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} bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 460 135 5933 4835 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4588 L 899 389 mt 899 416 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 811 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 944 4724 mt (0) s 1436 4615 mt 1436 4588 L 1436 389 mt 1436 416 L 1751 4615 mt 1751 4588 L 1751 389 mt 1751 416 L 1974 4615 mt 1974 4588 L 1974 389 mt 1974 416 L 2147 4615 mt 2147 4588 L 2147 389 mt 2147 416 L 2288 4615 mt 2288 4588 L 2288 389 mt 2288 416 L 2408 4615 mt 2408 4588 L 2408 389 mt 2408 416 L 2511 4615 mt 2511 4588 L 2511 389 mt 2511 416 L 2603 4615 mt 2603 4588 L 2603 389 mt 2603 416 L 2684 4615 mt 2684 4588 L 2684 389 mt 2684 416 L 2684 4615 mt 2684 4561 L 2684 389 mt 2684 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 2596 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 2729 4724 mt (1) s 3222 4615 mt 3222 4588 L 3222 389 mt 3222 416 L 3536 4615 mt 3536 4588 L 3536 389 mt 3536 416 L 3759 4615 mt 3759 4588 L 3759 389 mt 3759 416 L 3932 4615 mt 3932 4588 L 3932 389 mt 3932 416 L 4074 4615 mt 4074 4588 L 4074 389 mt 4074 416 L 4193 4615 mt 4193 4588 L 4193 389 mt 4193 416 L 4297 4615 mt 4297 4588 L 4297 389 mt 4297 416 L 4388 4615 mt 4388 4588 L 4388 389 mt 4388 416 L 4470 4615 mt 4470 4588 L 4470 389 mt 4470 416 L 4470 4615 mt 4470 4561 L 4470 389 mt 4470 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 4382 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 4515 4724 mt (2) s 5007 4615 mt 5007 4588 L 5007 389 mt 5007 416 L 5321 4615 mt 5321 4588 L 5321 389 mt 5321 416 L 5545 4615 mt 5545 4588 L 5545 389 mt 5545 416 L 5718 4615 mt 5718 4588 L 5718 389 mt 5718 416 L 5859 4615 mt 5859 4588 L 5859 389 mt 5859 416 L 5978 4615 mt 5978 4588 L 5978 389 mt 5978 416 L 6082 4615 mt 6082 4588 L 6082 389 mt 6082 416 L 6173 4615 mt 6173 4588 L 6173 389 mt 6173 416 L 6255 4615 mt 6255 4588 L 6255 389 mt 6255 416 L 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 6167 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 6300 4724 mt (3) s 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L /Helvetica /ISOLatin1Encoding 120 FMSR 631 4659 mt (0.15) s 899 4011 mt 953 4011 L 6255 4011 mt 6201 4011 L 698 4055 mt (0.2) s 899 3408 mt 953 3408 L 6255 3408 mt 6201 3408 L 631 3452 mt (0.25) s 899 2804 mt 953 2804 L 6255 2804 mt 6201 2804 L 698 2848 mt (0.3) s 899 2200 mt 953 2200 L 6255 2200 mt 6201 2200 L 631 2244 mt (0.35) s 899 1596 mt 953 1596 L 6255 1596 mt 6201 1596 L 698 1640 mt (0.4) s 899 993 mt 953 993 L 6255 993 mt 6201 993 L 631 1037 mt (0.45) s 899 389 mt 953 389 L 6255 389 mt 6201 389 L 698 433 mt (0.5) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 5356 0 899 4359 2 MP stroke DA 934 3 851 -22 934 38 852 -81 933 114 852 -111 899 4351 7 MP stroke DD 934 -682 851 -1733 934 -948 852 -221 933 -19 852 -19 899 4351 7 MP stroke gr DD 3114 294 mt (1 relevant feature) s 3068 4941 mt (Number of features) s 576 2628 mt -90 rotate (error) s 90 rotate SO end eplot epage end showpage %%EndDocument @endspecial -138 1560 a Fr(Figure)26 b(1.)h Fq(Plot)g(of)h(error)g (vs.)38 b(total)28 b(n)n(um)n(b)r(er)d(of)j(features)g(for)g(an)-138 1651 y(optimal)c(classi\014er)i(that)d(kno)n(ws)h(whic)n(h)g(feature)h (is)g(relev)l(an)n(t)e(to)i(a)-138 1743 y(classi\014cation)f(decision)e (\(solid\),)h(and)e(V)-6 b(oting)22 b(Gibbs)f(algorithms)-138 1834 y(using)h(di\013eren)n(t)e(priors)j(\(dash)e(and)g(dash-dot\).)32 b(Details)22 b(are)g(pro-)-138 1925 y(vided)j(in)h(Section)f(5.)-138 2098 y Fx(setting)d(\(Ng,)h(1998;)e(Littlestone,)i(1988;)f(Kivinen)f (and)h(W)-7 b(ar-)-138 2197 y(m)n(uth,)25 b(1994\).)34 b(That)23 b(this)h(result)f(is)h(not)f(merely)g(of)h(theoret-)-138 2297 y(ical)33 b(in)n(terest)g(is)g(demonstrated)f(b)n(y)h(the)h (empirical)f(results)-138 2397 y(sho)n(wn)26 b(in)h(Figure)f(1.)37 b(These)26 b(results,)h(whic)n(h)f(are)g(describ)r(ed)-138 2496 y(in)36 b(more)f(detail)h(in)g(Section)g(5,)i(sho)n(w)d (classi\014cation)f(error)-138 2596 y(rates)j(in)i(an)f(exp)r(erimen)n (t)g(in)h(whic)n(h)f(one)g(feature)g(is)g(rele-)-138 2696 y(v)-5 b(an)n(t)28 b(to)f(a)g(classi\014cation)g(decision.)36 b(The)28 b(solid)g(curv)n(e)e(plots)-138 2795 y(the)h(error)e(rate)h (for)h(an)f(algorithm)g(that)h(is)g(told)g(in)g(adv)-5 b(ance)-138 2895 y(whic)n(h)30 b(feature)g(is)g(relev)-5 b(an)n(t.)44 b(The)31 b(other)e(t)n(w)n(o)h(curv)n(es)f(sho)n(w)-138 2994 y(the)g(error)d(rates)h(for)h(V)-7 b(oting)28 b(Gibbs)g (algorithms)f(whic)n(h)h(are)-138 3094 y(not)e(told)g(whic)n(h)g (feature)g(is)g(relev)-5 b(an)n(t)25 b(and)h(whic)n(h)g(mak)n(e)f(use) -138 3194 y(of)40 b(di\013eren)n(t)h(priors.)73 b(The)41 b(high)f(degree)f(of)h(insensitivit)n(y)-138 3293 y(to)30 b(irrelev)-5 b(an)n(t)28 b(features)i(exhibited)g(b)n(y)f(the)i(lo)n(w) n(er)d(\(dashed\))-138 3393 y(curv)n(e)f(is)g(surprising)f(and)i(notew) n(orth)n(y)-7 b(.)-138 3542 y(The)33 b(remainder)f(of)g(this)h(pap)r (er)g(is)f(structured)h(as)f(follo)n(ws.)-138 3642 y(Section)e(2)f(pro) n(vides)g(a)g(formal)g(in)n(tro)r(duction)h(to)g(the)g(prob-)-138 3742 y(lem)j(and)g(the)h(algorithms.)51 b(Section)33 b(3)g(then)h(presen)n(ts)e(our)-138 3841 y(main)25 b(results)g(on)f (the)i(qualit)n(y)e(of)h(the)h(V)-7 b(oting)25 b(Gibbs)g(classi-)-138 3941 y(\014er)j(in)i(the)f(case)f(of)h(a)f(correctly)g(sp)r(eci\014ed)h (prior,)f(and)g(Sec-)-138 4041 y(tion)21 b(4)f(go)r(es)f(on)i(to)f (discuss)g(V)-7 b(oting)20 b(Gibbs)h(in)g(the)g(con)n(text)f(of)-138 4140 y(missp)r(eci\014ed)34 b(priors,)g(with)g(application)f(to)g (feature)h(selec-)-138 4240 y(tion.)52 b(Lastly)-7 b(,)33 b(Section)g(5)f(presen)n(ts)g(exp)r(erimen)n(tal)g(results,)-138 4339 y(and)27 b(Section)h(6)f(closes)g(with)h(our)f(conclusions.)-138 4577 y Fw(2.)38 b(Problem)d(de\014nition)i(and)h(notation)-138 4739 y Fz(2.1)f(Ba)m(y)m(esian)c(classi\014cation)-138 4894 y Fx(W)-7 b(e)31 b(are)e(concerned)g(with)i(the)g(problem)f(of)g (Ba)n(y)n(esian)e(clas-)-138 4994 y(si\014cation)d(in)g(the)g (discriminativ)n(e)g(setting,)g(where)g(giv)n(en)f(an)-138 5094 y(input)j Fv(x)d Fp(2)f Fv(X)7 b Fx(,)26 b(w)n(e)g(wish)g(to)g (predict)h(the)f(corresp)r(onding)f(la-)-138 5193 y(b)r(el)30 b Fv(y)e Fp(2)d(f)p Fx(0)p Fv(;)14 b Fx(1)p Fp(g)p Fx(.)40 b(F)-7 b(ormally)g(,)28 b(w)n(e)h(assume)f(a)h(family)g(of)g(prob-)-138 5293 y(abilistic)34 b(binary)g(predictors)g Fp(f)p Fv(f)925 5305 y Fs(\022)996 5293 y Fx(:)h Fv(X)41 b Fp(7!)34 b Fx([0)p Fv(;)14 b Fx(1])34 b Fp(j)h Fv(\022)i Fp(2)e Fx(\002)p Fp(g)1962 -24 y Fx(parameterized)h(b)n(y)h Fv(\022)42 b Fp(2)e Fx(\002,)g(where)d Fv(f)3232 -12 y Fs(\022)3269 -24 y Fx(\()p Fv(x)p Fx(\))i(is)f(in)n(terpreted)1962 75 y(as)k(the)i(probabilit)n(y)e(that)h Fv(y)j Fx(is)c(1)h(giv)n(en)f Fv(x)p Fx(.)84 b(F)-7 b(or)42 b(exam-)1962 175 y(ple,)53 b(for)47 b(Ba)n(y)n(esian)e(logistic)i(regression,)j(w)n(e)d(w)n(ould)g (use)1962 275 y Fv(f)2003 287 y Fs(\022)2040 275 y Fx(\()p Fv(x)p Fx(\))f(=)f(1)p Fv(=)p Fx(\(1)26 b(+)h(exp\()p Fp(\000)p Fv(w)2868 245 y Fs(T)2921 275 y Fv(x)h Fp(\000)f Fv(\014)t Fx(\)\),)45 b(parameterized)39 b(b)n(y)1962 374 y Fv(\022)25 b Fx(=)e(\()p Fv(w)r(;)14 b(\014)t Fx(\).)37 b(Since)25 b(w)n(e)f(are)g(in)h(a)f(Ba)n(y)n(esian)f(setting,)i(w)n(e)f (also)1962 474 y(ha)n(v)n(e)i(a)i(prior)e(distribution)i Fv(p)2921 486 y Ft(\002)2976 474 y Fx(\()p Fp(\001)p Fx(\))g(o)n(v)n(er)e Fv(\022)r Fx(.)1962 623 y(W)-7 b(e)37 b(also)f(assume)g(a)g(\014xed)h(distribution)g Fv(D)i Fx(o)n(v)n(er)c Fv(X)43 b Fx(from)1962 723 y(whic)n(h)30 b(training)g(examples)g(are)f(dra)n(wn)g(iid.)46 b(W)-7 b(e)31 b(are)e(giv)n(en)1962 823 y(a)34 b(training)g(set)h Fv(S)40 b Fx(=)34 b Fp(f)p Fx(\()p Fv(x)2807 835 y Fs(i)2835 823 y Fv(;)14 b(y)2913 835 y Fs(i)2940 823 y Fx(\))p Fp(g)3014 793 y Fs(m)3014 844 y(i)p Ft(=1)3161 823 y Fx(of)34 b Fv(m)h Fx(examples,)h(gen-)1962 922 y(erated)28 b(b)n(y)g(\014rst)h(sampling)f Fv(\022)2899 892 y Fo(\003)2966 922 y Fx(according)f(to)i(the)g(prior)e Fv(p)3834 934 y Ft(\002)3889 922 y Fx(,)1962 1022 y(then)33 b(sampling)f Fv(x)2559 1034 y Fs(i)2620 1022 y Fx(iid)h(according)e(to)i Fv(D)r Fx(,)h(and)f(\014nally)f(set-)1962 1122 y(ting)e(eac)n(h)g Fv(y)2366 1134 y Fs(i)2424 1122 y Fx(indep)r(enden)n(tly)h(to)f(1)g(or) f(0)h(according)f(to)h(the)1962 1221 y(probabilities)k(Pr)o([)p Fv(y)2598 1233 y Fs(i)2661 1221 y Fx(=)i(1)p Fp(j)p Fv(x)2874 1233 y Fs(i)2901 1221 y Fv(;)14 b(\022)2979 1191 y Fo(\003)3018 1221 y Fx(])35 b(=)g Fv(f)3217 1233 y Fs(\022)3251 1217 y Fn(\003)3289 1221 y Fx(\()p Fv(x)3368 1233 y Fs(i)3397 1221 y Fx(\).)60 b(Finally)-7 b(,)37 b(let)1962 1321 y Fv(p)2004 1333 y Ft(\002)2059 1321 y Fx(\()p Fv(\022)r Fp(j)p Fv(S)5 b Fx(\))39 b(denote)f(the)h(p)r(osterior)e(distribution)i (of)f Fv(\022)j Fx(giv)n(en)1962 1420 y(the)31 b(dataset)f Fv(S)5 b Fx(.)46 b(W)-7 b(e)31 b(explicitly)g(allo)n(w)f(the)h(case)f (of)g Fv(m)f Fx(=)e(0)1962 1520 y(training)e(examples,)g(in)g(whic)n(h) h(case)e(the)i(p)r(osterior)e(reduces)1962 1620 y(to)j(the)h(prior.) 1962 1769 y(A)j(classi\014er)e(is)i(an)n(y)f(\(p)r(ossibly)g(sto)r(c)n (hastic\))g(map)h Fv(h)c Fx(:)i Fv(X)34 b Fp(7!)1962 1869 y(f)p Fx(0)p Fv(;)14 b Fx(1)p Fp(g)p Fx(.)77 b(F)-7 b(or)41 b(example,)k(the)d(familiar)f(Ba)n(y)n(esian)e(optimal)1962 1968 y(classi\014er)26 b Fv(h)2343 1980 y Fs(B)2428 1968 y Fx(is)h(obtained)h(b)n(y)f(calculating)2308 2198 y(Pr)o([)p Fv(y)f Fx(=)c(1)p Fp(j)p Fv(x;)14 b(S)5 b Fx(])23 b(=)2913 2085 y Fm(Z)2959 2274 y Fs(\022)3010 2198 y Fv(f)3051 2210 y Fs(\022)3088 2198 y Fx(\()p Fv(x)p Fx(\))p Fv(p)3241 2210 y Ft(\002)3297 2198 y Fx(\()p Fv(\022)r Fp(j)p Fv(S)5 b Fx(\))p Fv(d\022)243 b Fx(\(1\))1962 2437 y(and)34 b(then)g(predicting)g Fv(h)2772 2449 y Fs(B)2829 2437 y Fx(\()p Fv(x)p Fp(j)p Fv(S)5 b Fx(\))34 b(=)f Fv(h)3199 2449 y Fs(B)3256 2437 y Fx(\()p Fv(x)p Fx(\))i(=)e(1)h(if)g(Pr)o([)p Fv(y)i Fx(=)1962 2537 y(1)p Fp(j)p Fv(x;)14 b(S)5 b Fx(])23 b Fp(\025)f Fx(0)p Fv(:)p Fx(5,)27 b(and)h(predicting)f(0)g(otherwise.) 1962 2746 y Fz(2.2)37 b(The)32 b(V)-8 b(oting)31 b(Gibbs)h (classi\014er)1962 2902 y Fx(W)-7 b(e)23 b(let)30 b(^)-49 b Fv(p)2257 2914 y Ft(\002)2334 2902 y Fx(denote)23 b(a)f(prior)f(used) h(b)n(y)h(a)f(learning)f(pro)r(cedure.)1962 3001 y(When)28 b(^)-49 b Fv(p)2239 3013 y Ft(\002)2317 3001 y Fp(6)p Fx(=)23 b Fv(p)2447 3013 y Ft(\002)2501 3001 y Fx(,)f(w)n(e)e(sa)n(y)27 b(^)-49 b Fv(p)2840 3013 y Ft(\002)2915 3001 y Fx(is)20 b(a)g(missp)r(eci\014ed)h(prior.)33 b(Note)1962 3101 y(that)19 b(when)g(w)n(e)g(refer)f(to)h(the)h(\\Ba)n(y)n(esian)c (optimal)j(classi\014er,")1962 3201 y(w)n(e)27 b(alw)n(a)n(ys)f(mean)h (the)h(classi\014er)f(that)g(uses)h Fv(p)3446 3213 y Ft(\002)3500 3201 y Fx(.)1962 3350 y(When)40 b(a)f(Gibbs)h (classi\014er)f Fv(h)2944 3362 y Fs(G)3039 3350 y Fx(using)46 b(^)-49 b Fv(p)3310 3362 y Ft(\002)3405 3350 y Fx(is)40 b(required)e(to)1962 3450 y(classify)26 b Fv(x)p Fx(,)h(it)g(\014rst)f (samples)2913 3428 y(^)2906 3450 y Fv(\022)i Fx(according)d(to)h(the)h (\(p)r(ossibly)1962 3549 y(missp)r(eci\014ed\))32 b(p)r(osterior)e (distribution)39 b(^)-49 b Fv(p)3310 3561 y Ft(\002)3365 3549 y Fx(\()p Fp(\001j)p Fv(S)5 b Fx(\),)33 b(then)f(fur-)1962 3649 y(ther)j(samples)40 b(^)-48 b Fv(y)38 b Fx(so)c(that)41 b(^)-47 b Fv(y)38 b Fx(=)d(1)f(with)i(probabilit)n(y)e Fv(f)3769 3661 y Ft(^)3763 3676 y Fs(\022)3800 3649 y Fx(\()p Fv(x)p Fx(\))1962 3749 y(and)f(^)-48 b Fv(y)26 b Fx(=)d(0)k(with)h(probabilit)n(y)f(1)18 b Fp(\000)g Fv(f)3151 3761 y Ft(^)3145 3776 y Fs(\022)3182 3749 y Fx(\()p Fv(x)p Fx(\).)38 b(Finally)-7 b(,)28 b(its)g(pre-)1962 3848 y(diction)g(is)f Fv(h)2370 3860 y Fs(G)2426 3848 y Fx(\()p Fv(x)p Fp(j)p Fv(S)5 b Fx(\))24 b(=)e Fv(h)2775 3860 y Fs(G)2831 3848 y Fx(\()p Fv(x)p Fx(\))i(=)k(^)-47 b Fv(y)r Fx(.)1962 3998 y(W)-7 b(e)27 b(are)f(in)n(terested)h(in)g(the) g(p)r(erformance)f(of)h(the)h(extension)1962 4097 y(of)h(Gibbs)f (classi\014ers)f(to)i(the)g(full)g(Mon)n(te)g(Carlo)e(setting,)i(in) 1962 4197 y(whic)n(h)j(m)n(ultiple)g(samples)f(are)g(tak)n(en)g(from)g (the)h(p)r(osterior.)1962 4297 y(W)-7 b(e)32 b(call)g(this)g(the)h(V)-7 b(oting)32 b(Gibbs)g(\(V)n(G\))h(classi\014er.)49 b(When)1962 4396 y(ask)n(ed)39 b(to)h(predict)f(a)h(lab)r(el)g(for)f(an)h(input)g Fv(x)p Fx(,)k(the)c(V)-7 b(oting)1962 4496 y(Gibbs)37 b(classi\014er)f(V)n(G\()p Fv(N)9 b Fx(\))38 b(\014rst)f(dra)n(ws)e Fv(N)46 b Fx(\(a)37 b(parameter\))1962 4595 y(iid)32 b(samples)f Fv(\022)2441 4565 y Ft(1)2478 4595 y Fv(;)14 b(:)g(:)g(:)g(;)g(\022)2704 4565 y Fs(N)2799 4595 y Fx(according)30 b(to)h(the)h(p)r(osterior)e(dis-)1962 4695 y(tribution)j(^)-49 b Fv(p)2353 4707 y Ft(\002)2408 4695 y Fx(\()p Fp(\001j)p Fv(S)5 b Fx(\).)37 b(Then,)26 b(it)h(further)f(samples)f Fv(y)3584 4665 y Ft(1)3621 4695 y Fv(;)14 b(:)g(:)g(:)f(;)h(y)3849 4665 y Fs(N)1962 4795 y Fx(indep)r(enden)n(tly)-7 b(,)31 b(setting)f Fv(y)2837 4765 y Fs(i)2891 4795 y Fx(=)d(1)i(with)i (probabilit)n(y)d Fv(f)3713 4810 y Fs(\022)3747 4794 y Fl(i)3777 4795 y Fx(\()p Fv(x)p Fx(\),)1962 4894 y(and)23 b Fv(y)2163 4864 y Fs(i)2214 4894 y Fx(=)f(0)i(with)g(probabilit)n(y)e (1)10 b Fp(\000)g Fv(f)3140 4910 y Fs(\022)3174 4893 y Fl(i)3204 4894 y Fx(\()p Fv(x)p Fx(\).)36 b(Finally)24 b(it)g(pic)n(ks)1962 4994 y(its)f(output)h(b)n(y)f(taking)g(a)f(ma)5 b(jorit)n(y-v)n(ote)21 b(of)i(the)h Fv(y)3556 4964 y Fs(i)3583 4994 y Fx(,)g(predict-)1962 5094 y(ing)38 b(1)f(if)44 b(^)-48 b Fv(y)43 b Fx(=)d(\(1)p Fv(=)-5 b(N)9 b Fx(\))2698 5031 y Fm(P)2785 5052 y Fs(N)2785 5118 y(i)p Ft(=1)2910 5094 y Fv(y)2954 5063 y Fs(i)3022 5094 y Fp(\025)39 b Fx(0)p Fv(:)p Fx(5,)h(and)e(0)f(otherwise.)1962 5193 y(\(Alternativ)n(ely)-7 b(,)33 b(one)e(ma)n(y)h(also)f(skip)g(the)i (second)e(stage)g(of)1962 5293 y(sampling)g(and)h(predict)f Fv(h)2816 5308 y Fk(V)11 b Fs(G)p Ft(\()p Fs(N)6 b Ft(\))3044 5293 y Fx(\()p Fv(x)p Fp(j)p Fv(S)f Fx(\))30 b(=)g Fv(h)3407 5308 y Fk(V)11 b Fs(G)o Ft(\()p Fs(N)6 b Ft(\))3634 5293 y Fx(\()p Fv(x)p Fx(\))31 b(=)e(1)p eop %%Page: 3 3 3 2 bop -138 -24 a Fx(when)39 b(\(1)p Fv(=)-5 b(N)9 b Fx(\))323 -86 y Fm(P)424 -24 y Fv(f)465 -8 y Fs(\022)499 -25 y Fl(i)529 -24 y Fx(\()p Fv(x)p Fx(\))43 b Fp(\025)e Fx(0)p Fv(:)p Fx(5,)h(and)c(0)h(otherwise.)70 b(This)-138 75 y(algorithm,)36 b(whic)n(h)e(skips)h(one)f(step)h(of)g (randomization,)g(is)-138 175 y(probably)25 b(more)h(app)r(ealing)f(to) h(man)n(y)-7 b(,)26 b(and)g(is)g(also)f(b)r(e)i(used)-138 275 y(in)34 b(some)f(of)h(our)f(exp)r(erimen)n(ts.)55 b(All)34 b(of)g(our)f(analyses)f(and)-138 374 y(results)27 b(also)g(apply)g(to)g(it.\))-138 524 y(V)-7 b(oting)20 b(Gibbs)h(uses)f(a)f(Mon)n(te)h(Carlo)f(appro)n(ximation)g(to)h(the) -138 623 y(Ba)n(y)n(esian)37 b(optimal)i(classi\014er,)h(and)f(V)n (G\(1\))g(is)g(the)g(Gibbs)-138 723 y(classi\014er.)d(V)-7 b(oting)28 b(Gibbs)g(should)g(b)r(e)g(though)n(t)f(of)h(as)f(using)-138 823 y(samples)33 b(to)g(obtain)g(a)g(Mon)n(te)g(Carlo)f(estimate)i(of)f (Pr)o([)p Fv(y)i Fx(=)-138 922 y(1)p Fp(j)p Fv(S;)14 b(x)p Fx(],)42 b(and)c(then)i(thresholding)e(its)h(estimate)g(at)f(0.5) g(to)-138 1022 y(mak)n(e)32 b(its)i(prediction.)53 b(F)-7 b(olk)33 b(wisdom)g(suggests)f(that)h(only)-138 1122 y(a)26 b(small)g(n)n(um)n(b)r(er)g(of)g(Mon)n(te)g(Carlo)f(samples)g (are)g(needed)i(in)-138 1221 y(order)g(to)h(do)h(w)n(ell)f(in)h(the)g (Ba)n(y)n(esian)d(classi\014cation)h(setting.)-138 1321 y(W)-7 b(e)24 b(seek)f(to)g(in)n(v)n(estigate)f(the)h(degree)g(to)g (whic)n(h)g(this)h(is)f(true.)-138 1470 y(An)29 b(imp)r(ortan)n(t)f (feature)g(of)h(the)g(V)-7 b(oting)28 b(Gibbs)h(classi\014er)e(is)-138 1570 y(that)i(it)h(can)f(dra)n(w)f(its)h(samples)g Fv(\022)959 1540 y Ft(1)996 1570 y Fv(;)14 b(:)g(:)g(:)g(;)g(\022)1222 1540 y Fs(N)1314 1570 y Fx(o\013-line)29 b(b)r(efore)-138 1670 y(the)38 b(new)g(input)g(v)n(ector)e(is)h(presen)n(ted.)67 b(When)38 b(exp)r(ensiv)n(e)-138 1769 y(metho)r(ds)k(suc)n(h)g(as)f (Mark)n(o)n(v)e(c)n(hain)j(Mon)n(te)f(Carlo)g(\(Gilks)-138 1869 y(et)19 b(al.,)i(1996\))c(or)h(rejection)g(sampling)h(\(Ripley)-7 b(,)21 b(1987\))c(are)h(re-)-138 1968 y(quired)23 b(to)g(dra)n(w)g(the) h(samples,)f(this)h(enables)f(us)g(to)h(p)r(erform)-138 2068 y(the)32 b(exp)r(ensiv)n(e)g(sampling)f(o\017ine,)i(making)f(the)g (algorithm)-138 2168 y(subsequen)n(tly)23 b(able)g(to)h(classify)f (individual)g(inputs)i(quic)n(kly)-7 b(.)-138 2377 y Fz(2.3)37 b(Error)32 b(metrics)-138 2533 y Fx(Giv)n(en)41 b(a)g(training)f(set)h Fv(S)5 b Fx(,)45 b(the)d(exp)r(ected)f (generalization)-138 2632 y(error)52 b(of)h(h)n(yp)r(othesis)h Fv(h)f Fx(on)h(a)f(particular)f(input)j Fv(x)67 b Fp(2)-138 2732 y Fv(X)55 b Fx(is)48 b Fv(")129 2744 y Fs(x)171 2732 y Fx(\()p Fv(h)p Fx(\))58 b(=)g(Pr)o([)p Fv(h)p Fx(\()p Fv(x)p Fx(\))h Fp(6)p Fx(=)e Fv(y)s Fp(j)p Fv(x;)14 b(S)5 b Fx(])58 b(=)1327 2665 y Fm(R)1366 2761 y Fs(\022)1417 2732 y Fx(Pr[)p Fv(h)p Fx(\()p Fv(x)p Fx(\))g Fp(6)p Fx(=)-138 2831 y Fv(y)s Fp(j)p Fv(x;)14 b(\022)r Fx(])p Fv(p)119 2843 y Ft(\002)174 2831 y Fx(\()p Fv(\022)r Fp(j)p Fv(S)5 b Fx(\))p Fv(d\022)r Fx(,)23 b(where)18 b(the)i(probabilit)n(y)f(is)g(o)n(v)n(er)f(an)n(y)g(ran-)-138 2931 y(domization)31 b(in)h Fv(h)g Fx(and)g(in)g(the)g(uncertain)n(t)n (y)f(in)i Fv(y)h Fx(giv)n(en)d Fv(S)5 b Fx(.)-138 3031 y(Note)36 b(that)h(this)f(is)h(the)f Fu(Bayesian)k(exp)l(e)l(cte)l(d)d (err)l(or)g Fx(and)f(is)-138 3130 y(a)n(v)n(eraged)24 b(o)n(v)n(er)g Fv(\022)r Fx(,)j(whic)n(h)g(di\013ers)f(from)g(the)h(P) -7 b(A)n(C)26 b(notion)g(of)-138 3230 y(error)35 b(in)h(whic)n(h)h (misclassi\014cation)e(is)h(measured)g(with)h(re-)-138 3330 y(sp)r(ect)28 b(to)f(a)h(single)f(\\true")f Fv(\022)k Fx(\(V)-7 b(alian)n(t,)28 b(1984\).)-138 3479 y(In)j(some)g(cases)f(it) i(ma)n(y)e(b)r(e)i(appropriate)d(to)i(view)g(the)h(test)-138 3579 y(set)j(as)f(generated)g(from)g(a)h(distribution)g(that)g(is)g (di\013eren)n(t)-138 3678 y(from)g(the)g(training)g(distribution.)59 b(W)-7 b(e)36 b(assume)e(a)h(testing)-138 3778 y(distribution)20 b Fv(D)378 3748 y Fo(0)420 3778 y Fx(o)n(v)n(er)e(the)i(input)g(space)f Fv(X)7 b Fx(.)33 b(W)-7 b(e)20 b(then)g(de\014ne)-138 3877 y(the)28 b(generalization)e(error)f(of)j(a)f(classi\014er)f Fv(h)i Fx(to)f(b)r(e)-2 4090 y Fv(")p Fx(\()p Fv(h)p Fx(\))d(=)e Fv(")299 4102 y Fs(D)355 4086 y Fn(0)377 4102 y Fs(;S)445 4090 y Fx(\()p Fv(h)p Fx(\))i(=)e(E)725 4102 y Fs(x)p Fo(\030)p Fs(D)871 4086 y Fn(0)911 4090 y Fx([Pr)o([)p Fv(h)p Fx(\()p Fv(x)p Fx(\))i Fp(6)p Fx(=)f Fv(y)s Fp(j)p Fv(x;)14 b(S)5 b Fx(]])136 b(\(2\))-138 4302 y(where)30 b(the)h(subscript)f Fv(x)e Fp(\030)g Fv(D)849 4272 y Fo(0)903 4302 y Fx(means)i(the)h(exp)r(ectation)f(is) -138 4402 y(with)e(resp)r(ect)f(to)g Fv(x)h Fx(distributed)g(according) e(to)h Fv(D)1482 4371 y Fo(0)1506 4402 y Fx(,)g(and)g(w)n(e)-138 4501 y(ha)n(v)n(e)f(again)h(used)g(a)h(Ba)n(y)n(esian)d(notion)i(of)h (error.)-138 4651 y(In)23 b(this)f(pap)r(er,)h(w)n(e)f(are)g(concerned) f(with)i(ho)n(w)f(w)n(ell)g(the)h(V)-7 b(ot-)-138 4750 y(ing)20 b(Gibbs)h(classi\014er)e(appro)n(ximates)g(the)i(Ba)n(y)n (esian)d(optimal)-138 4850 y(classi\014er)32 b Fv(h)249 4862 y Fs(B)306 4850 y Fx(.)56 b(There)33 b(are)g(t)n(w)n(o)g(standard) g(w)n(a)n(ys)f(to)i(quan-)-138 4950 y(tify)d(this.)44 b(Giv)n(en)29 b(a)h(classi\014er)e Fv(h)p Fx(,)j(w)n(e)e(de\014ne)h (its)g Fu(additional)-138 5049 y(absolute)23 b(err)l(or)29 b Fx(\(compared)19 b(to)g(the)h(Ba)n(y)n(esian)e(optimal)h(clas-)-138 5149 y(si\014er\))27 b(to)h(b)r(e)449 5293 y Fv(")488 5305 y Fs(D)544 5289 y Fn(0)566 5305 y Fs(;S)634 5293 y Fx(\()p Fv(h)p Fx(\))19 b Fp(\000)f Fv(")887 5305 y Fs(D)943 5289 y Fn(0)965 5305 y Fs(;S)1033 5293 y Fx(\()p Fv(h)1113 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b(1994\),)g(whic)n(h)g(b)r(ounds)g(the)h(c)n(hance)e (of)h(the)g(mean)g(of)1962 1561 y Fv(N)45 b Fx(iid)37 b(random)e(v)-5 b(ariables)35 b(b)r(eing)h(far)g(from)g(the)h(exp)r (ected)1962 1661 y(v)-5 b(alue.)82 b(This)43 b(pro)n(v)n(es)e(Equation) g(\(7\).)83 b(T)-7 b(aking)42 b(exp)r(ecta-)1962 1761 y(tions)k(on)f(b)r(oth)h(sides)f(with)i(resp)r(ect)e(to)h Fv(x)53 b Fp(\030)g Fv(D)3667 1731 y Fo(0)3736 1761 y Fx(giv)n(es)1962 1860 y Fv(")2001 1872 y Fs(D)2057 1856 y Fn(0)2079 1872 y Fs(;S)2147 1860 y Fx(\()p Fv(h)2227 1875 y Fk(V)11 b Fs(G)p Ft(\()p Fs(N)6 b Ft(\))2455 1860 y Fx(\))p Fp(\000)p Fv(")2591 1872 y Fs(D)2647 1856 y Fn(0)2670 1872 y Fs(;S)2738 1860 y Fx(\()p Fv(h)2818 1872 y Fs(B)2875 1860 y Fx(\))23 b Fp(\024)g Fx(1)p Fv(=)3102 1790 y Fp(p)p 3170 1790 115 4 v 3170 1860 a Fv(eN)8 b Fx(,)21 b(whic)n(h)d(completes)1962 1960 y(the)28 b(\014rst)f(part)g (of)h(the)g(pro)r(of.)1962 2093 y(F)-7 b(or)28 b(the)g Fv(O)r Fx(\(1)p Fv(=)-5 b(N)9 b Fx(\))29 b(b)r(ound,)f(assume)g(as)f(b) 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Fo(0)2052 2696 y Fx(.)f(\(It)23 b(is)g(easily)f(v)n(eri\014ed)g(that)i(this)f(is) g(satis\014ed)f(b)n(y)h(pic)n(king)1962 2788 y Fv(B)2029 2758 y Fo(0)2095 2788 y Fx(=)43 b(2)p Fv(B)t(=\016)s Fx(,)e(since)e(otherwise)g(the)h(densit)n(y)f Fv(p)p Fx(\()p 3582 2742 V Fv(y)s Fx(\))h(w)n(ould)1962 2871 y(in)n(tegrate)29 b(to)h(greater)e(than)j(1,)f(a)g(con)n(tradiction.\)) 43 b(W)-7 b(e)30 b(can)1962 2954 y(write)d(the)h(additional)f(absolute) g(error)f(as)2059 3060 y Ff(Z)2136 3080 y Fj(0)p Fg(:)p Fj(5)2102 3230 y(0)2220 3162 y Fq(\(Pr[)p Fi(h)2397 3174 y Fc(V)10 b Fg(G)q Fj(\()p Fg(N)5 b Fj(\))2603 3162 y Fq(\()p Fi(x)p Fq(\))21 b Fh(6)p Fq(=)g Fi(y)s Fh(j)p 2871 3120 41 4 v Fi(y)r Fq(])d Fh(\000)p 3026 3120 V 16 w Fi(y)s Fq(\))p Fi(p)p Fq(\()p 3166 3120 V Fi(y)r Fq(\))p Fi(d)p 3276 3120 V(y)2204 3377 y Fh(\024)2346 3275 y Ff(Z)2423 3295 y Fj(0)p Fg(:)p Fj(5)2389 3445 y(0)2507 3377 y Fq(\()p 2537 3335 V Fi(y)i Fq(+)c(\(1)i Fh(\000)e Fq(2)p 2871 3335 V Fi(y)s Fq(\)Pr)q([)5 b(^)-43 b Fi(y)24 b Fh(\025)d Fq(0)p Fi(:)p Fq(5)p Fh(j)p 3307 3335 V Fi(y)t Fq(])c Fh(\000)p 3464 3335 V 17 w Fi(y)r Fq(\))p Fi(p)p Fq(\()p 3603 3335 V Fi(y)r Fq(\))p Fi(d)p 3713 3335 V(y)2204 3592 y Fq(=)2346 3490 y Ff(Z)2423 3510 y Fj(0)p Fg(:)p Fj(5)2389 3660 y(0)2507 3592 y Fq(\(1)g Fh(\000)g Fq(2)p 2707 3550 V Fi(y)s Fq(\)Pr[)5 b(^)-43 b Fi(y)24 b Fh(\025)e Fq(0)p Fi(:)p Fq(5)p Fh(j)p 3143 3550 V Fi(y)t Fq(])p Fi(p)p Fq(\()p 3275 3550 V Fi(y)r Fq(\))p Fi(d)p 3385 3550 V(y)2204 3809 y Fh(\024)2346 3707 y Ff(Z)2423 3728 y Fj(0)p Fg(:)p Fj(5)p Fb(\000)p Fg(\016)2389 3877 y Fj(0)2598 3809 y Fq(Pr[)5 b(^)-43 b Fi(y)24 b Fh(\025)d Fq(0)p Fi(:)p Fq(5)p Fh(j)p 2962 3767 V Fi(y)t Fq(])p Fi(p)p Fq(\()p 3094 3767 V Fi(y)s Fq(\))p Fi(d)p 3205 3767 V(y)2346 4024 y Fq(+)2419 3922 y Ff(Z)2496 3943 y Fj(0)p Fg(:)p Fj(5)2462 4092 y(0)p Fg(:)p Fj(5)p Fb(\000)p Fg(\016)2623 4024 y Fq(\(1)d Fh(\000)f Fq(2)p 2824 3982 V Fi(y)r Fq(\)Pr)q([)5 b(^)-43 b Fi(y)24 b Fh(\025)d Fq(0)p Fi(:)p Fq(5)p Fh(j)p 3259 3982 V Fi(y)t Fq(])p Fi(p)p Fq(\()p 3391 3982 V Fi(y)r Fq(\))p Fi(d)p 3501 3982 V(y)1962 4251 y Fx(If)28 b(w)n(e)f(can)g(sho)n (w)g(that)h(eac)n(h)f(of)g(the)h(ab)r(o)n(v)n(e)e(t)n(w)n(o)h(in)n (tegrals)f(is)1962 4351 y Fv(O)r Fx(\(1)p Fv(=)-5 b(N)9 b Fx(\),)30 b(then)f(w)n(e)g(are)f(done.)40 b(The)29 b(\014rst)g(is)f(easy)-7 b(.)40 b(F)-7 b(or)p 3779 4305 44 4 v 29 w Fv(y)27 b Fp(\024)1962 4451 y Fx(0)p Fv(:)p Fx(5)20 b Fp(\000)g Fv(\016)s Fx(,)33 b(Pr)o([)6 b(^)-48 b Fv(y)32 b Fp(\025)c Fx(0)p Fv(:)p Fx(5)p Fp(j)p 2678 4405 V Fv(y)r Fx(])h Fp(\024)g Fx(exp\()p Fp(\000)p Fx(2)p Fv(\016)3173 4421 y Ft(2)3210 4451 y Fv(N)9 b Fx(\))31 b(\(b)n(y)g(the)h(Ho)r(e\013d-)1962 4566 y(ing)g(inequalit)n(y)g (again\))f(so)2859 4500 y Fm(R)2914 4520 y Ft(0)p Fs(:)p Ft(5)p Fo(\000)p Fs(\016)2898 4596 y Ft(0)3103 4566 y Fx(Pr)o([)6 b(^)-48 b Fv(y)33 b Fp(\025)d Fx(0)p Fv(:)p Fx(5)p Fp(j)p 3514 4521 V Fv(y)r Fx(])p Fv(p)p Fx(\()p 3654 4521 V Fv(y)s Fx(\))p Fv(d)p 3773 4521 V(y)j Fp(\024)1962 4666 y Fx(exp\()p Fp(\000)p Fx(2)p Fv(\016)2268 4636 y Ft(2)2304 4666 y Fv(N)9 b Fx(\))34 b(=)f Fv(O)r Fx(\(1)p Fv(=)-5 b(N)9 b Fx(\).)55 b(F)-7 b(or)33 b(the)h(second)f(in)n(tegral,) h(w)n(e)1962 4766 y(can)27 b(again)g(apply)g(the)h(Ho)r(e\013ding)g (inequalit)n(y)-7 b(,)27 b(to)h(get:)2012 4893 y Fm(Z)2095 4914 y Ft(0)p Fs(:)p Ft(5)2058 5082 y(0)p Fs(:)p Ft(5)p Fo(\000)p Fs(\016)2232 5006 y Fx(\(1)18 b Fp(\000)g Fx(2)p 2449 4960 V Fv(y)r Fx(\)Pr[)6 b(^)-48 b Fv(y)26 b Fp(\025)c Fx(0)p Fv(:)p Fx(5)p Fp(j)p 2921 4960 V Fv(y)r Fx(])p Fv(p)p Fx(\()p 3061 4960 V Fv(y)s Fx(\))p Fv(d)p 3180 4960 V(y)2150 5252 y Fp(\024)2298 5139 y Fm(Z)2381 5159 y Ft(0)p Fs(:)p Ft(5)2344 5327 y(0)p Fs(:)p Ft(5)p Fo(\000)p Fs(\016)2532 5252 y Fx(2\(0)p Fv(:)p Fx(5)17 b Fp(\000)p 2813 5206 V 18 w Fv(y)s Fx(\))d(exp\()p Fp(\000)p Fx(2\(0)p Fv(:)p Fx(5)j Fp(\000)p 3408 5206 V 18 w Fv(y)s Fx(\))3484 5218 y Ft(2)3521 5252 y Fv(N)9 b Fx(\))p Fv(B)3696 5218 y Fo(0)3720 5252 y Fv(d)p 3763 5206 V(y)p eop %%Page: 5 5 5 4 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b(there)g(b)r(e)h(a)f(classi\014cation)f(problem)g(where)h (the)h(inputs)-138 3711 y Fv(X)32 b Fx(ha)n(v)n(e)25 b Fv(f)34 b Fx(features,)26 b(of)g(whic)n(h)f(an)h(unkno)n(wn)f(subset) h(is)g(rel-)-138 3811 y(ev)-5 b(an)n(t.)43 b(More)29 b(sp)r(eci\014cally)-7 b(,)30 b(let)g Fv(R)e Fp(2)f(f)p Fx(0)p Fv(;)14 b Fx(1)p Fp(g)1278 3781 y Fs(f)1348 3811 y Fx(b)r(e)31 b(a)e(random)-138 3911 y(v)-5 b(ariable)18 b(that)i(is)g(a)f(string)g(of)g Fv(f)28 b Fx(bits)20 b(that)g(indicates)f(whether)-138 4010 y(eac)n(h)f(of)g(the)h Fv(f)28 b Fx(features)18 b(is)g(relev)-5 b(an)n(t.)33 b(Our)18 b(prior)25 b(^)-50 b Fv(p)1446 4022 y Ft(\002)1520 4010 y Fx(assumes)-138 4110 y(that)24 b(the)g(subset)g(of)f(relev)-5 b(an)n(t)23 b(features)g(is)h(pic)n(k)n(ed)f(randomly)-138 4210 y(according)j(to)h(the)h(follo)n(wing)f(pro)r(cedure:)-78 4400 y(1.)41 b(First,)28 b(the)h(n)n(um)n(b)r(er)e Fv(r)k Fx(of)d(relev)-5 b(an)n(t)28 b(features)f(is)h(c)n(hosen)28 4500 y(uniformly)f(from)h Fp(f)p Fx(0)p Fv(;)14 b Fx(1)p Fv(;)g(:)g(:)g(:)e(;)i(f)9 b Fp(g)p Fx(.)-78 4654 y(2.)41 b(Second,)35 b(one)e(of)g(the)747 4586 y Fm(\000)785 4624 y Fs(f)785 4666 y(r)822 4586 y Fm(\001)893 4654 y Fx(subsets)g(of)h Fv(r)i Fx(out)d(of)h(the)f Fv(f)28 4753 y Fx(features)27 b(is)h(c)n(hosen)e(randomly)-7 b(.)-138 4944 y(Note)30 b(therefore)e(that)i(a)f Fu(p)l(articular)i Fx(feature)e(subset)h(of)f(size)-138 5044 y Fv(r)h Fx(has)d(c)n(hance)g (1)p Fv(=)445 4977 y Fm(\000)482 5014 y Fs(f)482 5056 y(r)519 4977 y Fm(\001)571 5044 y Fx(\()p Fv(f)g Fx(+)18 b(1\))28 b(of)f(b)r(eing)h(c)n(hosen.)-138 5193 y(Next,)53 b(w)n(e)47 b(also)f(assume)h(that,)53 b(conditioned)47 b(on)g Fv(R)q Fx(,)52 b(w)n(e)-138 5293 y(ha)n(v)n(e)39 b(some)h(prior)46 b(^)-49 b Fv(p)545 5305 y Ft(\002)600 5293 y Fx(\()p Fv(\022)r Fp(j)p Fv(R)q Fx(\))40 b(\(so)g(that,)k(e.g.,) f(for)d(all)g Fv(\022)i Fx(to)1962 -24 y(whic)n(h)36 b(^)-49 b Fv(p)2243 -12 y Ft(\002)2298 -24 y Fx(\()p Fv(\022)r Fp(j)p Fv(R)q Fx(\))29 b(assigns)f(p)r(ositiv)n(e)h (probabilit)n(y)-7 b(,)28 b(the)h(classi-)1962 75 y(\014er)k Fv(f)2152 87 y Fs(\022)2189 75 y Fx(\()p Fp(\001)p Fx(\))h(examines)f (the)h Fv(i)p Fx(-th)f(feature)g(of)g(its)h(inputs)f(only)1962 175 y(if)e Fv(R)2104 187 y Fs(i)2160 175 y Fx(=)d(1\).)45 b(F)-7 b(or)30 b(instance,)h(for)g(logistic)e(regression)g(where)1962 275 y Fv(f)2003 287 y Fs(\022)2040 275 y Fx(\()p Fv(x)p Fx(\))37 b(=)e Fv(f)2329 287 y Fs(w)r(;\014)2442 275 y Fx(\()p Fv(x)p Fx(\))i(=)e(1)p Fv(=)p Fx(\(1)22 b(+)h(exp\()p Fp(\000)p Fv(w)3243 245 y Fs(T)3296 275 y Fv(x)g Fp(\000)g Fv(\014)t Fx(\)\),)38 b(w)n(e)d(ma)n(y)1962 374 y(ha)n(v)n(e)j(^)-49 b Fv(p)p Fx(\()p Fv(w)r(;)14 b(\014)t Fp(j)p Fv(R)q Fx(\))34 b(dra)n(wing)d Fv(w)2914 386 y Fs(i)2974 374 y Fx(from)h(a)g(Normal)o (\(0)p Fv(;)14 b(\033)3680 344 y Ft(2)3677 395 y(1)3718 374 y Fx(\))32 b(dis-)1962 474 y(tribution)c(if)g Fv(R)2452 486 y Fs(i)2503 474 y Fx(=)22 b(1,)27 b(and)h Fv(w)2903 486 y Fs(i)2954 474 y Fx(=)23 b(0)k(otherwise.)1962 623 y(W)-7 b(e)34 b(are)e(in)n(terested)h(in)h(ev)-5 b(aluating)33 b(ho)n(w)g(w)n(ell)g(a)g(V)n(G)h(algo-)1962 723 y(rithm)41 b(can)g(p)r(erform)f(feature)h(selection.)76 b(Therefore,)43 b(w)n(e)1962 823 y(w)n(an)n(t)f(to)g(compare)g(its)g(p)r(erformance)g (against)f(that)i(of)f(a)1962 922 y(\\Ba)n(y)n(es)31 b(optimal)j(classi\014er")d(that)j(kno)n(ws)f(in)h(adv)-5 b(ance)33 b(ex-)1962 1022 y(actly)18 b(whic)n(h)h(features)f(are)g (relev)-5 b(an)n(t.)33 b(So,)20 b(for)f(some)f Fv(R)3676 992 y Fo(\003)3714 1022 y Fx(|the)1962 1122 y(\\true")27 b(set)h(of)f(relev)-5 b(an)n(t)28 b(features|let)f Fv(p)3289 1134 y Ft(\002)3344 1122 y Fx(\()p Fv(\022)r Fx(\))d(=)30 b(^)-49 b Fv(p)3603 1134 y Ft(\002)3658 1122 y Fx(\()p Fv(\022)r Fp(j)p Fv(R)3818 1091 y Fo(\003)3857 1122 y Fx(\).)1962 1221 y(Ho)n(w)42 b(w)n(ell)g(do)r(es)h(V)-7 b(oting)42 b(Gibbs)h(using)f(the)h(missp)r(eci\014ed)1962 1321 y(prior)33 b(^)-49 b Fv(p)28 b Fx(do?)1962 1493 y Fz(Theorem)i(2)42 b Fu(L)l(et)35 b(any)i Fv(m)2845 1505 y Ft(0)2882 1493 y Fv(;)14 b(N)44 b Fu(and)37 b Fx(0)d Fv(<)g(\017)g(<)f Fx(1)j Fu(b)l(e)g(\014xe)l(d,)1962 1592 y(and)k(assume)g(the)f(tr)l(aining)h(and)g(testing)g (distributions)g Fv(D)1962 1692 y Fu(and)32 b Fv(D)2196 1662 y Fo(0)2252 1692 y Fu(ar)l(e)g(the)g(same.)45 b(A)n(lso)32 b(let)g Fv(R)3159 1662 y Fo(\003)3229 1692 y Fu(b)l(e)g(\014xe)l(d,)g (and)g(let)g Fv(r)3873 1662 y Fo(\003)1962 1792 y Fu(b)l(e)24 b(the)h(numb)l(er)e(of)j(r)l(elevant)e(fe)l(atur)l(es)g(\()p Fv(r)3255 1761 y Fo(\003)3317 1792 y Fx(=)3405 1729 y Fm(P)3493 1816 y Fs(i)3534 1792 y Fv(R)3598 1761 y Fo(\003)3597 1813 y Fs(i)3636 1792 y Fu(\).)37 b(L)l(et)24 b(a)1962 1891 y(tr)l(aining)e(set)g Fv(S)k Fu(of)d(size)f Fv(m)g Fu(b)l(e)g(given,)i(wher)l(e)f Fv(m)e Fu(is)h(distribute)l(d)1962 1991 y(uniformly)36 b(in)g Fp(fd)p Fx(\(1)22 b Fp(\000)g Fv(\017)p Fx(\))p Fv(m)2855 2003 y Ft(0)2892 1991 y Fp(e)p Fv(;)14 b Fp(d)p Fx(\(1)22 b Fp(\000)g Fv(\017)p Fx(\))p Fv(m)3325 2003 y Ft(0)3362 1991 y Fp(e)h Fx(+)f(1)p Fv(;)14 b(:)g(:)g(:)f(;)h(m)3808 2003 y Ft(0)3845 1991 y Fp(g)p Fu(.)1962 2090 y(Then)1980 2235 y Fx(E[)p Fv(")p Fx(\()p Fv(h)2179 2250 y Fk(V)d Fs(G)o Ft(\()p Fs(N)6 b Ft(\))2406 2235 y Fx(\)])24 b Fp(\024)2054 2326 y Fm(\022)2115 2443 y Fx(1)18 b(+)g Fv(O)2338 2326 y Fm(\022)2460 2387 y Fx(1)p 2409 2424 145 4 v 2409 2441 a Fp(p)p 2478 2441 76 4 v 70 x Fv(N)2564 2326 y Fm(\023\023)2700 2301 y( )2765 2443 y Fx(E[)p Fv(")p Fx(\()p Fv(h)2964 2455 y Fs(B)3021 2443 y Fx(\)])h(+)f Fv(O)3257 2301 y Fm( )3323 2317 y(r)p 3406 2317 283 4 v 3416 2387 a Fv(r)3455 2363 y Fo(\003)3508 2387 y Fx(log)c Fv(f)p 3416 2424 263 4 v 3492 2500 a(m)3565 2512 y Ft(0)3688 2301 y Fm(!!)3806 2443 y Fx(\(8\))1962 2658 y Fu(wher)l(e)40 b(the)f(exp)l(e)l(ctations)h(ar)l(e)f(over)h(the) g(r)l(andom)g(tr)l(aining)1962 2757 y(set.)1962 2929 y Fz(Corollary)32 b(3)41 b Fu(T)-6 b(o)72 b(ensur)l(e)f(that)g Fx(E[)p Fv(")p Fx(\()p Fv(h)3357 2944 y Fk(V)11 b Fs(G)o Ft(\()p Fs(N)6 b Ft(\))3584 2929 y Fx(\)])72 b Fu(is)g(at)1962 3029 y(most)c(some)h(c)l(onstant)e Fv(\017)2856 3041 y Ft(0)2986 3029 y Fv(>)92 b Fx(0)68 b Fu(mor)l(e)g(than)g Fx(\(1)46 b(+)1962 3128 y Fv(O)r Fx(\(1)p Fv(=)2143 3058 y Fp(p)p 2212 3058 76 4 v 70 x Fv(N)9 b Fx(\)\)E[)p Fv(")p Fx(\()p Fv(h)2551 3140 y Fs(B)2608 3128 y Fx(\)])p Fu(,)77 b(it)67 b(su\016c)l(es)g(to)g(cho)l(ose)h Fv(m)3720 3140 y Ft(0)3847 3128 y Fx(=)1962 3228 y(\012\()p Fv(r)2093 3198 y Fo(\003)2146 3228 y Fx(log)14 b Fv(f)9 b Fx(\))p Fu(.)1962 3400 y Fz(Remark)47 b(\(random)h(training)g(set)f(size\).)79 b Fx(The)42 b(the-)1962 3500 y(orem)36 b(con)n(tains)h(a)f(minor)h(tec) n(hnical)g(assumption)g(that)g Fv(m)1962 3599 y Fx(ha)n(v)n(e)23 b(some)g(small)g(amoun)n(t)g(of)h(randomization)e(around)h Fv(m)3852 3611 y Ft(0)3889 3599 y Fx(.)1962 3699 y(This)29 b(is)f(a)g(condition)g(that)h(treats)f(the)h(training)f(set)h(size)f (as)1962 3798 y(random)20 b(\(usually)g(not)h(an)f(unrealistic)g (assumption\),)i(and)e(is)1962 3898 y(needed)g(in)g(the)h(pro)r(of)e (of)h(theorem.)34 b(\(See)20 b(the)h(full)f(pap)r(er)g(\(Ng)1962 3998 y(&)27 b(Jordan,)f(2001\))g(for)i(details.\))1962 4147 y(Note)21 b(that)h(to)f(state)h(the)f(simplest)h(p)r(ossible)f (result,)h(w)n(e)f(ha)n(v)n(e)1962 4247 y(giv)n(en)35 b(the)i(theorem)e(only)h(in)g(terms)g(of)g(a)g(1)p Fv(=)3471 4176 y Fp(p)p 3539 4176 V 3539 4247 a Fv(N)45 b Fx(con)n(v)n(er-)1962 4346 y(gence)36 b(rate.)64 b(Note)37 b(also)f(that)h(that)g(b)n(y)g (letting)g Fv(N)47 b Fx(=)38 b Fp(1)p Fx(,)1962 4446 y(this)31 b(result)g(also)f(giv)n(es)f(a)i(b)r(ound)g(for)g(the)g (setting)g(of)g(exact)1962 4546 y(Ba)n(y)n(esian)26 b(inference)h (using)g(missp)r(eci\014ed)h(priors.)1962 4695 y(The)52 b(corollary)-7 b(,)56 b(whic)n(h)c(re-states)e(the)i(error)e(b)r(ound)j (in)1962 4795 y(the)33 b(Theorem)e(in)i(terms)f(of)g(a)g(sample)g (complexit)n(y)g(result,)1962 4894 y(sho)n(ws)24 b(that)g(if)i(the)f (\(appro)n(ximate\))e(training)h(set)h(size)f Fv(m)3794 4906 y Ft(0)3856 4894 y Fx(is)1962 4994 y(\012\()p Fv(r)2093 4964 y Fo(\003)2146 4994 y Fx(log)14 b Fv(f)9 b Fx(\))27 b(\(and)f(if)h Fv(N)35 b Fx(is)27 b(not)f(unreasonably)f(small\),)i (then)1962 5094 y(w)n(e)32 b(will)g(do)g(nearly)f(as)h(w)n(ell)g(as)g (if)g(w)n(e)g(had)g(kno)n(wn)g(exactly)1962 5193 y(whic)n(h)g(features) g(are)f(relev)-5 b(an)n(t.)51 b(This)32 b(is)g(the)h(sample)f(com-)1962 5293 y(plexit)n(y)38 b(of)h(Ba)n(y)n(esian)d(feature)j(selection,)h (and)f(since)f(it)h(is)p eop %%Page: 6 6 6 5 bop -138 -24 a Fx(only)28 b Fu(lo)l(garithmic)k Fx(in)d Fv(f)9 b Fx(,)29 b(the)g(total)f(n)n(um)n(b)r(er)h(of)g(features,)g(it) -138 75 y(means)k(that)i(Ba)n(y)n(esian)c(feature)j(selection)f(using)h (the)g(par-)-138 175 y(ticular)24 b(prior)g(describ)r(ed)g(earlier)g (is)g(v)n(ery)g(insensitiv)n(e)g(to)h(the)-138 275 y(presence)41 b(of)g(irrelev)-5 b(an)n(t)40 b(features.)78 b(This)41 b(result)h(also)e(re-)-138 374 y(co)n(v)n(ers)35 b(the)i(b)r(est)g(kno) n(wn)f(suc)n(h)h(rates)e(\(Littlestone,)40 b(1988;)-138 474 y(Kivinen)33 b(&)g(W)-7 b(arm)n(uth,)34 b(1994;)g(Ng,)g(1998\),)f (and)g(has)g(sam-)-138 574 y(ple)27 b(complexit)n(y)g(that)g(b)r(eats)g (that)h(of)f(the)g(common)g(\\wrap-)-138 673 y(p)r(er")36 b(mo)r(del)g(\(Koha)n(vi)f(&)h(John,)j(1997\))34 b(feature)j(selection) -138 773 y(algorithm)29 b(\(see)i(the)g(analysis)e(in)i(Ng,)h(1998\).) 44 b(Indeed,)32 b(the)-138 872 y(logarithmic)g(dep)r(endence)i (suggests)e(that)h(w)n(e)g(can,)h(for)f(in-)-138 972 y(stance,)d Fu(squar)l(e)f Fx(the)h(total)f(n)n(um)n(b)r(er)h(of)f (features,)h(and)f(need)-138 1072 y(only)e Fu(twic)l(e)g Fx(as)g(m)n(uc)n(h)g(training)f(data)h(as)f(a)h(result.)37 b(Alterna-)-138 1171 y(tiv)n(ely)-7 b(,)35 b(w)n(e)e(can)g(also)g(view) g(this)h(as)f(sa)n(ying)f(that)i(Ba)n(y)n(esian)-138 1271 y(feature)23 b(selection)h(can)f(handle)g Fu(exp)l(onential)t(ly)i Fx(man)n(y)e(irrele-)-138 1371 y(v)-5 b(an)n(t)20 b(features)f(as)h(w)n (e)f(ha)n(v)n(e)g(training)g(examples.)34 b(W)-7 b(e)20 b(b)r(eliev)n(e)-138 1470 y(this)33 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h(learning)e(and)i(indeed)f(suc)n(h)-138 3317 y(b)r(ounds)37 b(w)n(ere)f(the)i(inspiration)e(for)h(the)g(theorem.)65 b(F)-7 b(or)36 b(a)-138 3416 y(closely)30 b(related)h(result)f(in)i (the)f(w)n(orst-case)e(setting)i(for)g(ex-)-138 3516 y(act)42 b(Ba)n(y)n(esian)f(inference)h(\(corresp)r(onding)f(to)h Fv(N)56 b Fx(=)48 b Fp(1)p Fx(\),)-138 3616 y(see)27 b(\(Barron)f(et)i(al.,)f(1993\).)-138 3765 y Fz(Pro)s(of)45 b(of)g(Theorem)f(2.)72 b Fx(The)39 b(result)g(is)g(easily)g(sho)n(wn) -138 3865 y(using)27 b(Equation)g(\(9\),)h(b)n(y)f(observing)f(that)244 4032 y(^)-49 b Fv(p)279 4044 y Ft(\002)334 4032 y Fx(\()p Fv(\022)r Fx(\))84 b(=)671 3954 y Fm(X)705 4132 y Fs(R)811 4032 y Fx(^)-49 b Fv(p)p Fx(\()p Fv(R)q Fx(\))7 b(^)-49 b Fv(p)1016 4044 y Ft(\002)1071 4032 y Fx(\()p Fv(\022)r Fp(j)p Fv(R)q Fx(\))401 b(\(10\))523 4235 y Fp(\025)90 b Fx(^)-49 b Fv(p)o Fx(\()p Fv(R)808 4201 y Fo(\003)847 4235 y Fx(\))7 b(^)-49 b Fv(p)921 4247 y Ft(\002)976 4235 y Fx(\()p Fv(\022)r Fp(j)p Fv(R)1136 4201 y Fo(\003)1174 4235 y Fx(\))458 b(\(11\))523 4399 y(=)869 4343 y(1)p 681 4380 419 4 v 681 4460 a(\()p Fv(f)27 b Fx(+)18 b(1\))952 4393 y Fm(\000)990 4430 y Fs(f)990 4472 y(r)1023 4456 y Fn(\003)1061 4393 y Fm(\001)1116 4399 y Fx(^)-49 b Fv(p)1151 4411 y Ft(\002)1206 4399 y Fx(\()p Fv(\022)r Fp(j)p Fv(R)1366 4365 y Fo(\003)1404 4399 y Fx(\))228 b(\(12\))523 4632 y(=)869 4576 y(1)p 681 4613 V 681 4694 a(\()p Fv(f)27 b Fx(+)18 b(1\))952 4627 y Fm(\000)990 4664 y Fs(f)990 4706 y(r)1023 4690 y Fn(\003)1061 4627 y Fm(\001)1109 4632 y Fv(p)1151 4644 y Ft(\002)1206 4632 y Fx(\()p Fv(\022)r Fx(\))p Fv(:)330 b Fx(\(13\))-138 4840 y(This)28 b(implies)f(that)-3 5035 y(KL\()p Fv(p)188 5047 y Ft(\002)243 5035 y Fp(jj)7 b Fx(^)-49 b Fv(p)331 5047 y Ft(\002)386 5035 y Fx(\))83 b(=)649 4922 y Fm(Z)695 5110 y Fs(\022)746 5035 y Fv(p)788 5047 y Ft(\002)843 5035 y Fx(\()p Fv(\022)r Fx(\))14 b(log)1094 4979 y Fv(p)1136 4991 y Ft(\002)1191 4979 y Fx(\()p Fv(\022)r Fx(\))p 1094 5016 203 4 v 1101 5092 a(^)-49 b Fv(p)1136 5104 y Ft(\002)1191 5092 y Fx(\()p Fv(\022)r Fx(\))1307 5035 y Fv(d\022)275 b Fx(\(14\))501 5259 y Fp(\024)649 5146 y Fm(Z)695 5334 y Fs(\022)746 5259 y Fv(p)788 5271 y Ft(\002)843 5259 y Fx(\()p Fv(\022)r Fx(\))14 b(log)1084 5191 y Fm(\000)1122 5259 y Fx(\()p Fv(f)27 b Fx(+)18 b(1\))1393 5191 y Fm(\000)1431 5229 y Fs(f)1431 5271 y(r)1464 5255 y Fn(\003)1502 5191 y Fm(\001\001)1592 5259 y Fv(d\022)-10 b Fx(\(15\))2601 -24 y(=)83 b(log)2870 -91 y Fm(\000)2908 -24 y Fx(\()p Fv(f)27 b Fx(+)18 b(1\))3179 -91 y Fm(\000)3217 -54 y Fs(f)3217 -12 y(r)3250 -28 y Fn(\003)3288 -91 y Fm(\001\001)3764 -24 y Fx(\(16\))2601 100 y Fp(\024)83 b Fx(\()p Fv(r)2820 66 y Fo(\003)2877 100 y Fx(+)18 b(1\))c(log\()p Fv(f)28 b Fx(+)18 b(1\))p Fv(;)328 b Fx(\(17\))1962 305 y(whic)n(h)29 b(when)g(substituted)g(bac) n(k)f(in)n(to)g(Equation)g(\(9\),)h(giv)n(es)1962 404 y(the)f(theorem.)1411 b Fa(\003)1962 554 y Fx(It)41 b(is)g(also)e(in)n (teresting)h(to)h(note)g(that)g(if)g(w)n(e)f(had)h(used)f(a)1962 653 y(more)34 b(\\naiv)n(e")e(c)n(hoice)i(of)g(prior,)h(for)e(instance) h(if)h(w)n(e)f(ha)n(v)n(e)1962 753 y(a)g(prior)f(whic)n(h)h(p)r(osits)g (that)g(eac)n(h)g(feature)f(indep)r(enden)n(tly)1962 852 y(has)47 b(a)g(some)h(\014xed)f(probabilit)n(y)g(of)h(b)r(eing)f (relev)-5 b(an)n(t)47 b(\(so)1962 952 y Fv(R)38 b Fx(is)g(a)f(sequence) f(of)i Fv(f)46 b Fx(indep)r(enden)n(t)38 b(coin)f(tosses\),)j(then)1962 1052 y(an)49 b(argumen)n(t)f(similar)h(to)g(the)g(one)g(ab)r(o)n(v)n(e) f(w)n(ould)h(giv)n(e)1962 1151 y(KL)o(\()p Fv(p)2152 1163 y Ft(\002)2208 1151 y Fp(jj)7 b Fx(^)-49 b Fv(p)2296 1163 y Ft(\002)2351 1151 y Fx(\))37 b(=)g Fv(O)r Fx(\()p Fv(f)9 b Fx(\).)64 b(This)36 b(giv)n(es)f(an)h(upp)r(er-b)r(ound)g(on) 1962 1251 y(the)42 b(sample)g(complexit)n(y)f(of)h(feature)f(selection) g(of)h Fv(O)r Fx(\()p Fv(f)9 b Fx(\),)1962 1351 y(whic)n(h)35 b(is)h(v)-5 b(astly)35 b(inferior)g(to)g Fv(O)r Fx(\()p Fv(r)3089 1320 y Fo(\003)3143 1351 y Fx(log)14 b Fv(f)9 b Fx(\))35 b(when)h Fv(r)3645 1320 y Fo(\003)3720 1351 y Fp(\034)g Fv(f)9 b Fx(.)1962 1450 y(Our)33 b(exp)r(erimen)n(ts)h(in)h (the)f(next)g(section)g(will)h(also)e(empir-)1962 1550 y(ically)40 b(compare)e(these)j(t)n(w)n(o)e(t)n(yp)r(es)h(of)g(priors)f (for)g(feature)1962 1650 y(selection.)1962 1888 y Fw(5.)f(Exp)s(erimen) m(ts)1962 2049 y Fz(5.1)f(The)32 b(case)g(of)g(correct)h(priors)1962 2204 y Fx(Our)28 b(\014rst)g(exp)r(erimen)n(t)g(compares)e(V)n(G\()p Fv(N)9 b Fx(\))29 b(and)f(the)h(Ba)n(y)n(es)1962 2304 y(optimal)20 b(classi\014er)f(in)i(a)f(simple)h(setting)f(that)h(w)n (as)e(c)n(hosen)h(so)1962 2404 y(that)28 b(exact)e(Ba)n(y)n(esian)g (inference)h(is)g(feasible,)g(whic)n(h)g(allo)n(ws)1962 2503 y(rep)r(eated)h(comparison)f(b)r(et)n(w)n(een)i(the)g(t)n(w)n(o)f (metho)r(ds.)41 b(Con-)1962 2603 y(sider)23 b(a)f(parameter)g Fv(\022)j Fx(uniformly)e(distributed)h(in)f([0)p Fv(;)14 b Fx(1],)23 b(and)1962 2703 y(let)34 b(the)f(target)g(output)h(on)f (input)h Fv(x)g Fx(\(also)e(uniformly)h(dis-)1962 2802 y(tributed)23 b(in)h([0)p Fv(;)14 b Fx(1]\))22 b(b)r(e)h(1)g(if)g Fv(x)h Fp(\025)e Fv(\022)r Fx(,)j(and)d(0)h(otherwise.)34 b(Using)1962 2902 y(correct)23 b(priors)31 b(^)-49 b Fv(p)2512 2914 y Ft(\002)2590 2902 y Fx(=)22 b Fv(p)2719 2914 y Ft(\002)2774 2902 y Fx(,)k(eac)n(h)e(classi\014er)f(\(b)r(oth)j (Ba)n(y)n(es)d(op-)1962 3001 y(timal)29 b(and)f(V)n(G\))h(w)n(as)e (trained)i(with)g Fv(m)f Fx(training)g(examples)1962 3101 y Fp(f)p Fx(\()p Fv(x)2083 3113 y Fs(i)2111 3101 y Fv(;)14 b(y)2189 3113 y Fs(i)2216 3101 y Fx(\))p Fp(g)26 b Fx(with)h(noisy)f(lab)r(els)g(that)h(w)n(ere)e(corrupted)g(at)i(the) 1962 3201 y(\(kno)n(wn\))j(noise)g(rate)g(of)h(0.2,)f(so)g(that)h Fv(y)3264 3213 y Fs(i)3320 3201 y Fx(=)c(1)j(with)i(proba-)1962 3300 y(bilit)n(y)25 b(0.8)e(when)i Fv(x)2567 3312 y Fs(i)2618 3300 y Fp(\025)e Fv(\022)r Fx(,)i(and)f Fv(y)2994 3312 y Fs(i)3045 3300 y Fx(=)e(1)j(with)f(probabilit)n(y)g(0.2)1962 3400 y(when)31 b Fv(x)2229 3412 y Fs(i)2286 3400 y Fv(<)d(\022)r Fx(.)47 b(On)31 b(eac)n(h)f(trial,)h(b)r(oth)g(classi\014ers)f(observ)n (ed)1962 3500 y(exactly)d(the)h(same)f(data)g(sample.)1962 3649 y(Figure)d(2\(a\))g(presen)n(ts)f(a)h(plot)h(of)f(the)h (generalization)d(errors)1962 3749 y(of)40 b(the)h(Ba)n(y)n(es)d (optimal)i(classi\014er)f(and)h(of)h(V)-7 b(oting)40 b(Gibbs)1962 3848 y(with)25 b Fv(N)31 b Fx(=)23 b(1)p Fv(;)14 b Fx(7)23 b(and)h(51,)g(plotted)g(as)g(a)f(function)i(of)f (training)1962 3948 y(set)c(size.)34 b(V)n(G\(1\)'s)20 b(error)e(seems)h(somewhat)h(larger)e(than)i(the)1962 4048 y(Ba)n(y)n(es)26 b(optimal)i(classi\014er's,)f(V)n(G\(7\))h(app)r (ears)f(to)h(b)r(e)h(trac)n(k-)1962 4147 y(ing)20 b(it)g(quite)h(w)n (ell,)g(and)f(V)n(G\(51\)'s)g(p)r(erformance)f(is)h(virtually)1962 4247 y(indistinguishable)27 b(from)f(that)i(of)f(the)g(Ba)n(y)n(es)e (optimal)i(clas-)1962 4346 y(si\014er.)1962 4496 y(Since)i(our)g(b)r (ounds)g(are)g(on)g(the)g(additional)g(relativ)n(e)f(error,)1962 4595 y(w)n(e)21 b(also)g(plot)g(the)h(additional)f(error)f(as)h(a)g (function)h(of)g Fv(N)9 b Fx(,)23 b(for)1962 4695 y(a)32 b(training)g(set)g(of)g(size)g Fv(m)f Fx(=)g(10.)50 b(\(See)33 b(Figure)e(2\(b\).\))52 b(As)1962 4795 y(exp)r(ected,)40 b(the)d(additional)g(relativ)n(e)f(error)f(do)r(es)i(decrease)1962 4894 y(quite)32 b(rapidly)f(with)h Fv(N)9 b Fx(.)50 b(If)32 b(the)g(additional)f(relativ)n(e)g(error)1962 4994 y(deca)n(ys)f(as)h Fv(O)r Fx(\(1)p Fv(=)-5 b(N)9 b Fx(\),)33 b(then)f(w)n(e)f(w)n(ould)g (also)g(exp)r(ect)g(on)h(the)1962 5094 y(log-log)20 b(scale)h(of)h(the) g(plot)g(to)g(see)f(a)h(line)g(with)g(slop)r(e)g(appro)n(x-)1962 5193 y(imately)27 b(-1.)36 b(Ignoring)25 b(the)i(single)f(p)r(oin)n(t)h (corresp)r(onding)e(to)1962 5293 y Fv(N)35 b Fx(=)26 b(1)j(\(see)g(caption\),)h(Figure)f(2\(b\))g(seems)g(to)h(almost)f(ex-) p eop %%Page: 7 7 7 6 bop -138 952 a @beginspecial 56 @llx 200 @lly 551 @urx 600 @ury 1555 @rwi @setspecial %%BeginDocument: plots/exptsampf.ps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 460 135 5944 4798 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 866 4761 mt (0) s 1435 4615 mt 1435 4561 L 1435 389 mt 1435 443 L 1369 4761 mt (20) s 1970 4615 mt 1970 4561 L 1970 389 mt 1970 443 L 1904 4761 mt (40) s 2506 4615 mt 2506 4561 L 2506 389 mt 2506 443 L 2440 4761 mt (60) s 3041 4615 mt 3041 4561 L 3041 389 mt 3041 443 L 2975 4761 mt (80) s 3577 4615 mt 3577 4561 L 3577 389 mt 3577 443 L 3477 4761 mt (100) s 4113 4615 mt 4113 4561 L 4113 389 mt 4113 443 L 4013 4761 mt (120) s 4648 4615 mt 4648 4561 L 4648 389 mt 4648 443 L 4548 4761 mt (140) s 5184 4615 mt 5184 4561 L 5184 389 mt 5184 443 L 5084 4761 mt (160) s 5719 4615 mt 5719 4561 L 5719 389 mt 5719 443 L 5619 4761 mt (180) s 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L 6155 4761 mt (200) s 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L 798 4659 mt (0) s 899 4087 mt 953 4087 L 6255 4087 mt 6201 4087 L 631 4131 mt (0.02) s 899 3559 mt 953 3559 L 6255 3559 mt 6201 3559 L 631 3603 mt (0.04) s 899 3030 mt 953 3030 L 6255 3030 mt 6201 3030 L 631 3074 mt (0.06) s 899 2502 mt 953 2502 L 6255 2502 mt 6201 2502 L 631 2546 mt (0.08) s 899 1974 mt 953 1974 L 6255 1974 mt 6201 1974 L 698 2018 mt (0.1) s 899 1446 mt 953 1446 L 6255 1446 mt 6201 1446 L 631 1490 mt (0.12) s 899 917 mt 953 917 L 6255 917 mt 6201 917 L 631 961 mt (0.14) s 899 389 mt 953 389 L 6255 389 mt 6201 389 L 631 433 mt (0.16) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 27 6 27 -2 26 -1 27 2 27 3 27 1 26 4 27 3 27 -4 27 4 27 1 26 -2 27 3 27 -1 27 3 26 -2 27 7 27 2 27 -1 27 4 26 -1 27 2 27 0 27 5 26 -1 27 -1 27 4 27 -2 27 5 26 -1 27 5 27 2 27 1 27 1 26 5 27 -3 27 0 27 3 26 3 27 3 27 5 27 -5 27 6 26 1 27 0 27 2 27 2 26 1 27 0 27 5 27 0 27 3 26 1 27 8 27 -6 27 7 26 0 27 4 27 -2 27 8 27 -1 26 0 27 4 27 2 27 1 26 10 27 -4 27 6 27 -2 27 0 26 9 27 3 27 -4 27 6 26 5 27 0 27 4 27 7 27 4 26 -2 27 3 27 3 27 6 27 4 26 1 27 7 27 -1 27 5 26 0 27 7 27 -2 27 6 27 5 26 5 27 9 27 -7 27 8 26 4 27 6 3604 4174 100 MP stroke 27 -1 27 10 27 -5 26 6 27 9 27 0 27 6 26 6 27 13 27 2 27 0 27 6 26 4 27 7 27 4 27 8 26 7 27 7 27 -2 27 16 27 12 26 -7 27 12 27 9 27 -9 26 25 27 14 27 -4 27 10 27 7 26 6 27 22 27 -11 27 22 27 -3 26 20 27 15 27 -2 27 2 26 17 27 13 27 10 27 10 27 22 26 10 27 2 27 11 27 24 26 10 27 4 27 46 27 -8 27 29 26 13 27 12 27 17 27 21 26 19 27 29 27 8 27 25 27 26 26 12 27 34 27 27 27 11 26 48 27 28 27 26 27 30 27 51 26 45 27 24 27 61 27 12 26 55 27 71 27 70 27 42 27 51 26 53 27 75 27 68 27 94 27 84 26 100 27 115 27 118 27 178 26 120 27 118 1167 1740 92 MP stroke DD 27 10 27 -3 26 -2 27 0 27 2 27 7 26 9 27 2 27 -4 27 7 27 -5 26 5 27 -2 27 1 27 6 26 -5 27 12 27 -2 27 -1 27 8 26 -3 27 0 27 6 27 6 26 -3 27 0 27 3 27 2 27 4 26 1 27 1 27 13 27 -5 27 -1 26 8 27 -6 27 5 27 -1 26 7 27 3 27 8 27 -2 27 5 26 -1 27 0 27 1 27 9 26 -3 27 0 27 8 27 -2 27 11 26 -4 27 12 27 -7 27 8 26 -3 27 7 27 2 27 4 27 1 26 4 27 8 27 -3 27 5 26 10 27 -5 27 12 27 -4 27 2 26 10 27 8 27 -12 27 13 26 4 27 7 27 6 27 10 27 -4 26 -1 27 2 27 7 27 17 27 1 26 1 27 16 27 -13 27 12 26 -1 27 15 27 -11 27 9 27 10 26 3 27 12 27 -2 27 5 26 2 27 26 3604 3974 100 MP stroke 27 -7 27 16 27 -7 26 0 27 10 27 3 27 9 26 8 27 22 27 9 27 -9 27 23 26 -1 27 10 27 5 27 6 26 13 27 9 27 6 27 5 27 30 26 -19 27 21 27 14 27 -15 26 40 27 10 27 -8 27 18 27 10 26 3 27 44 27 -26 27 42 27 2 26 25 27 19 27 -3 27 6 26 36 27 18 27 0 27 11 27 43 26 -1 27 23 27 25 27 3 26 25 27 -8 27 63 27 4 27 50 26 8 27 23 27 34 27 26 26 8 27 40 27 17 27 65 27 15 26 -6 27 65 27 23 27 26 26 78 27 30 27 51 27 63 27 69 26 35 27 25 27 120 27 -6 26 91 27 91 27 79 27 87 27 69 26 80 27 104 27 95 27 137 27 75 26 193 27 112 27 194 27 216 26 176 27 155 1167 576 92 MP stroke 27 7 27 -2 26 -1 27 2 27 3 27 2 26 4 27 5 27 -6 27 4 27 1 26 -2 27 3 27 1 27 1 26 -1 27 9 27 0 27 -2 27 5 26 0 27 2 27 1 27 4 26 -1 27 -1 27 6 27 -3 27 4 26 -1 27 6 27 4 27 -1 27 2 26 6 27 -6 27 2 27 3 26 2 27 5 27 6 27 -6 27 5 26 2 27 0 27 2 27 2 26 2 27 -1 27 6 27 0 27 4 26 1 27 9 27 -9 27 8 26 0 27 7 27 -4 27 9 27 0 26 -1 27 5 27 3 27 0 26 12 27 -7 27 9 27 -4 27 1 26 11 27 2 27 -4 27 7 26 4 27 -1 27 4 27 9 27 3 26 0 27 4 27 0 27 11 27 1 26 2 27 9 27 -1 27 4 26 0 27 11 27 -3 27 5 27 4 26 5 27 11 27 -7 27 9 26 5 27 7 3604 4130 100 MP stroke 27 -3 27 14 27 -8 26 6 27 11 27 3 27 3 26 8 27 12 27 6 27 -4 27 8 26 4 27 7 27 7 27 8 26 4 27 12 27 -2 27 16 27 16 26 -12 27 16 27 12 27 -15 26 31 27 12 27 -5 27 14 27 6 26 5 27 29 27 -14 27 24 27 -5 26 22 27 14 27 3 27 -1 26 20 27 18 27 10 27 10 27 20 26 9 27 8 27 11 27 28 26 11 27 0 27 50 27 -3 27 26 26 23 27 6 27 18 27 27 26 21 27 31 27 9 27 23 27 29 26 13 27 35 27 38 27 10 26 47 27 36 27 28 27 32 27 57 26 54 27 20 27 74 27 13 26 61 27 73 27 78 27 40 27 65 26 52 27 86 27 72 27 102 27 89 26 118 27 126 27 128 27 195 26 140 27 131 1167 1448 92 MP stroke 27 6 27 -2 26 -1 27 2 27 4 27 1 26 3 27 4 27 -4 27 3 27 2 26 -2 27 3 27 -1 27 2 26 -1 27 7 27 2 27 -1 27 4 26 -1 27 1 27 1 27 5 26 -1 27 -2 27 5 27 -2 27 5 26 -1 27 5 27 2 27 1 27 1 26 5 27 -3 27 0 27 3 26 3 27 3 27 6 27 -6 27 6 26 1 27 0 27 2 27 2 26 1 27 0 27 5 27 1 27 3 26 1 27 7 27 -6 27 8 26 0 27 4 27 -2 27 8 27 -1 26 0 27 4 27 2 27 1 26 10 27 -3 27 5 27 -2 27 0 26 9 27 3 27 -4 27 6 26 5 27 1 27 3 27 8 27 3 26 -2 27 4 27 2 27 6 27 4 26 1 27 8 27 -1 27 4 26 0 27 8 27 -3 27 7 27 5 26 5 27 9 27 -7 27 8 26 5 27 6 3604 4167 100 MP stroke 27 -2 27 11 27 -5 26 5 27 10 27 0 27 6 26 6 27 13 27 2 27 1 27 5 26 4 27 7 27 4 27 8 26 7 27 8 27 -3 27 16 27 13 26 -7 27 12 27 9 27 -8 26 25 27 13 27 -4 27 11 27 8 26 5 27 21 27 -9 27 21 27 -3 26 21 27 15 27 -2 27 3 26 16 27 14 27 9 27 11 27 22 26 10 27 3 27 11 27 23 26 11 27 4 27 47 27 -8 27 28 26 14 27 12 27 18 27 20 26 20 27 29 27 9 27 25 27 27 26 11 27 35 27 27 27 12 26 49 27 29 27 25 27 31 27 51 26 46 27 24 27 62 27 12 26 56 27 71 27 71 27 44 27 52 26 54 27 77 27 68 27 95 27 86 26 103 27 115 27 121 27 180 26 123 27 119 1167 1696 92 MP stroke gr DD 2722 4904 mt (m \(number of training examples\)) s 576 3027 mt -90 rotate (Generalization error) s 90 rotate 1981 294 mt (Comparison of Voting Gibbs and the Bayes optimal classifier) s SO end eplot epage end showpage %%EndDocument @endspecial 1377 w @beginspecial 57 @llx 197 @lly 550 @urx 600 @ury 1555 @rwi @setspecial %%BeginDocument: plots/expt10llf.ps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 469 135 5924 4835 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4588 L 899 389 mt 899 416 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 811 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 944 4724 mt (0) s 1436 4615 mt 1436 4588 L 1436 389 mt 1436 416 L 1751 4615 mt 1751 4588 L 1751 389 mt 1751 416 L 1974 4615 mt 1974 4588 L 1974 389 mt 1974 416 L 2147 4615 mt 2147 4588 L 2147 389 mt 2147 416 L 2288 4615 mt 2288 4588 L 2288 389 mt 2288 416 L 2408 4615 mt 2408 4588 L 2408 389 mt 2408 416 L 2511 4615 mt 2511 4588 L 2511 389 mt 2511 416 L 2603 4615 mt 2603 4588 L 2603 389 mt 2603 416 L 2684 4615 mt 2684 4588 L 2684 389 mt 2684 416 L 2684 4615 mt 2684 4561 L 2684 389 mt 2684 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 2596 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 2729 4724 mt (1) s 3222 4615 mt 3222 4588 L 3222 389 mt 3222 416 L 3536 4615 mt 3536 4588 L 3536 389 mt 3536 416 L 3759 4615 mt 3759 4588 L 3759 389 mt 3759 416 L 3932 4615 mt 3932 4588 L 3932 389 mt 3932 416 L 4074 4615 mt 4074 4588 L 4074 389 mt 4074 416 L 4193 4615 mt 4193 4588 L 4193 389 mt 4193 416 L 4297 4615 mt 4297 4588 L 4297 389 mt 4297 416 L 4388 4615 mt 4388 4588 L 4388 389 mt 4388 416 L 4470 4615 mt 4470 4588 L 4470 389 mt 4470 416 L 4470 4615 mt 4470 4561 L 4470 389 mt 4470 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 4382 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 4515 4724 mt (2) s 5007 4615 mt 5007 4588 L 5007 389 mt 5007 416 L 5321 4615 mt 5321 4588 L 5321 389 mt 5321 416 L 5545 4615 mt 5545 4588 L 5545 389 mt 5545 416 L 5718 4615 mt 5718 4588 L 5718 389 mt 5718 416 L 5859 4615 mt 5859 4588 L 5859 389 mt 5859 416 L 5978 4615 mt 5978 4588 L 5978 389 mt 5978 416 L 6082 4615 mt 6082 4588 L 6082 389 mt 6082 416 L 6173 4615 mt 6173 4588 L 6173 389 mt 6173 416 L 6255 4615 mt 6255 4588 L 6255 389 mt 6255 416 L 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 6167 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 6300 4724 mt (3) s 899 4615 mt 926 4615 L 6255 4615 mt 6228 4615 L 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 4659 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 4585 mt (-3) s 899 4191 mt 926 4191 L 6255 4191 mt 6228 4191 L 899 3943 mt 926 3943 L 6255 3943 mt 6228 3943 L 899 3767 mt 926 3767 L 6255 3767 mt 6228 3767 L 899 3630 mt 926 3630 L 6255 3630 mt 6228 3630 L 899 3519 mt 926 3519 L 6255 3519 mt 6228 3519 L 899 3425 mt 926 3425 L 6255 3425 mt 6228 3425 L 899 3343 mt 926 3343 L 6255 3343 mt 6228 3343 L 899 3271 mt 926 3271 L 6255 3271 mt 6228 3271 L 899 3206 mt 926 3206 L 6255 3206 mt 6228 3206 L 899 3206 mt 953 3206 L 6255 3206 mt 6201 3206 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 3250 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 3176 mt (-2) s 899 2782 mt 926 2782 L 6255 2782 mt 6228 2782 L 899 2534 mt 926 2534 L 6255 2534 mt 6228 2534 L 899 2358 mt 926 2358 L 6255 2358 mt 6228 2358 L 899 2222 mt 926 2222 L 6255 2222 mt 6228 2222 L 899 2110 mt 926 2110 L 6255 2110 mt 6228 2110 L 899 2016 mt 926 2016 L 6255 2016 mt 6228 2016 L 899 1934 mt 926 1934 L 6255 1934 mt 6228 1934 L 899 1862 mt 926 1862 L 6255 1862 mt 6228 1862 L 899 1798 mt 926 1798 L 6255 1798 mt 6228 1798 L 899 1798 mt 953 1798 L 6255 1798 mt 6201 1798 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 1842 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 1768 mt (-1) s 899 1374 mt 926 1374 L 6255 1374 mt 6228 1374 L 899 1126 mt 926 1126 L 6255 1126 mt 6228 1126 L 899 950 mt 926 950 L 6255 950 mt 6228 950 L 899 813 mt 926 813 L 6255 813 mt 6228 813 L 899 702 mt 926 702 L 6255 702 mt 6228 702 L 899 607 mt 926 607 L 6255 607 mt 6228 607 L 899 526 mt 926 526 L 6255 526 mt 6228 526 L 899 453 mt 926 453 L 6255 453 mt 6228 453 L 899 389 mt 926 389 L 6255 389 mt 6228 389 L 899 389 mt 953 389 L 6255 389 mt 6201 389 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 433 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 359 mt (0) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 4 2 4 3 4 5 4 0 4 6 4 4 4 4 4 1 4 6 4 2 4 4 4 1 5 2 4 6 4 2 4 2 4 3 5 6 4 4 4 3 5 3 4 2 4 5 5 4 4 6 5 2 4 3 5 4 4 4 5 5 4 4 5 -1 5 5 4 5 5 5 5 2 4 5 5 5 5 4 5 2 5 4 5 6 5 1 5 4 5 8 5 6 5 4 5 4 5 3 5 4 5 5 6 3 5 6 5 6 5 6 6 5 5 3 6 0 5 7 6 4 5 4 6 4 6 7 5 6 6 6 6 3 6 5 6 6 5 6 6 7 6 5 7 -3 6 4 6 6 6 9 6 6 7 5 6 3 6 5 7 9 6 6 7 9 6 5 7 5 7 6 7 5 7 6 6 6 7 5 8 7 7 7 7 3 7 9 7 1 8 7 7 6 8 3 7 7 8 11 5011 3797 100 MP stroke 8 3 8 6 8 5 8 12 8 6 8 2 8 4 8 4 9 11 8 4 9 9 9 10 8 3 9 7 9 8 9 6 10 7 9 1 10 11 9 6 10 8 10 10 10 8 10 8 10 12 10 9 11 4 10 9 11 11 11 9 11 9 11 5 12 7 11 12 12 15 12 9 12 16 12 8 13 6 13 13 12 12 14 14 13 12 14 9 13 9 15 10 14 12 15 6 14 16 16 13 15 12 16 9 16 11 17 12 16 13 18 11 17 16 18 9 19 13 19 14 19 23 20 20 20 18 21 12 22 13 22 24 23 11 23 16 25 22 25 12 25 17 27 21 28 15 29 21 29 14 31 34 33 27 33 10 36 21 37 32 38 28 41 42 43 27 46 33 48 38 52 45 56 49 59 40 65 53 70 51 78 53 86 82 97 84 111 74 130 100 155 110 195 147 261 188 396 263 1751 1361 100 MP stroke gr /Helvetica /ISOLatin1Encoding 120 FMSR 585 3535 mt -90 rotate (Additional relative error of Voting Gibbs) s 90 rotate 3533 4941 mt (N) s 2589 294 mt (Experiment with 10 training examples) s gs 899 389 5357 4227 MR c np DO 852 459 899 902 2 MP stroke SO gr end eplot epage end showpage %%EndDocument @endspecial 1377 w @beginspecial 57 @llx 197 @lly 550 @urx 600 @ury 1555 @rwi @setspecial %%BeginDocument: plots/expt50llf.ps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 469 135 5924 4835 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4588 L 899 389 mt 899 416 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 811 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 944 4724 mt (0) s 1436 4615 mt 1436 4588 L 1436 389 mt 1436 416 L 1751 4615 mt 1751 4588 L 1751 389 mt 1751 416 L 1974 4615 mt 1974 4588 L 1974 389 mt 1974 416 L 2147 4615 mt 2147 4588 L 2147 389 mt 2147 416 L 2288 4615 mt 2288 4588 L 2288 389 mt 2288 416 L 2408 4615 mt 2408 4588 L 2408 389 mt 2408 416 L 2511 4615 mt 2511 4588 L 2511 389 mt 2511 416 L 2603 4615 mt 2603 4588 L 2603 389 mt 2603 416 L 2684 4615 mt 2684 4588 L 2684 389 mt 2684 416 L 2684 4615 mt 2684 4561 L 2684 389 mt 2684 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 2596 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 2729 4724 mt (1) s 3222 4615 mt 3222 4588 L 3222 389 mt 3222 416 L 3536 4615 mt 3536 4588 L 3536 389 mt 3536 416 L 3759 4615 mt 3759 4588 L 3759 389 mt 3759 416 L 3932 4615 mt 3932 4588 L 3932 389 mt 3932 416 L 4074 4615 mt 4074 4588 L 4074 389 mt 4074 416 L 4193 4615 mt 4193 4588 L 4193 389 mt 4193 416 L 4297 4615 mt 4297 4588 L 4297 389 mt 4297 416 L 4388 4615 mt 4388 4588 L 4388 389 mt 4388 416 L 4470 4615 mt 4470 4588 L 4470 389 mt 4470 416 L 4470 4615 mt 4470 4561 L 4470 389 mt 4470 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 4382 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 4515 4724 mt (2) s 5007 4615 mt 5007 4588 L 5007 389 mt 5007 416 L 5321 4615 mt 5321 4588 L 5321 389 mt 5321 416 L 5545 4615 mt 5545 4588 L 5545 389 mt 5545 416 L 5718 4615 mt 5718 4588 L 5718 389 mt 5718 416 L 5859 4615 mt 5859 4588 L 5859 389 mt 5859 416 L 5978 4615 mt 5978 4588 L 5978 389 mt 5978 416 L 6082 4615 mt 6082 4588 L 6082 389 mt 6082 416 L 6173 4615 mt 6173 4588 L 6173 389 mt 6173 416 L 6255 4615 mt 6255 4588 L 6255 389 mt 6255 416 L 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 6167 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 6300 4724 mt (3) s 899 4615 mt 926 4615 L 6255 4615 mt 6228 4615 L 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 4659 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 4585 mt (-3) s 899 4191 mt 926 4191 L 6255 4191 mt 6228 4191 L 899 3943 mt 926 3943 L 6255 3943 mt 6228 3943 L 899 3767 mt 926 3767 L 6255 3767 mt 6228 3767 L 899 3630 mt 926 3630 L 6255 3630 mt 6228 3630 L 899 3519 mt 926 3519 L 6255 3519 mt 6228 3519 L 899 3425 mt 926 3425 L 6255 3425 mt 6228 3425 L 899 3343 mt 926 3343 L 6255 3343 mt 6228 3343 L 899 3271 mt 926 3271 L 6255 3271 mt 6228 3271 L 899 3206 mt 926 3206 L 6255 3206 mt 6228 3206 L 899 3206 mt 953 3206 L 6255 3206 mt 6201 3206 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 3250 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 3176 mt (-2) s 899 2782 mt 926 2782 L 6255 2782 mt 6228 2782 L 899 2534 mt 926 2534 L 6255 2534 mt 6228 2534 L 899 2358 mt 926 2358 L 6255 2358 mt 6228 2358 L 899 2222 mt 926 2222 L 6255 2222 mt 6228 2222 L 899 2110 mt 926 2110 L 6255 2110 mt 6228 2110 L 899 2016 mt 926 2016 L 6255 2016 mt 6228 2016 L 899 1934 mt 926 1934 L 6255 1934 mt 6228 1934 L 899 1862 mt 926 1862 L 6255 1862 mt 6228 1862 L 899 1798 mt 926 1798 L 6255 1798 mt 6228 1798 L 899 1798 mt 953 1798 L 6255 1798 mt 6201 1798 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 1842 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 1768 mt (-1) s 899 1374 mt 926 1374 L 6255 1374 mt 6228 1374 L 899 1126 mt 926 1126 L 6255 1126 mt 6228 1126 L 899 950 mt 926 950 L 6255 950 mt 6228 950 L 899 813 mt 926 813 L 6255 813 mt 6228 813 L 899 702 mt 926 702 L 6255 702 mt 6228 702 L 899 607 mt 926 607 L 6255 607 mt 6228 607 L 899 526 mt 926 526 L 6255 526 mt 6228 526 L 899 453 mt 926 453 L 6255 453 mt 6228 453 L 899 389 mt 926 389 L 6255 389 mt 6228 389 L 899 389 mt 953 389 L 6255 389 mt 6201 389 L /Helvetica /ISOLatin1Encoding 120 FMSR 640 433 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 773 359 mt (0) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 4 6 4 -2 4 2 4 5 4 5 4 5 4 5 4 0 4 3 4 3 4 5 4 3 5 -3 4 5 4 4 4 4 4 5 5 4 4 0 4 3 5 5 4 3 4 4 5 6 4 0 5 5 4 7 5 3 4 0 5 3 4 2 5 4 5 1 4 4 5 4 5 1 4 3 5 5 5 5 5 4 5 5 5 6 5 2 5 5 5 7 5 4 5 5 5 4 5 3 5 5 5 2 6 6 5 6 5 4 5 2 6 7 5 5 6 4 5 3 6 2 5 7 6 3 6 6 5 3 6 3 6 9 6 0 6 6 5 4 6 5 6 1 7 7 6 1 6 1 6 5 6 6 7 5 6 1 6 8 7 4 6 2 7 7 6 6 7 6 7 8 7 4 7 7 6 7 7 -1 8 8 7 6 7 1 7 8 7 6 8 5 7 3 8 5 7 12 8 8 5011 3864 100 MP stroke 8 4 8 1 8 8 8 7 8 3 8 7 8 6 8 4 9 9 8 7 9 11 9 2 8 3 9 8 9 5 9 6 10 8 9 13 10 6 9 8 10 2 10 9 10 10 10 4 10 8 10 7 11 10 10 13 11 7 11 13 11 9 11 12 12 12 11 13 12 12 12 4 12 9 12 3 13 12 13 11 12 9 14 15 13 12 14 10 13 15 15 14 14 7 15 14 14 9 16 13 15 7 16 11 16 18 17 11 16 16 18 19 17 18 18 10 19 17 19 18 19 12 20 11 20 11 21 18 22 16 22 11 23 27 23 10 25 26 25 21 25 22 27 25 28 26 29 23 29 21 31 37 33 18 33 24 36 27 37 25 38 33 41 29 43 42 46 17 48 33 52 40 56 28 59 50 65 59 70 63 78 58 86 58 97 72 111 99 130 104 155 104 195 129 261 210 396 285 1751 1371 100 MP stroke gr /Helvetica /ISOLatin1Encoding 120 FMSR 585 3535 mt -90 rotate (Additional relative error of Voting Gibbs) s 90 rotate 3533 4941 mt (N) s 2589 294 mt (Experiment with 50 training examples) s gs 899 389 5357 4227 MR c np DO 852 493 899 878 2 MP stroke SO gr end eplot epage end showpage %%EndDocument @endspecial 457 1052 a Fx(\(a\))1269 b(\(b\))i(\(c\))-138 1220 y Fr(Figure)26 b(2.)h Fq(\(a\))20 b(Plot)i(of)f(error)g(vs.)33 b(n)n(um)n(b)r(er)18 b(of)k(training)f(examples)f Fi(m)g Fq(for)i(the)e(Ba)n(y)n(es)h(optimal)f(classi\014er)j(\(solid\))e(and)f 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y(p)r(erformance)i(at)h(a)g(rate)f(of)h Fv(O)r Fx(\(1)p Fv(=)-5 b(N)9 b Fx(\).)36 b(When)23 b(it)h(is)f(tractable)1962 4486 y(to)37 b(sample)g(from)g(the)h(p)r(osterior)d(distribution)j(of)f (the)h(pa-)1962 4585 y(rameters,)26 b(this)i(indicates)g(that)g(V)-7 b(oting)27 b(Gibbs)h(can)g(indeed)p 1962 4639 V 1962 4716 a Fq(quence)i(\(run)g(using)i(the)e(\\correct")j(prior)38 b(^)-45 b Fi(p)p Fq(\))31 b(w)n(as)g(used)g(to)g(ap-)1962 4799 y(pro)n(ximate)39 b(b)r(oth)g Fi(h)2578 4807 y Fg(B)2671 4799 y Fq(and)g(the)g(ground-truth)f(p)r(osterior)j(dis-)1962 4882 y(tributions.)55 b(Lastly)-6 b(,)35 b(these)e(exp)r(erimen)n(ts)e (w)n(ere)i(run)f(using)h(the)1962 4965 y(alternativ)n(e)45 b(v)n(ersion)f(of)h(V)n(G\()p Fi(N)8 b Fq(\))44 b(describ)r(ed)g(in)g (Section)h(2.2,)1962 5048 y(that)32 b(skips)h(the)f(second)h(stage)h (of)f(sampling)g(\(in)n(v)n(olving)f(dra)n(w-)1962 5140 y(ing)f Fi(y)2136 5108 y Fg(i)2162 5140 y Fq('s)g(from)f(Bernoulli\()p Fi(f)2802 5156 y Fg(\022)2833 5142 y Fl(i)2865 5140 y Fq(\()p Fi(x)p Fq(\)\)\),)h(and)g(predicts)f(1)h(whenev)n(er)1962 5223 y(\(1)p Fi(=)l(N)8 b Fq(\))2177 5167 y Ff(P)2270 5223 y Fi(f)2307 5239 y Fg(\022)2338 5225 y Fl(i)2369 5223 y Fq(\()p Fi(x)p Fq(\))21 b Fh(\025)g Fq(0)p Fi(:)p Fq(5.)p eop %%Page: 8 8 8 7 bop -138 1479 a @beginspecial 56 @llx 197 @lly 550 @urx 600 @ury 2332 @rwi @setspecial %%BeginDocument: plots/1varyNf.ps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 460 135 5933 4835 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4588 L 899 389 mt 899 416 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 811 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 944 4724 mt (0) s 1436 4615 mt 1436 4588 L 1436 389 mt 1436 416 L 1751 4615 mt 1751 4588 L 1751 389 mt 1751 416 L 1974 4615 mt 1974 4588 L 1974 389 mt 1974 416 L 2147 4615 mt 2147 4588 L 2147 389 mt 2147 416 L 2288 4615 mt 2288 4588 L 2288 389 mt 2288 416 L 2408 4615 mt 2408 4588 L 2408 389 mt 2408 416 L 2511 4615 mt 2511 4588 L 2511 389 mt 2511 416 L 2603 4615 mt 2603 4588 L 2603 389 mt 2603 416 L 2684 4615 mt 2684 4588 L 2684 389 mt 2684 416 L 2684 4615 mt 2684 4561 L 2684 389 mt 2684 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 2596 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 2729 4724 mt (1) s 3222 4615 mt 3222 4588 L 3222 389 mt 3222 416 L 3536 4615 mt 3536 4588 L 3536 389 mt 3536 416 L 3759 4615 mt 3759 4588 L 3759 389 mt 3759 416 L 3932 4615 mt 3932 4588 L 3932 389 mt 3932 416 L 4074 4615 mt 4074 4588 L 4074 389 mt 4074 416 L 4193 4615 mt 4193 4588 L 4193 389 mt 4193 416 L 4297 4615 mt 4297 4588 L 4297 389 mt 4297 416 L 4388 4615 mt 4388 4588 L 4388 389 mt 4388 416 L 4470 4615 mt 4470 4588 L 4470 389 mt 4470 416 L 4470 4615 mt 4470 4561 L 4470 389 mt 4470 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 4382 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 4515 4724 mt (2) s 5007 4615 mt 5007 4588 L 5007 389 mt 5007 416 L 5321 4615 mt 5321 4588 L 5321 389 mt 5321 416 L 5545 4615 mt 5545 4588 L 5545 389 mt 5545 416 L 5718 4615 mt 5718 4588 L 5718 389 mt 5718 416 L 5859 4615 mt 5859 4588 L 5859 389 mt 5859 416 L 5978 4615 mt 5978 4588 L 5978 389 mt 5978 416 L 6082 4615 mt 6082 4588 L 6082 389 mt 6082 416 L 6173 4615 mt 6173 4588 L 6173 389 mt 6173 416 L 6255 4615 mt 6255 4588 L 6255 389 mt 6255 416 L 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 6167 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 6300 4724 mt (3) s 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L /Helvetica /ISOLatin1Encoding 120 FMSR 631 4659 mt (0.15) s 899 4011 mt 953 4011 L 6255 4011 mt 6201 4011 L 698 4055 mt (0.2) s 899 3408 mt 953 3408 L 6255 3408 mt 6201 3408 L 631 3452 mt (0.25) s 899 2804 mt 953 2804 L 6255 2804 mt 6201 2804 L 698 2848 mt (0.3) s 899 2200 mt 953 2200 L 6255 2200 mt 6201 2200 L 631 2244 mt (0.35) s 899 1596 mt 953 1596 L 6255 1596 mt 6201 1596 L 698 1640 mt (0.4) s 899 993 mt 953 993 L 6255 993 mt 6201 993 L 631 1037 mt (0.45) s 899 389 mt 953 389 L 6255 389 mt 6201 389 L 698 433 mt (0.5) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np DA 934 3 851 -22 934 38 852 -81 933 114 852 -111 899 4351 7 MP stroke DD 934 -682 851 -1733 934 -948 852 -221 933 -19 852 -19 899 4351 7 MP stroke DA 934 -3 851 -29 934 36 852 -80 933 110 852 -112 899 4348 7 MP stroke DD 934 -608 851 -1765 934 -1009 852 -236 933 -25 852 -19 899 4348 7 MP stroke DA 934 -34 851 -24 934 30 852 -78 933 102 852 -118 899 4343 7 MP stroke DD 934 -564 851 -1692 934 -1101 852 -260 933 -42 852 -26 899 4343 7 MP stroke DA 934 -127 851 -37 934 9 852 -70 933 91 852 -143 899 4325 7 MP stroke DD 934 -489 851 -1528 934 -1208 852 -334 933 -90 852 -34 899 4325 7 MP stroke gr DD 3114 294 mt (1 relevant feature) s 3068 4941 mt (Number of features) s 576 2628 mt -90 rotate (error) s 90 rotate SO end eplot epage end showpage %%EndDocument @endspecial 2106 w @beginspecial 56 @llx 197 @lly 550 @urx 600 @ury 2332 @rwi @setspecial %%BeginDocument: plots/3relevantf.ps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 460 135 5933 4835 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6914 5188 PR 6 w 0 4226 5356 0 0 -4226 899 4615 4 MP PP -5356 0 0 4226 5356 0 0 -4226 899 4615 5 MP stroke 4 w DO SO 6 w 0 sg 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L 899 4615 mt 6255 4615 L 899 4615 mt 899 389 L 899 4615 mt 899 4588 L 899 389 mt 899 416 L 899 4615 mt 899 4561 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 811 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 944 4724 mt (0) s 1436 4615 mt 1436 4588 L 1436 389 mt 1436 416 L 1751 4615 mt 1751 4588 L 1751 389 mt 1751 416 L 1974 4615 mt 1974 4588 L 1974 389 mt 1974 416 L 2147 4615 mt 2147 4588 L 2147 389 mt 2147 416 L 2288 4615 mt 2288 4588 L 2288 389 mt 2288 416 L 2408 4615 mt 2408 4588 L 2408 389 mt 2408 416 L 2511 4615 mt 2511 4588 L 2511 389 mt 2511 416 L 2603 4615 mt 2603 4588 L 2603 389 mt 2603 416 L 2684 4615 mt 2684 4588 L 2684 389 mt 2684 416 L 2684 4615 mt 2684 4561 L 2684 389 mt 2684 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 2596 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 2729 4724 mt (1) s 3222 4615 mt 3222 4588 L 3222 389 mt 3222 416 L 3536 4615 mt 3536 4588 L 3536 389 mt 3536 416 L 3759 4615 mt 3759 4588 L 3759 389 mt 3759 416 L 3932 4615 mt 3932 4588 L 3932 389 mt 3932 416 L 4074 4615 mt 4074 4588 L 4074 389 mt 4074 416 L 4193 4615 mt 4193 4588 L 4193 389 mt 4193 416 L 4297 4615 mt 4297 4588 L 4297 389 mt 4297 416 L 4388 4615 mt 4388 4588 L 4388 389 mt 4388 416 L 4470 4615 mt 4470 4588 L 4470 389 mt 4470 416 L 4470 4615 mt 4470 4561 L 4470 389 mt 4470 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 4382 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 4515 4724 mt (2) s 5007 4615 mt 5007 4588 L 5007 389 mt 5007 416 L 5321 4615 mt 5321 4588 L 5321 389 mt 5321 416 L 5545 4615 mt 5545 4588 L 5545 389 mt 5545 416 L 5718 4615 mt 5718 4588 L 5718 389 mt 5718 416 L 5859 4615 mt 5859 4588 L 5859 389 mt 5859 416 L 5978 4615 mt 5978 4588 L 5978 389 mt 5978 416 L 6082 4615 mt 6082 4588 L 6082 389 mt 6082 416 L 6173 4615 mt 6173 4588 L 6173 389 mt 6173 416 L 6255 4615 mt 6255 4588 L 6255 389 mt 6255 416 L 6255 4615 mt 6255 4561 L 6255 389 mt 6255 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 6167 4798 mt (10) s /Helvetica /ISOLatin1Encoding 80 FMSR 6300 4724 mt (3) s 899 4615 mt 953 4615 L 6255 4615 mt 6201 4615 L /Helvetica /ISOLatin1Encoding 120 FMSR 698 4659 mt (0.2) s 899 3911 mt 953 3911 L 6255 3911 mt 6201 3911 L 631 3955 mt (0.25) s 899 3206 mt 953 3206 L 6255 3206 mt 6201 3206 L 698 3250 mt (0.3) s 899 2502 mt 953 2502 L 6255 2502 mt 6201 2502 L 631 2546 mt (0.35) s 899 1798 mt 953 1798 L 6255 1798 mt 6201 1798 L 698 1842 mt (0.4) s 899 1093 mt 953 1093 L 6255 1093 mt 6201 1093 L 631 1137 mt (0.45) s 899 389 mt 953 389 L 6255 389 mt 6201 389 L 698 433 mt (0.5) s 899 4615 mt 6255 4615 L 899 389 mt 6255 389 L 899 4615 mt 899 389 L 6255 4615 mt 6255 389 L gs 899 389 5357 4227 MR c np 4504 0 1751 4425 2 MP stroke DA 934 -125 851 -244 934 -244 852 -36 933 -115 1751 4404 6 MP stroke DD 934 -1096 851 -934 934 -1097 852 -340 933 -135 1751 4405 6 MP stroke gr DD 3084 294 mt (3 relevant features) s 3068 4941 mt (Number of features) s 576 2628 mt -90 rotate (error) s 90 rotate SO end eplot epage end showpage %%EndDocument @endspecial 781 1578 a Fx(\(a\))1998 b(\(b\))-138 1746 y Fr(Figure)26 b(3.)h Fq(\(a\))33 b(Plot)h(of)f(errors)h(of)g(V)n (G\(1\),)h(V)n(G\(3\),)g(V)n(G\(7\),)g(V)n(G\(15\))f(using)f(the)g (\\go)r(o)r(d")h(\(dash\))f(and)g(\\naiv)n(e")h(\(dash-dot\))e(priors.) -138 1838 y(Higher)26 b(lines)g(corresp)r(ond)g(to)g(lo)n(w)n(er)h(v)l (alues)f(of)g Fi(N)8 b Fq(.)35 b(\(b\))25 b(Same)g(as)h(Figure)g(1,)h (but)e(with)g(3)h(instead)g(of)h(1)f(relev)l(an)n(t)f(features.)-138 2106 y Fx(pro)n(vide)g(a)h(go)r(o)r(d,)g(practical)f(w)n(a)n(y)g(to)i (appro)n(ximate)d(optimal)-138 2205 y(Ba)n(y)n(esian)e (classi\014cation.)34 b(In)24 b(the)g(con)n(text)f(of)g(feature)h (selec-)-138 2305 y(tion,)k(w)n(e)g(also)f(sho)n(w)n(ed)g(that)i(V)-7 b(oting)28 b(Gibbs)h(has)e(v)n(ery)g(high)-138 2404 y(tolerance)34 b(to)i(the)f(presence)g(of)g(irrelev)-5 b(an)n(t)35 b(features,)i(with) -138 2504 y(b)r(ounds)24 b(comparable)e(to)i(those)f(of)h(the)g(b)r (est)g(kno)n(wn)f(feature)-138 2604 y(selection)k(algorithms.)-138 2842 y Fw(Ac)m(kno)m(wledgemen)m(ts)-138 3003 y Fx(W)-7 b(e)47 b(thank)g(Nando)g(de)g(F)-7 b(reitas,)51 b(V)-7 b(assilis)46 b(P)n(apa)n(v)-5 b(assiliou)-138 3103 y(and)50 b(Hanna)f(P)n(asula)g(for)g(helpful)i(con)n(v)n(ersations)c(ab)r(out) -138 3202 y(this)33 b(w)n(ork.)50 b(This)33 b(w)n(ork)e(w)n(as)g(supp)r (orted)i(b)n(y)f(ONR)h(MURI)-138 3302 y(N00014-00-1-0637)21 b(and)28 b(NSF)g(gran)n(t)e(I)r(IS-9988642.)-138 3540 y Fw(References)-138 3690 y Fx(Barron,)j(A.,)j(Clark)n(e,)e(B.,)h(&)f (Haussler,)h(D.)g(\(1993\).)44 b(Infor-)-55 3789 y(mation)31 b(b)r(ounds)h(for)f(the)h(risk)f(of)g(Ba)n(y)n(esian)f(predictions)-55 3889 y(and)39 b(the)h(redundancy)f(of)h(Univ)n(ersal)e(co)r(des.)72 b Fu(Pr)l(o)l(c)l(e)l(e)l(d-)-55 3988 y(ings)24 b(of)g(the)g (International)g(Symp)l(osium)g(on)g(Information)-55 4088 y(The)l(ory)p Fx(.)-138 4257 y(Denison,)f(D.,)h(&)e(Mallic)n(k,)g (B.)g(\(2000\).)27 b(Classi\014cation)21 b(trees.)-55 4357 y(In)29 b(D.)h(Dey)-7 b(,)30 b(S.)g(Ghosh)f(and)g(B.)g(Mallic)n(k) f(\(Eds.\),)i Fu(Gener-)-55 4456 y(alize)l(d)i(Line)l(ar)g(Mo)l(dels:) 43 b(A)30 b(Bayesian)j(p)l(ersp)l(e)l(ctive)p Fx(,)e(365{)-55 4556 y(372.)26 b(Marcel-Dekk)n(er.)-138 4725 y(Gilks,)38 b(W.)e(R.,)i(Ric)n(hardson,)e(S.,)j(&)c(Spiegelhalter,)i(D.)f(J.)-55 4825 y(\(Eds.\).)23 b(\(1996\).)29 b Fu(Markov)e(Chain)g(Monte)f(Carlo) h(in)f(Pr)l(ac-)-55 4924 y(tic)l(e)p Fx(.)37 b(Chapman)27 b(and)h(Hall.)-138 5094 y(Green,)40 b(P)-7 b(.)37 b(J.)g(\(1995\).)65 b(Rev)n(ersible)36 b(jump)i(Mark)n(o)n(v)e(c)n(hain)-55 5193 y(Mon)n(te)d(Carlo)e(computation)i(and)f(Ba)n(y)n(esian)f(mo)r (del)i(de-)-55 5293 y(termination.)j Fu(Biometrika)p Fx(,)30 b Fu(82)p Fx(,)f(711{732.)1962 2106 y(Kearns,)22 b(M.,)j(&)d(V)-7 b(azirani,)23 b(U.)h(V.)f(\(1994\).)28 b Fu(A)n(n)d(intr)l(o)l(duction)2045 2205 y(to)30 b(c)l(omputational)h (le)l(arning)f(the)l(ory)p Fx(.)37 b(MIT)28 b(Press.)1962 2338 y(Kivinen,)c(J.,)h(&)e(W)-7 b(arm)n(uth,)25 b(M.)f(K.)f(\(1994\).) 30 b Fu(Exp)l(onentiate)l(d)2045 2438 y(gr)l(adient)24 b(versus)f(gr)l(adient)h(desc)l(ent)f(for)h(line)l(ar)g(pr)l(e)l (dictors)2045 2537 y Fx(\(T)-7 b(ec)n(hnical)22 b(Rep)r(ort)g (UCSC-CRL-94-16\).)k(Univ.)d(of)f(Cal-)2045 2637 y(ifornia)h(San)n(ta)g (Cruz,)h(Computer)f(Researc)n(h)f(Lab)r(oratory)-7 b(.)1962 2770 y(Koha)n(vi,)29 b(R.,)j(&)e(John,)g(G.)h(H.)g(\(1997\).)43 b(W)-7 b(rapp)r(ers)30 b(for)g(fea-)2045 2869 y(ture)20 b(subset)g(selection.)j Fu(A)n(rti\014cial)g(Intel)t(ligenc)l(e)p Fx(,)g Fu(97)p Fx(,)f(273{)2045 2969 y(324.)1962 3102 y(Littlestone,)34 b(N.)g(\(1988\).)51 b(Learning)32 b(quic)n(kly)g (when)h(irrele-)2045 3201 y(v)-5 b(an)n(t)30 b(attributes)h(ab)r(ound:) 42 b(A)31 b(new)g(linear-threshold)e(al-)2045 3301 y(gorithm.)36 b Fu(Machine)c(L)l(e)l(arning)p Fx(,)c Fu(2)p Fx(,)g(285{318.)1962 3434 y(Ng,)44 b(A.)e(Y.)f(\(1998\).)75 b(On)41 b(F)-7 b(eature)41 b(Selection:)63 b(Learning)2045 3534 y(with)20 b(exp)r(onen)n(tially)g(man)n(y)f(irrelev)-5 b(an)n(t)18 b(features)i(as)f(train-)2045 3633 y(ing)29 b(examples.)40 b Fu(Pr)l(o)l(c)l(e)l(e)l(dings)31 b(of)h(the)f(Fifte)l(enth)h (Interna-)2045 3733 y(tional)40 b(Confer)l(enc)l(e)h(on)e(Machine)j(L)l (e)l(arning)c Fx(\(pp.)h(404{)2045 3832 y(412\).)d(Morgan)26 b(Kaufmann.)1962 3965 y(Ng,)83 b(A.)73 b(Y.,)84 b(&)72 b(Jordan,)82 b(M.)73 b(I.)f(\(2001\).)169 b(Con-)2045 4065 y(v)n(ergence)68 b(rates)h(of)g(the)h(V)-7 b(oting)70 b(Gibbs)f(classi\014er,)2045 4164 y(with)d(application)e(to)h(Ba)n(y)n (esian)f(feature)g(selection.)2045 4264 y Fq(www.cs.b)r(erk)n(eley)-6 b(.edu/~ang/pap)r(ers/icml01-vg-long.ps)p Fx(.)1962 4397 y(Ripley)f(,)27 b(B.)f(D.)h(\(1987\).)33 b Fu(Sto)l(chastic)d (simulation)p Fx(.)35 b(John)26 b(Wi-)2045 4497 y(ley)-7 b(.)1962 4629 y(Syk)i(acek,)26 b(P)-7 b(.)26 b(\(2000\).)34 b(On)26 b(input)i(selection)e(with)h(rev)n(ersible)2045 4729 y(jump)45 b(Mark)n(o)n(v)d(c)n(hain)i(Mon)n(te)g(Carlo)f (sampling.)86 b Fu(A)l(d-)2045 4829 y(vanc)l(es)24 b(in)f(Neur)l(al)g (Information)h(Pr)l(o)l(c)l(essing)g(Systems)f(12)2045 4928 y Fx(\(pp.)28 b(638{644\).)1962 5061 y(V)-7 b(alian)n(t,)24 b(L.)f(G.)g(\(1984\).)28 b(A)c(theory)e(of)h(the)g(learnable.)29 b Fu(Com-)2045 5161 y(munic)l(ations)h(of)g(the)g(A)n(CM)p Fx(,)e Fu(25)p Fx(,)h(1134{1142.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF