Review - Probabilistic Classification

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1 Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw - Probablstc Classfcaton p( C ), P (C ) a. Paramtr stmaton - assum a known form for th p.d.f. and stmat th ncssary paramtrs (man and varanc for normal dstrbutons) b. Dnsty stmaton - stmat th non-paramtrc p.d.f. s from th gvn sampls.. Evaluat th dscrmnant 3. Assgn accordngly. ˆp( C ) ˆP (C ) ˆp( C ) ˆP (C ) stmatd valus C C

2 Rvw Mamum Lklhood Paramtr Estmaton Choos as stmats th valus of th paramtrs that mamz th lklhood (probablty) of th obsrvd st of tranng sampls (lablld). ˆP ML ˆ ML m ˆΣ ML ( ˆ)( ˆ) T basd ˆΣ u [ ] ˆΣ ML ( m)( m) T unbasd 3 Bays Estmaton (Bays Larnng) Wth ML stmaton, w vwd th paramtrs as fd valus but unknown. Usng th tranng sampls, w found th valus that mamz th lklhood of obtanng thos obsrvd sampls. In Baysan stmaton, w try to nfr th probablstc dstrbuton of th paramtrs from an a pror guss of th dstrbuton and th tranng sampls. ots for th followng pags: - Each class s approachd sparatly. So th dnsts nd to b calculatd for ach class. - Sampls ar assumd to b ndpndnt of ach othr. 4

3 Gvn an assumd form for th p.d.f. of : wth som paramtr (g. man): p( ) and assumd a pror dstrbuton: (an ntal guss that rflcts our knowldg and our uncrtanty about th paramtrs) Thn Bays thorm says: p( { p({ } ) p({ Th bottom trm s a normalzng factor that s ndpndnt of thta. Th drawng of ach sampl s ndpndnt, and so: p( { α p( ) 5 Eampl: -D Gaussan wth Known Varanc Consdr th cas whr th class man s th only unknown paramtr. Assum th dstrbuton s Gaussan. p( ) (, ) ( ) π W hav som pror knowldg of th man - a guss that ts man and varanc ar: (, ) π 6

4 Charls Robrtson YES ESTIMATIO ES ESTIMATIO May 9, 5. BAYES ESTIMATIO (, ) Bays Estmaton (, ). BAYES ESTIMATIO π. BAYES ESTIMATIO (, ) p( ) π π p p (p( {, ) α π π ( (, ) (, ) π α p Usng pth pror knowldg, p( { α p a postror ) dnsty th fnd th usng (, π π BAYES ESTIMATIO ( (,gathrd ) p( { sampls. α p p. from th nformaton ESTIMATIO ( π π. BAYES π π α p ( α p p( { α p( ) α pπ ( π ( α p } ) p({ p p( { αα p( { ) p (, p (, ) p({ p. BAYES ESTIMATIO p p( { p( { α p π π π π π ( ( p( { α p p α p π π π π α p( ( pαp( { ) p( α p( { ) p p( { ( p( ) p( { α π 88/988 -Spcal Topcs Engnrng: Pattrn Rcognton EG n Computr ( p( { p 7 α p π α( p, ) α p ( ) α p( p( { α p π m p( { p π p ( { αp( { p p p( { p p π α π π π α π p p ( p( { m p( { p p( ) p( { α π π ( π (, ) m p( ) ( α p α p α p ( ) ( π ( ( p α p p α p( { α p π π αp ( m m α p do thta Thosfactors nto th that not dpnd on hav bnabsorbd m constants. ( p( { p π α p ofa s an ponntal functon Bcausth form quadratc, th p( { p dstrbuton sπ a normal dnsty Th factors can b rwrttn to ft th p( { p followng form: π p( { p π ESTIMATIO. BAYES ESTIMATIO m m Rcognton Solvng EG 88/988 - Spcal Topcs n Computr Engnrng: Pattrn m m m 8

5 [ ( Ths ylds: m (m s th sampl man) Solvng: m Thus th man of th paramtr thta aftr sampls s a wghtd avrag of th sampl man and ntal stmat of th man. Th varanc of th paramtr aftr sampls s affctd by th varanc of th class, th ntal stmat of th paramtr varanc, and th numbr of sampls. 9 What happns as gts vry larg? lm m lm m p(,,..., n ) Fgur 3.(partal) From Pattrn Classfcaton by Duda, Hart, and Stork

6 So Bays stmaton allows larnng, allowng for ntal stmats and thn succssv modfcatons to ths as sampls ar gathrd and codd. Som trms: - f paramtr varanc s, th pror crtanty s so strong that th pror stmat of th man wll always b usd for th condtond man: m - f paramtr varanc s much largr than th class varanc, thn th sampl man wll b usd for th condtond man. Th fnal stp s to us th nformaton about th man (th a postror dnsty) to fnd th class-condtonal dnsty. p( { p( )p( { d { } (, ) (proof skppd) Compar wth: p( ) (, ) Th condtonal man s tratd as th ral man, and th known varanc s ncrasd to account for th uncrtanty of th man. As th numbr of sampls s ncrasd, th dstrbuton of th classcondtonal dnsty approachs th tru dstrbuton.

7 Dnsty Estmaton In paramtr stmaton, w assumd a form for th dstrbuton and stmatd th paramtrs, thr as spcfc valus (ML) or dstrbutons (Bays ). Howvr, f w do not know th spcfc form of th pdf, thn w nd a dffrnt approach. Dnsty stmaton s a non-paramtrc tchnqu that uss varatons of hstogram appromaton. Smplst cas s parttonng th fatur spac nto bns. 3 -D bns Tak th -as and dvd nto bns of lngth h. Estmat th probablty of a sampl n ach bn. P k k s th numbr of sampls n th bn 4

8 Dnsty Estmaton As an altrnatv to bns, w can tak wndows of unt volum and apply ths wndows to ach sampl. Th ovrlap of th wndows dfns th stmatd p.d.f. Ths tchnqu s known as Parzn wndows or krnls. 5

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