Overview. COMP14112: Artificial Intelligence Fundamentals. Lecture 0 Very Brief Overview. Structure of this course

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1 OMP: Arificial Inelligence Fundamenals Lecure 0 Very Brief Overview Lecurer: Xiao-Jun Zeng x.zeng@mancheser.ac.uk Overview This course will focus mainly on probabilisic mehods in AI We shall presen probabiliy heory as a general heory of reasoning We shall develop recapiulae) he mahemaical deails of ha heory We shall invesigae wo applicaions of ha probabiliy heory in AI which rely on hese mahemaical deails: Robo localizaion Speech recogniion Srucure of his course Par One -The firs weeks + week 0 Week.. 0 Lecure Robo localizaion I Robo localizaion II Foundaions of probabiliy Brief hisory of AI Par Revision Examples class Probabiliy I Probabiliy II Probabiliy III Turing s paper.. Revision Laboraory exercise. Robo localizaion I.D Robo localizaion II OMP: Arificial Inelligence Fundamenals Lecure - Probabilisic Robo Localizaion I Probabilisic Robo Localizaion I Ouline Background Inroducion of probabiliy Definiion of probabiliy disribuion Properies of probabiliy disribuion Robo localizaion problem Background onsider a mobile robo operaing in a relaively saic hence, known) environmen cluered wih obsacles.

2 Background The robo is equipped wih noisy) sensors and unreliable) acuaors, enabling i o perceive is environmen and move around wihin i. We consider perhaps he mos basic quesion in mobile roboics: how does he robo know where i is? This is a classic case of reasoning under uncerainy: almos none of he informaion he robo has abou is posiion is sensor repors and he acions i has ried o execue) yield cerain informaion. The heory we shall use o manage his uncerainy is probabiliy heory. Inroducion of Probabiliy Sample space Definiion. The sample space, Ω, is he se of all possible oucomes of an experimen. Example. Assume ha he experimen is o check he exam marks of sudens in he school of S, hen he sample space is Ω = { s s is a S suden } In he following, le Ms) represens he exam mark for suden s and a be a naural number in [0, 00], we define { a} { s M s) = a} { a} { s M s) a} { < a} { s M s) a} {[ a, b]} { s a M s) b} 6 7 Inroducion of Probabiliy Even: Definiion. An even, E, is any subse of he sample space Ω. Examples Even ={0}, i.e., he se of all sudens wih 0% mark Even ={ 0}, i.e., he se of all sudens who have passed Definiion of probabiliy disribuion Probabiliy disribuion Definiion. Given sample space Ω, a probabiliy disribuion is a funcion which assigns a real number in [0,] for each even E in Ω and saisfies i) P Ω) = K) ii) If E E = φ i.e., if E and E are muually exclusive ), hen p E = + K) where E E means E and E; E E means E or E. Example. Le as he percenage of sudens whose marks are in E, hen is a probabiliy disribuion on Ω. 8 9 Definiion of Probabiliy Disribuion Some noes on probabiliy disribuion For an even E, we refer o as he probabiliy of E Think of as represening one s degree of belief in E: if =, hen E is regarded as cerainly rue; if = 0., hen E is regarded as jus as likely o be rue as false; if = 0, hen E is regarded as cerainly false. So he probabiliy can be experimenal or subjecive based dependen on he applicaions Properies of Probabiliy Disribuion Basic Properies of probabiliy disribuion: Le be he probabiliy on Ω, hen For any even E, E ) =, where E is complemenary i.e., no E ) even; If evens E F, hen ; For any wo evens E and F, E = + E For empy even i.e., empy se) φ, φ) = 0 0

3 Properies of Probabiliy Disribuion Example: Le Ω = { s s is a S suden} be he sample space for checking he exam marks of sudens, Le: E = { 0}, i.e., he even of pass F = { 70), i.e., he even of s class Then as F E If = 0.7 = 0.0 hen he probabiliy of even G = {< 0} i.e., he even of fail ) is G) = E ) = = 0.7 = 0. Properies of Probabiliy Disribuion Example coninue) : Assume E = { 0} F = { 70) = 0.7, = 0.0 Quesion: wha is he probabiliy of even H = { < 0} { 70} i.e., he even of fail or s class Answer: As E F = { < 0} { 70} = φ, Then he probabili y of H ) = E F ) = E = = 0. even H is ) + F ) Properies of Probabiliy Disribuion Properies of Probabiliy Disribuion The following noions help us o perform basic calculaions wih probabiliies. Definiion: Evens are muually exclusive if E, E,..., E n E E ) = 0 i j i j Evens E, E,..., E n are joinly exhausive if Evens E, E,..., E n form a pariion of ) if hey are muually exclusive and joinly exhausive. Example. Evens E i = { i}, i = 0,,...,00 form a pariion of Ω which is defined in he previous examples. for E E... En ) = Ω Propery relaed o pariion: If Evens E, E,..., E n are muually exclusive, hen p E E... E ) = E ) + E ) E ) n n If evens E, E,..., E n form a pariion E ) + E ) En ) = Properies of Probabiliy Disribuion Example: onsider he following evens E = { 0}, i.e., he even of fail E = {[,9]}, i.e., he even of pass bu less han. E = {[60,69]}, i.e., he even of. E ]}, i.e., he even of s = {[70,00 class Then < E, E form a pariion of Ω E, E, If p E ) = 0., E ) = 0., E ) 0., hen = E ) = [ + E ) + ] = 0.0 Furher le E = { 0} i.e., he even of pass ), hen p E ) = E E E ) = E ) + E ) + E ) = 0.7 The robo localizaion problem is one of he mos basic problems in he design of auonomous robos: Le a robo equipped wih various sensors move freely wihin a known, saic environmen. Deermine, on he basis of he robo s sensor readings and he acions i has performed, is curren pose posiion and orienaion). In oher words: Where am I? 6 7

4 Any such robo mus be equipped wih sensors o perceive is environmen and acuaors o change i, or move abou wihin i. Some ypes of acuaor D moors aached o wheels) Sepper moors aached o gripper arms) Hydraulic conrol Arificial muscles... Some ypes of sensor ameras) Tacile sensors Bumpers Whiskers Range-finders Infra-red Sonar Laser range-finders ompasses Ligh-deecors 8 9 In performing localizaion, he robo has he following o work wih: knowledge abou is iniial siuaion knowledge of wha is sensors have old i knowledge of wha acions i has performed The knowledge abou is iniial siuaion may be incomplee or inaccurae). The sensor readings are ypically noisy. The effecs of rying o perform acions are ypically unpredicable. 0 We shall consider robo localizaion in a simplified seing: a single robo equipped wih rangefinder sensors moving freely in a square arena cluered wih recangular obsacles: In his exercise, we shall model he robo as a poin objec occupying some posiion in he arena no conained in one of he obsacles. We impose a square grid on he arena, wih he grid lines numbered 0 o 99 saring a he near lef-hand corner). We divide he circle ino 00 unis of π/0 radians each, again numbered 0 o 99 measured clockwise from he posiive x-axis. We ake ha he robo always o be locaed a one of hese grid inersecions, and o have one of hese orienaions. The robo s pose can hen be represened by a riple of inegers i, ) in he range [0, 99], hus: as indicaed. Now le us apply he ideas of probabiliy heory o his siuaion. Le L i, be he even ha he robo has pose i, ) where i, are inegers in he range [0,99]). The collecion of evens {L i, 0 i, < 00} forms a pariion! The robo will represen is beliefs abou is locaion as a probabiliy disribuion Thus, L i, ) is he robo s degree of belief ha is curren pose is i, ).

5 The probabiliies L i, ) can be sored in a marix. Bu his is hard o display visually, so we proceed as follows. Le L i,j be he even ha he robo has posiion i, j), and le L be he even ha he robo has orienaion Thus L i, j Li. j. L Li, As he evens L i, j, form a pariion, hen, based on he propery relaed o pariion, we have Li, j ) = P Li, j, P = P Li, ) L ) = P Li, = P Li, j ) P, These summaions can be viewed graphically: The robo s degrees of belief concerning is posiion can hen be viewed as a surface, and is degrees of belief concerning is orienaion can be viewed as a curve, hus: The quesion before us is: how should he robo assign hese degrees of belief? 6 7 There are wo specific problems: wha probabiliies does he robo sar wih? how should he robo s probabiliies change? A reasonable answer o he firs quesion is o assume all poses equally likely, excep hose which correspond o posiions occupied by obsacles: We shall invesigae he second quesion in he nex lecure. 8