte ote tht / s mesure of precso ths should be red s Byes ML pror Byes precso s greter th both tht of ML d pror 7

28 8 Observtos ote tht precso creses wth vrce goes to zero we re gurteed tht the lmt of fte dt we hve covergece to sgle estmte for lrge the lkelhood term domtes the pror term the soluto s equvlet to tht of ML for smll the pror domtes ths lwys hppes for Byes solutos [] ˆ X

29 Observtos 3 for gve ˆ [] f >>.e. we relly do t kow wht s pror the ML o the other hd f <<.e. we re very cert pror the summry Byes estmte combes the pror belefs wth the evdece provded by the dt very tutve mer 9

30 3

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