Introduction to Bayesian Estimation. McGill COMP 765 Sept 12 th, 2017

Transcription

1 Inrodcion o Baesian Esimaion McGill COM 765 Sep 2 h 207

2 Where am I? or firs core problem Las class: We can model a robo s moions and he world as spaial qaniies These are no perfec and herefore i is p o algorihms o compensae Toda: Represening moion and sensing probabilisicall Formlaion of localiaion as Baesian inference Describe and anale a firs simple algorihm

3

4 Eample: Landing on mars

5 Complemenar inp sorces A GS ells or global posiion wih consan noise +/- 5m: appears as jier arond pah An IMU ell or relaive moion wih nknown aw drif: diverges over ime A good algorihm will fse hese inps inelligenl and recover a pah which bes eplains boh: Smooher han GS Less drif han IMU

6 Eample: Self-driving Sorce: Dave Fergson Solve for X alk Jl 203 hp://

7 Raw Sensor Daa Sonar Laser Measred disances for epeced disance of 300 cm. I is crcial ha we measre he noise well o inegrae. 7

8 robabilisic Roboics Ke idea: Eplici represenaion of ncerain sing he calcls of probabili heor ercepion = sae esimaion Acion = ili opimiaion 8

9 Discree Random Variables X denoes a random variable. X can ake on a conable nmber of vales in { 2 n }. X= i or i is he probabili ha he random variable X akes on vale i.. is called probabili mass fncion. E.g. Room

10 Coninos Random Variables X akes on vales in he coninm. px= or p is a probabili densi fncion. E.g. r a b p d p b a 0

11 Join and Condiional robabili X= and Y= = If X and Y are independen hen = is he probabili of given = / = If X and Y are independen hen =

12 Law of Toal robabili Marginals Discree case Coninos case p d p p d p p p d 2

13 3 Baes Formla evidence prior likelihood

14 4 Normaliaion a : a a : Algorihm:

15 5 Condiioning Law of oal probabili: d d d

16 Baes Rle wih Backgrond Knowledge 6

17 7 Condiioning Toal probabili: d d d

18 8 Condiional Independence eqivalen o and

19 Simple Eample of Sae Esimaion Sppose a robo obains measremen Wha is open? 9

20 Casal vs. Diagnosic Reasoning open is diagnosic. open is casal. Ofen casal knowledge is easier o obain. Baes rle allows s o se con casal freqencies! knowledge: open open open 20

21 Eample open = 0.6 open = 0.3 open = open = 0.5 open open open open p open open p open open raises he probabili ha he door is open. 2

22 Combining Evidence Sppose or robo obains anoher observaion 2. How can we inegrae his new informaion? More generall how can we esimae... n? 22

23 23 Recrsive Baesian Updaing n n n n n n Assmpion: n is independen of... n- if we know n i i n n n n n n n n

24 24 Eample: Second Measremen 2 open = open = 0.6 open =2/ open open open open open open open 2 lowers he probabili ha he door is open.

25 A Tpical ifall Two possible locaions and 2 = =0.09 =0.07 Inegrae same repeaedl Wha are we doing wrong? p d p2 d p d Nmber of inegraions 25

26 Acions Ofen he world is dnamic since acions carried o b he robo acions carried o b oher agens or js he ime passing b change he world. How can we incorporae sch acions? 26

27 Tpical Acions The robo rns is wheels o move The robo ses is maniplaor o grasp an objec lans grow over ime Acions are never carried o wih absole cerain. In conras o measremens acions generall increase he ncerain. 27

28 Modeling Acions To incorporae he ocome of an acion ino he crren belief we se he condiional pdf This erm specifies he pdf ha eecing changes he sae from o. 28

29 Eample: Closing he door 29

30 Sae Transiions for = close door : open closed 0 If he door is open he acion close door scceeds in 90% of all cases. 30

31 Inegraing he Ocome of Acions Coninos case: ' ' d' Discree case: ' ' 3

32 32 Eample: The Resling Belief ' ' ' ' closed closed closed open open open open open open closed closed closed open open closed closed closed

33 Baes Filers: Framework Given: Sream of observaions and acion daa : Sensor model. Acion model. rior probabili of he ssem sae. d { Waned: Esimae of he sae X of a dnamical ssem. The poserior of he sae is also called Belief: Bel } 33

34 Markov Assmpion p 0 : : : p p : : : p Underling Assmpions Saic world Independen noise erfec model no approimaion errors 34

35 d Bel Baes Filers Baes = observaion = acion = sae Bel Markov Markov d d Toal prob. Markov d

36 Bel Bel d Baes Filer Algorihm. Algorihm Baes_filer Beld : If d is a percepal daa iem hen 4. For all do For all do Else if d is an acion daa iem hen 0. For all do. 2. Rern Bel Bel ' Bel Bel' Bel ' Bel ' Bel ' ' Bel ' d' 36

37 We have he mah wha s lef? Choose a daa srcre o represen Bel Creae pdae rles ha inegrae he moions and measremens Feed in daa and o come or resls For he res of his secion we will consider a hge varie of mehods ha roghl fall wihin his framework. Firs eamples: Fied-resolion discreiaion of Bel : Markov Localiaion Variable-resolion discreiaion of Bel : aricle Filers Resricion of Bel o a simple parameric famil : Kalman Filers

38 Discree Baes Filer Algorihm. Algorihm Discree_Baes_filer Beld : If d is a percepal daa iem hen 4. For all do For all do Else if d is an acion daa iem hen 0. For all do. 2. Rern Bel Bel ' Bel Bel' Bel ' ' Bel ' Bel ' ' Bel '

39 iecewise Consan Bel

40 iecewise Consan Wha abo Angle? Bel

41 Implemenaion To pdae he belief pon sensor inp and o carr o he normaliaion one has o ierae over all cells of he grid. Especiall when he belief is peaked which is generall he case dring posiion racking one wans o avoid pdaing irrelevan aspecs of he sae space. One approach is no o pdae enire sb-spaces of he sae space. This however reqires o monior wheher he robo is delocalied or no. To achieve his one can consider he likelihood of he observaions given he acive componens of he sae space.

42 Implemenaion 2 To efficienl pdae he belief pon robo moions one picall assmes a bonded Gassian model for he moion ncerain. This redces he pdae cos from On 2 o On where n is he nmber of saes. The pdae can also be realied b shifing he daa in he grid according o he measred moion. In a second sep he grid is hen convolved sing a separable Gassian Kernel. Two-dimensional eample: /6 /8 /6 /4 /8 /4 /8 /2 + /4 /2 /4 /6 /8 /6 /4 Fewer arihmeic operaions Easier o implemen

43 Grid-based Localiaion 43

44 Sonars and Occpanc Grid Map 44

45 Conclsion: Baesian esimaion gives a framework o inegrae evidence assming onl local knowledge. The mah ells s he correc operaions o fse he knowledge properl represening he poserior disribion Discree Baes Filers are never sed: we ms hink of smarer was o represen or disribions and o compe pdaes ne ime!