Localiaio MEM456/800 Localiaio: Baes Filer Where am I? Week 4 i Hsieh Evirome Sesors cuaors Sofware Ucerai is Everwhere Level of ucerai deeds o he alicaio How do we hadle ucerai? Eamle roblem Esimaig a robo s coordiaes relaive o a eeral referece frame Give: Evirome w/ 3 doors Robo is give a ma of his evirome ask: Fid ou where o his ma i is hrough sesig & moio. alculus of robabili heor o Rerese Ucerai 3 4
5 Robo Localiaio Eamle 6 Robo Localiaio Eamle 7 Sae Esimaio ore Idea: Esimae sae from sesor daa. Esimaio of quaiies from sesor daa ha are o direcl observable bu ca be iferred. Buildig ief disribuios over ossible world saes. robabilisic iferece: rocess of calculaig ossible values of radom variables ha are derived from oher radom variables & observed daa. 8 Baes Formula evidece rior likelihood
3 9 ormaliaio au : au au : lgorihm: 0 odiioig Law of oal robabili: d d d Baes Rule wih Backgroud owledge ssumios o Sae is comlee if i is he bes redicor of he fuure owledge of as saes measuremes or corols DOES O imrove accurac of redicio Sas ohig abou wheher fuure is deermiisic or sochasic If fuure is sochasic he rocesses ha mee hese codiios are called Markov chais I heor we assume comlee saes I racice o sae is reall comlee
robabilisic Geeraive Laws Evoluio of sae ad measuremes is govered b robabilisic laws Measuremes / Observaios/ reces : = + + Measureme robabili cios Ofe he world is damic sice acios carried ou b he robo acios carried ou b oher ages or jus he ime assig b chage he world. How ca we icororae such acios? 3 4 ical cios he robo urs is wheels o move he robo uses is maiulaor o gras a objec las grow over ime cios are ever carried ou wih absolue cerai. I coras o measuremes acios geerall icrease he ucerai. Modelig cios o icororae he oucome of a acio u io he curre ief we use he codiioal df u his erm secifies he df ha eecuig u chages he sae from o. 5 6 4
Simle Eamle of Sae Esimaio Suose a robo obais measureme Wha is oe? ausal vs. Diagosic Reasoig oe is diagosic. oe is causal. Ofe causal kowledge is easier o obai. Baes rule allows us o use causal kowledge: oe oe oe 7 8 Eamle oe = 0.6 oe = 0.3 oe = oe = 0.5 oe? oe oe oe oe oe oe oe 0.60.5 oe 0.67 0.60.5 0.30.5 3 ombiig Evidece Suose our robo obais aoher observaio. How ca we iegrae his ew iformaio? More geerall how ca we esimae...? raises he robabili ha he door is oe 9 0 5
6 Recursive Baesia Udaig Markov assumio: is ideede of... - if we kow....... i i Eamle: Secod Measureme oe = 0.5 oe = 0.6 oe =/3 0.65 8 5 3 5 3 3 3 oe oe oe oe oe oe oe lowers he robabili ha he door is oe. 3 cios Ofe he world is damic sice acios carried ou b he robo acios carried ou b oher ages or jus he ime assig b chage he world. How ca we icororae such acios? 4 Modelig cios o icororae he oucome of a acio u io he curre ief we use he codiioal df u his erm secifies he df ha eecuig u chages he sae from o.
Eamle: losig he door Sae rasiios u for u = close door : 0.9 0. oe closed 0 If he door is oe he acio close door succeeds i 90% of all cases. Sae rasiio robabili - 5 6 Iegraig he Oucome of cios oiuous case: u u ' ' d' Discree case: u u ' ' closed u oe u Eamle: he Resulig Belief closed u oe oe closed u closed closed 9 0 closed u ' ' 5 3 5 8 8 6 oe u ' ' oe u oe oe oe u closed closed 5 0 3 0 8 8 6 closed u 7 8 7
Baes Filers: Framework Give: Sream of observaios ad acio daa u: d u u } { Sesor model. cio model u. rior robabili of he ssem sae. Waed: Esimae of he sae of a damical ssem. he oserior of he sae is also called Belief: Bel u u Markov ssumio 0 : : u : : : u: u Uderlig ssumios Saic world Ideede oise erfec model o aroimaio errors 9 30 Baes Filer. lgorihm Baes_filer Beld :. 0 3. If d is a erceual daa iem he 4. For all do 5. 6. 7. Bel Bel' Bel' For all do 8. 9. Bel' Bel' Else if d is a acio daa iem u he 0. For all do.. Bel u ' Bel ' d' Reur Bel Bel Baes Filers are Familiar! u Bel d alma filers aricle filers Hidde Markov models Damic Baesia eworks ariall Observable Markov Decisio rocesses OMDs 3 3 8
9 33 Summar Baes rule allows us o comue robabiliies ha are hard o assess oherwise. Uder he Markov assumio recursive Baesia udaig ca be used o efficiel combie evidece. Baes filers are a robabilisic ool for esimaig he sae of damic ssems. 34 alma Filer lgorihm. lgorihm alma_filer - - u :. redicio: 3. 4. 5. orrecio: 6. 7. 8. 9. Reur u B R I 35 Gaussias : e - Uivariae / / : μ Σ μ Σ μ Σ e Ν d Mulivariae 36 a b a Y b a Y roeries of Gaussias
0 37 We sa i he Gaussia world as log as we sar wih Gaussias ad erform ol liear rasformaios. B Y B Y Mulivariae Gaussias 38 Discree alma Filer u B Esimaes he sae of a discree-ime corolled rocess ha is govered b he liear sochasic differece equaio wih a measureme 39 omoes of a alma Filer Mari ha describes how he sae evolves from o - wihou corols or oise. Mari l ha describes how he corol u chages he sae from o -. B Mari k ha describes how o ma he sae o a observaio. Radom variables rereseig he rocess ad measureme oise ha are assumed o be ideede ad ormall disribued wih covariace R ad resecivel. 40 alma Filer Udaes i D wih I wih obs
4 alma Filer Udaes i D R B u ac a b u a 4 0 0 0 0 Liear Gaussia Ssems: Iiialiaio Iiial ief is ormall disribued: 43 Damics are liear fucio of sae ad corol lus addiive oise: u B Liear Gaussia Ssems: Damics R B u u R B u d u 44 Observaios are liear fucio of sae lus addiive oise: Liear Gaussia Ssems: Observaios
45 Liear Gaussia Ssems: Observaios wih e e I 46 alma Filer lgorihm. lgorihm alma_filer - - u :. redicio: 3. 4. 5. orrecio: 6. 7. 8. 9. Reur u B R I