( ) [ ] MAP Decision Rule
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1 Announcemens Bayes Decson Theory wh Normal Dsrbuons HW0 due oday HW o be assgned soon Proec descrpon posed Bomercs CSE 90 Lecure 4 CSE90, Sprng 04 CSE90, Sprng 04 Key Probables 4 ω class label X feaure vecor P (ω Pror probably of class P (x Probably dsrbuon funcon of feaure values. P(x ω Class condonal densy funcon or lklhood of feaure x gven ω P(ω x poseror probably densy funcon of he class gven he feaure. CSE90, Sprng 04 CSE90, Sprng 04 Paern Classfcaon, Chaper MAP Decson Rule 6 Bayesan Decson Theory Connuous Feaures 7 The Maxmum A Poseror (MAP decson rule s Le x be a vecor of feaures. ˆ ω ω P(ω x ω P(x ω P(ω P(x ω P(x ω P(ω ( ω ln P(x ω P(ω [ ] ω lnp(x ω + lnp(ω Bayes Rule Because P(x s no a funcon of ω Because ln s a monoonc funcon Paern Classfcaon, Chaper Le {ω, ω,, ω c } be he se of c saes of naure (or classes Le {α, α,, α a } be he se of possble acons Le λ(α ω be he loss for acon α when he sae of naure s ω Task s o fnd a decson funcon α ˆ (x ha akes n a feaure vecor x and reurns an acon. Paern Classfcaon, Chaper
2 Wha s he Expeced Loss for acon α? 8 0 For any gven x he expeced loss for acon α s c = = R( α x λ( α ω P( ω x = R(α x s called he Condonal Rsk (or Expeced Loss Gven a measured feaure vecor x, whch acon should we ake? Selec he acon α for whch R(α x s mnmum ˆ α (x = argmn α R(α x =c = argmn λ(α ω P(ω x α = Paern Classfcaon, Chaper Paern Classfcaon, Chaper Classfers, Dscrmnan Funcons and Decson Surfaces Dscrmnan Funcons: A generalzaon The mul-caegory case Consder a se of c dscrmnan funcons g (x, =,, c The classfer assgns a feaure vecor x o class ω f: g (x > g (x I In oher words ω g (x Desgnng a classfer amouns o specfyng he g (x Paern Classfcaon, Chaper Paern Classfcaon, Chaper Bayes Rsk as dscrmnan funcon. Le g (x = - R(α x (max. dscrmnan corresponds o mn. rsk! 3 Decson Regons 4 For he mnmum error rae, dscrmnan funcon s: g (x = P(ω x (max. dscrmnaon corresponds o max. poseror! g (x P(x ω P(ω Feaure space dvded no c decson regons R = {x : g (x > g (x } Any funcon F(r whch s monoonc over r>0 when appled o a se of dscrmnan funcons, yelds a new dscrmnan funcon wh he same decson regons/boundares. g (x = ln P(x ω + ln P(ω (ln: naural logarhm! Decson surfaces Boundary beween decson regons. {x:, g (x = g (x} We ll see hs form wh Normal dsrbuons Paern Classfcaon, Chaper Paern Classfcaon, Chaper
3 Wha s he form he Class Condonal Densy? Le s consder he case where s a normal dsrbuon. E.g., P(x ω s a Normal dsrbuon and see wha he decsons and boundares are lke. 5 Paern Classfcaon, Chaper The Unvarae Normal Densy Densy whch s analycally racable Connuous densy A lo of processes are asympocally Gaussan Handwren characers, speech sounds are deal or prooype corruped by random process (cenral lm heorem P( x = x exp, π σ σ Where: = mean (or expeced value of x σ = varance, expeced squared devaon 6 Paern Classfcaon, Chaper 7 Mulvarae densy 8 Mulvarae normal densy n d dmensons s: P( x = d / ( π Σ / exp ( x Σ ( x Paern Classfcaon, Chaper where: x = (x, x,, x d ( sands for he ranspose vecor form = (,,, d mean vecor Σ = d by d covarance marx Σ and Σ - are deermnan and nverse respecvely Paern Classfcaon, Chaper Covarance Marx Σ = E{x} Σ = E{(x (x } Σ s symmerc Σ s posve sem-defne De(Σ >0 Egenvalues of Σ are non-negave Egenvecors of Σ are orhogonal 9 0 Dscrmnan Funcons for he Normal Densy We saw ha he mnmum error-rae classfcaon (MAP can be acheved by he dscrmnan funcon g (x = ln P(x ω + ln P(ω Case of mulvarae normal for class condon densy (lkelhood funcon & P(x ω = exp (π d / Σ / (x Σ (x ' ( * + Equprobable conours are ellpsods whose axes are gven by Egenvecors of Σ and whose lenghs are gven by he Egenvalues. g (x = (x Σ (x d lnπ lnσ + lnp(ω = (x Σ (x lnσ + lnp(ω Paern Classfcaon, Chaper Paern Classfcaon, Chaper 3
4 Case : Σ = σ I (I sands for he deny marx A classfer ha uses lnear dscrmnan funcons s called a lnear machne g ( x = w x + w where : (lnear dscrmnan funcon w = ; w0 = + ln P( ω σ σ ( ω s called he hreshold for he h caegory! 0 0 The decson surfaces for a lnear machne are peces of hyperplanes defned by: g (x = g (x Paern Classfcaon, Chaper Paern Classfcaon, Chaper 3 The hyperplane separang R and R are gven by w ( x x0 = 0 w = σ x0 = ( + P( ω ln ( P( ω 4 always orhogonal o he lne lnkng he means! f P( ω = P( ω hen x0 = ( + Paern Classfcaon, Chaper Paern Classfcaon, Chaper 5 6 Paern Classfcaon, Chaper Paern Classfcaon, Chaper 4
5 7 8 Case : Σ = Σ (covarance of all classes are dencal bu arbrary! Hyperplane separang R and R w ( x x = 0 0 w = Σ ( [ P( ω / P( ω ] ln x 0 = ( + ( ( Σ ( Here he hyperplane separang R and R s generally no orhogonal o he lne beween he means! Paern Classfcaon, Chaper Paern Classfcaon, Chaper 9 Case 3: Σ = arbrary 30 The covarance marces are dfferen for each caegory where : W = Σ w = Σ w 0 g ( x = x W x + w x = w = Σ lnσ + ln P( ω 0 Here he separang surfaces are Hyperquadrcs whch are: hyperplanes, pars of hyperplanes, hyperspheres, hyperellpsods, hyperparabolods, hyperhyperbolods Paern Classfcaon, Chaper Paern Classfcaon, Chaper 3 3 Paern Classfcaon, Chaper Paern Classfcaon, Chaper 5
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