SA-1. Robotics Capstone. Intro to Probabilistic Techniques Probabilities Bayes rule Bayes filters
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1 SA- CSE-48 Roboics Capsoe Iro o robabilisic Techiqes robabiliies Baes rle Baes filers
2 robabilisic Roboics e idea: Eplici represeaio of cerai sig he calcls of probabili heor ercepio sae esimaio Acio Acio ili opimiaio 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 2
3 Aioms of robabili Theor ra deoes probabili ha proposiio A is re. 0 r A r Tre r False 0 r A B r A + r B r A B 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 3
4 A Closer Look a Aiom 3 r A B r A + r B r A B Tre A A B B B 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 4
5 Discree Radom Variables X deoes a radom variable. X ca ake o a coable mber of vales i { 2 }. X i or i is he probabili ha he radom variable X akes o vale i.. is called probabili mass fcio. E.g. Room /9/2006 CSE-48 Roboics Capsoe: rob Iro 5
6 Coios Radom Variables X akes o vales i he coim. px or p is a probabili desi fcio. r a b p d b a E.g. p 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 6
7 Joi ad Codiioal robabili X ad Y If X ad Y are idepede he is he probabili of give / If X ad Y are idepede he 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 7
8 Law of Toal robabili Margials Discree case Coios case p d p p d p p p d 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 8
9 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 9 Baes Baes Formla Formla evidece prior likelihood
10 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 0 Normaliaio Normaliaio η η Algorihm: a : a a : η η
11 4/9/2006 CSE-48 Roboics Capsoe: rob Iro Codiioig Codiioig Toal probabili: Toal probabili: d d d???
12 Codiioig Toal probabili: d Baes rle ad backgrod kowledge: 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 2
13 Codiioal Idepedece Eqivale o ad 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 3
14 Simple Eample of Sae Esimaio Sppose a robo obais measreme Wha is doorope? 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 4
15 Casal vs. Diagosic Reasoig ope ope is diagosic. ope ope is casal. Ofe Ofe casal kowledge is easier o obai. BaesBaes rle allows s o se casal kowledge: co freqecies! ope ope ope 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 5
16 Eample ope 0.6 ope 0.3 ope ope 0.5 ope ope ope ope p ope + ope p ope ope raises he probabili ha he door is ope. 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 6
17 Combiig Evidece Sppose Sppose or robo obais aoher observaio 2. How How ca we iegrae his ew iformaio? More More geerall how ca we esimae...? 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 7
18 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 8 Recrsive Baesia Updaig Recrsive Baesia Updaig Markov assmpio: is idepede of... - if we kow i i η η
19 Eample: Secod Measreme 2 ope 0.5 ope 2/3 2 ope 0.6 ope 2 2 ope 2 ope ope ope + ope 2 ope lowers he probabili ha he door is ope. 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 9
20 Baes Filers: Framework Give: Sream of observaios ad acio daa : d { 2 Sesor model. Acio model. rior probabili of he ssem sae. Waed: Esimae of he sae X of a damical ssem. The poserior of he sae is also called Belief: Bel 2 } 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 20
21 Markov Assmpio p 0 : : : p p : : : p Uderlig Assmpios Saic world Idepede oise erfec model o approimaio errors 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 2
22 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 22 observaio acio sae Baes Baes Filers Filers 2 Bel 2 2 η Baes Markov 2 η 2 2 d η Toal prob. Markov 2 d η d Bel η
23 Baes Filer Algorihm Bel d Bel η. Algorihm Baes_filer Beld : 2. η0 3. If d is a percepal daa iem he 4. For all do 5. ' For all do Else if d is a acio daa iem he 0. For all do. Bel' 2. Rer Bel Bel Bel η η + Bel' Bel' η Bel' ' Bel ' d' 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 23
24 Smmar BaesBaes rle allows s o compe probabiliies ha are hard o assess oherwise. Uder Uder he Markov assmpio recrsive Baesia pdaig ca be sed o efficiel combie evidece. BaesBaes filers are a probabilisic ool for esimaig he sae of damic ssems. 4/9/2006 CSE-48 Roboics Capsoe: rob Iro 24
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