%%Creator: %!PS-Adobe-2.0 dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: BayesLike.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica Helvetica-Bold Helvetica-Oblique %%+ Helvetica-BoldOblique Symbol %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips BayesLike.dvi -o BayesLike.ps -K -O 0.0in,0.5in %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2002.04.22:1359 %%BeginProcSet: texc.pro /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
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% /quotedblleft in TeX text % /asciicircum in ASCII % /dotaccent in TeX text % /grave accent in ANSI
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123 % /endash in TeX text % /emdash in TeX test 125 % /hungarumlaut in TeX text % /asciitilde in ASCII not used (see 168), use higher up instead 128 this position is unfortunate, but now too late to fix 129 130 132 135 136 137 139 142 145 146 147 148
153 155 158 159 % /space (no break space) 161
/currency % 164 /yen % 165 /brokenbar % /section % 167 /dieresis % 168 /copyright % /ordfeminine % /guillemotleft % /logicalnot % /sfthyphen % /registered % /macron % 175 /degree % 176 /plusminus % /twosuperior % /threesuperior % /acute % 180 /mu % 181 /paragraph % /periodcentered % /cedilla % 184 /onesuperior % /ordmasculine % /guillemotright % /onequarter % /onehalf % 189 /threequarters % /questiondown % /Agrave % 192 /Aacute % 193 /Acircumflex % /Atilde % 195 /Adieresis % /Aring % 197 /AE % 198 /Ccedilla % 199 /Egrave % 200 /Eacute % 201 /Ecircumflex % /Edieresis % /Igrave % 204 /Iacute % 205 /Icircumflex % /Idieresis % /Eth % 208 /Ntilde % 209 /Ograve % 210 /Oacute % 211 /Ocircumflex % /Otilde % 213 /Odieresis % /multiply % 215 /Oslash % 216 /Ugrave % 217 /Uacute % 218 /Ucircumflex % /Udieresis % /Yacute % 221 /Thorn % 222
166 169 170 171 172 173 % /hyphen (hanging hyphen) 174 177 178 179 182 183 185 186 187 188 190 191 194 196
202 203 206 207
212 214 % OE in T1
219 220
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223 226 228
234 235 238 239
244 246 % oe in T1
251 252 255 % germandbls in T1
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Fq(1)p eop %%Page: 2 2 2 1 bop 136 399 a Fp(\017)46 b Fq(A)31 b(probabilit)m(y)d(mo)s(del)h (is)g(just)h(that{)h(a)g Fj(mo)-5 b(del)136 586 y Fp(\017)46 b Fq(It)41 b(is)f(a)i(mathematical)f(appro)m(ximation)f(to)h(realit)m (y)g(that)g(simpli\014es)d(a)j(problem)e(b)m(y)i(ignoring)f(some)227 699 y(asp)s(ects)31 b(of)f(realit)m(y)136 887 y Fp(\017)46 b Fq(a)34 b(mo)s(del)e Fj(might)43 b Fq(help)32 b(us)h(gain)g(some)h (insigh)m(t)e(ab)s(out)h(a)h(particular)e(problem)g(but)h(then)g(again) g(it)g(ma)m(y)227 1000 y(not)136 1187 y Fp(\017)46 b Fq(All)29 b(mo)s(dels)g(are)i(wrong{)g(some)f(are)h(useful)1927 5956 y(2)p eop %%Page: 3 3 3 2 bop 0 399 a Fg(2.1)112 b(Example)37 b(1:)50 b(The)37 b(Basic)g(Linear)h(Mo)s(del)f(with)f(Normal)g(Disturbances)136 570 y Fp(\017)46 b Fq(Consider)38 b(a)i(set)g(of)g(measuremen)m(ts)f (of)h(household)e(income)h(\(that)i(w)m(e'll)e(denote)h Fo(x)p Fq(\))g(and)f(household)227 683 y(consumption)29 b(\(that)i(w)m(e'll)f(denote)h Fo(y)q Fq(\))g(for)f Fl(n)g Fq(households)f(at)i(a)g(single)e(p)s(oin)m(t)g(in)g(time)136 871 y Fp(\017)46 b Fq(W)-8 b(e)39 b(b)s(eliev)m(e)d(that)i(conditional) e(on)h(household)f(income)h(household)f(consumption)g(follo)m(ws)g(a)i (normal)227 984 y(distribution)27 b(and)j(that)h(the)f(observ)-5 b(ations)30 b(are)h(indep)s(enden)m(t)330 1171 y Fo({)45 b Fq(this)30 b(is)f(the)i(distributional)26 b(assumption)330 1317 y Fo({)45 b Fq(indep)s(endence)d(allo)m(ws)h(us)g(to)h(form)f(the) h(join)m(t)g(distribution)c(of)j(consumption)g(b)m(y)g(taking)h(the)427 1430 y(pro)s(duct)29 b(of)i(the)g(marginal)e(distributions)136 1618 y Fp(\017)46 b Fq(W)-8 b(e)28 b(b)s(eliev)m(e)e(that)h (consumption)e(v)-5 b(aries)26 b(linearly)e(in)i(income,)h(further)e(b) s(oth)h(a)h(p)s(ositiv)m(e)f(and)g(a)h(negativ)m(e)227 1731 y(relationship)h(are)j(theoretically)f(p)s(ossible)330 1918 y Fo({)45 b Fq(this)30 b(determines)f(the)h(parameter)h(space)136 2106 y Fp(\017)46 b Fq(Th)m(us)30 b(our)f(probabilit)m(y)f(mo)s(del)i (is)617 2375 y Fl(p)p Fq(\()p Fo(y)q Fp(j)p Fl(\014)830 2389 y Fn(0)871 2375 y Fl(;)15 b(\014)962 2389 y Fn(1)1002 2375 y Fl(;)g(\033)1097 2337 y Fn(2)1138 2375 y Fq(\))25 b(=)1331 2261 y Ff(n)1294 2288 y Fe(Y)1295 2484 y Ff(i)p Fn(=1)1425 2375 y Fl(f)1470 2389 y Fd(N)1542 2375 y Fq(\()p Fl(y)1622 2389 y Ff(i)1650 2375 y Fp(j)p Fl(\014)1726 2389 y Fn(0)1786 2375 y Fq(+)20 b Fl(x)1929 2389 y Ff(i)1957 2375 y Fl(\014)2008 2389 y Fn(1)2048 2375 y Fl(;)76 b(\033)2204 2337 y Fn(2)2244 2375 y Fq(\))p Fl(;)137 b(\014)2492 2389 y Fn(0)2557 2375 y Fp(2)25 b Fc(R)r Fl(;)16 b(\014)2794 2389 y Fn(1)2865 2375 y Fp(2)25 b Fc(R)r Fl(;)16 b(\033)3106 2337 y Fn(2)3177 2375 y Fp(2)24 b Fq(\(0)p Fl(;)15 b Fp(1)p Fq(\))227 2668 y(where)26 b Fl(f)531 2682 y Fd(N)602 2668 y Fq(\()p Fp(\001j)p Fl(\026;)15 b(\033)837 2635 y Fn(2)878 2668 y Fq(\))26 b(denotes)h(a)f(normal)g(densit)m(y)f(with)g (mean)h Fl(\026)g Fq(and)g(v)-5 b(ariance)26 b Fl(\033)3051 2635 y Fn(2)3116 2668 y Fq(and)g(the)g(last)g(p)s(ortion)227 2781 y(simply)i(means)i(that)h Fl(\014)1044 2795 y Fn(0)1115 2781 y Fq(and)e Fl(\014)1342 2795 y Fn(1)1413 2781 y Fq(can)h(b)s(e)g(an)m(y)h(real)f(n)m(um)m(b)s(er)f(and)g Fl(\033)2614 2748 y Fn(2)2684 2781 y Fq(can)i(b)s(e)f(an)m(y)h(p)s (ositiv)m(e)e(n)m(um)m(b)s(er.)136 2969 y Fp(\017)46 b Fq(Note)d(that)g(this)e(probabilit)m(y)e(mo)s(del)i(is)f(equiv)-5 b(alen)m(t)42 b(to)g(the)g(simple)e(linear)g(regression)i(mo)s(del)e (with)227 3082 y(normal)30 b(disturbances:)1392 3195
y Fl(y)1437 3209 y Ff(i)1490 3195 y Fq(=)25 b Fl(\014)1637 3209 y Fn(0)1697 3195 y Fq(+)20 b Fl(x)1840 3209 y Ff(i)1868 3195 y Fl(\014)1919 3209 y Fn(1)1979 3195 y Fq(+)g Fl(\017;)106 b(\017)25 b Fp(\030)g(N)13 b Fq(\(0)p Fl(;)i(\033)2659 3157 y Fn(2)2700 3195 y Fq(\))1927 5956 y(3)p eop %%Page: 4 4 4 3 bop 0 399 a Fg(2.2)112 b(Example)37 b(2:)50 b(Binomial)34 b(Sampling)136 570 y Fp(\017)46 b Fq(Consider)29 b(a)i(random)e(sample) h(of)g(size)g Fl(n)g Fq(from)g(the)h(p)s(opulation)d(of)i(registered)h (Seattle)g(v)m(oters.)136 758 y Fp(\017)46 b Fq(W)-8 b(e)40 b(observ)m(e)f(that)g Fl(y)i Fq(of)e(the)g Fl(n)f Fq(citizens)f(v)m(oted)j(in)d(the)i(1996)h(presiden)m(tial)c(election,) 41 b(the)e(remainder)227 871 y(abstained.)136 1058 y Fp(\017)46 b Fq(W)-8 b(e)43 b(assume)e(that)h(eac)m(h)g(citizen's)f (decision)f(to)i(v)m(ote)h(or)e(abstain)g(follo)m(ws)f(that)i (Bernoulli)d(distribu-)227 1171 y(tion,)33 b(the)g(probabilit)m(y)d(of) j(eac)m(h)g(citizen)f(v)m(oting)h(is)f(equal,)h(and)f(the)h(decisions)e (to)i(v)m(ote)h(or)f(abstain)f(are)227 1284 y(indep)s(enden)m(t)136 1472 y Fp(\017)46 b Fq(This)34 b(implies)f(that)j Fl(y)j Fq(follo)m(ws)c(the)g(binomial)f(distribution)e(with)i(sample)h(size)g Fl(n)g Fq(and)h(probabilit)m(y)d(of)227 1585 y(success)e Fl(\031)s Fq(.)330 1772 y Fo({)45 b Fq(this)30 b(is)f(the)i (distributional)26 b(assumption)136 1960 y Fp(\017)46 b Fq(The)33 b(binomial)e(sample)h(size)h(is)g(\014xed)f(at)j Fl(n)d Fq(and)h(the)h(parameter)f Fl(\031)k Fq(can)c(b)s(e)g(an)m(y)h (n)m(um)m(b)s(er)e(in)g(the)h([0,1])227 2073 y(in)m(terv)-5 b(al)330 2260 y Fo({)45 b Fq(this)30 b(determines)f(the)h(parameter)h (space)136 2448 y Fp(\017)46 b Fq(Th)m(us)30 b(our)f(probabilit)m(y)f (mo)s(del)i(is)1198 2697 y Fl(p)p Fq(\()p Fl(y)s Fp(j)p Fl(n;)15 b(\031)s Fq(\))26 b(=)1659 2569 y Fe(\022)1726 2636 y Fl(n)1729 2760 y(y)1780 2569 y Fe(\023)1847 2697 y Fl(\031)1902 2660 y Ff(y)1943 2697 y Fq(\(1)c Fp(\000)d Fl(\031)s Fq(\))2225 2660 y Fn(\()p Ff(n)p Fd(\000)p Ff(y)r Fn(\))2420 2697 y Fl(;)136 b(\031)29 b Fp(2)c Fq([0)p Fl(;)15 b Fq(1])1927 5956 y(4)p eop %%Page: 5 5 5 4 bop 0 399 a Fr(3)135 b(Lik)l(eliho)t(o)t(d)45 b(Inference)136 601 y Fp(\017)h Fq(Cen)m(tral)30 b(to)h(lik)m(eliho)s(o)s(d)d (inference)h(is)h(the)g(lik)m(eliho)s(o)s(d)d(function)i Fl(L)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))136 783 y Fp(\017)46 b Fq(Goal)31 b(is)e(to)j(\014nd)c(the)j(set)g(of)f (plausible)e(mo)s(del)h(parameters)i(giv)m(en)f(the)h(observ)m(ed)f (data)136 964 y Fp(\017)46 b Fq(the)29 b(lik)m(eliho)s(o)s(d)c (function)i Fj(do)-5 b(es)32 b(not)38 b Fq(giv)m(e)28 b(us)g(the)h(probabilit)m(y)c(of)k(parameter)g(v)-5 b(alues)27 b(or)i(sets)f(of)h(param-)227 1077 y(eter)i(v)-5 b(alues)30 b(conditional)f(on)h(the)h(observ)m(ed)f(data)330 1258 y Fo({)45 b Fq(W)-8 b(e'll)31 b(see)g(wh)m(y)e(this)h(is)f(the)i(case)g (when)f(w)m(e)g(talk)h(ab)s(out)f(Ba)m(y)m(esian)h(inference)136 1439 y Fp(\017)46 b Fq(the)d(lik)m(eliho)s(o)s(d)d(function)h(do)s(es)i (giv)m(e)g(us)f(information)f(ab)s(out)i(the)g Fj(r)-5 b(elative)50 b Fq(plausibilit)m(y)39 b(of)k(v)-5 b(arious)227 1552 y(parameter)31 b(v)-5 b(alues)30 b(conditional)f(on)h(the)g(same)h (observ)m(ed)g(data)136 1733 y Fp(\017)46 b Fq(if)36 b Fl(L)p Fq(\()p Fk(\022)468 1747 y Fn(1)507 1733 y Fp(j)p Fo(y)q Fq(\))h Fl(>)e(L)p Fq(\()p Fk(\022)917 1747 y Fn(2)957 1733 y Fp(j)p Fo(y)q Fq(\))j(w)m(e)e(can)h(sa)m(y)g(that)h Fk(\022)1844 1747 y Fn(1)1919 1733 y Fq(is)e(more)h(lik)m(ely)e(than)h Fk(\022)2769 1747 y Fn(2)2845 1733 y Fq(to)h(b)s(e)f(the)g(parameter)h
(v)-5 b(alue)227 1846 y(that)31 b(generated)h(the)e(observ)m(ed)h(data) 330 2028 y Fo({)45 b Fq(Note)32 b(w)m(ell)d(that)i(suc)m(h)f (comparisons)f(are)i(alw)m(a)m(ys)g(conditional)e(on)h(the)g(same)h (observ)m(ed)g(data)g(and)427 2141 y(the)g(same)g(family)d(of)j (sampling)d(densit)m(y)330 2280 y Fo({)45 b Fq(it)33 b(is)e(not)j(meaningful)c(to)k(compare)f(the)g Fl(L)p Fq(\()p Fk(\022)2038 2294 y Fn(1)2078 2280 y Fp(j)p Fo(y)q Fq(\))h(and)e Fl(L)p Fq(\()p Fk(\022)2558 2294 y Fn(2)2597 2280 y Fp(j)p Fo(y)2678 2247 y Fd(0)2703 2280 y Fq(\))h(where)f Fo(y)i Fq(and)f Fo(y)3361 2247 y Fd(0)3417 2280 y Fq(are)g(di\013eren)m (t)427 2393 y(datasets.)0 2634 y Fg(3.1)112 b(P)m(oin)m(t)36 b(Estimates)136 2805 y Fp(\017)46 b Fq(Often)30 b(w)m(e)h(are)g(in)m (terested)f(in)f(obtaining)g(a)i(p)s(oin)m(t)f(estimate)h(of)f(the)h (parameter)g(v)m(ector)h Fk(\022)136 2986 y Fp(\017)46 b Fq(Suc)m(h)30 b(a)h(p)s(oin)m(t)e(estimate)i(is)f(the)g Fm(Maximum)38 b(Likeliho)-5 b(o)g(d)37 b(Estimate)i Fq(or)30 b Fm(MLE)10 b Fq(.)136 3168 y Fp(\017)46 b Fq(The)30 b(MLE)g(of)h Fk(\022)i Fq(is)d(denoted)g(b)m(y)1400 3144 y(^)1395 3168 y Fk(\022)136 3349 y Fp(\017)232 3325 y Fq(^)227 3349 y Fk(\022)k Fq(is)29 b(the)i(v)-5 b(alue)29 b(of)i Fk(\022)i Fq(that)e(results)e(in)g Fl(L)p Fq(\()p Fk(\022)t Fp(j)p Fo(y)q Fq(\))i(ac)m(hieving)f(its)g(maxim)m(um)f(v)-5 b(alue.)136 3530 y Fp(\017)46 b Fq(Often)35 b(it)g(is)g(useful)e(to)j (w)m(ork)g(with)e(the)h(natural)g(logarithm)f(of)i(the)f(lik)m(eliho)s (o)s(d)e(function)h(whic)m(h)g(w)m(e'll)227 3643 y(denote)d Fl(`)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))h(and)e(call)f(the)i (log-lik)m(eliho)s(o)s(d)d(function)136 3824 y Fp(\017)46 b Fq(Since)28 b(the)g(MLE)h(maximizes)e(b)s(oth)h(the)g(lik)m(eliho)s (o)s(d)d(function)j(and)f(the)i(log-lik)m(eliho)s(o)s(d)d(function)h (this)g(is)227 3937 y(harmless)136 4119 y Fp(\017)46 b Fq(W)-8 b(orking)26 b(with)e(the)h(log-lik)m(eliho)s(o)s(d)d (function)i(is)h(useful)e(for)i(computational)g(reasons)h(and)e(is)g (also)i(useful)227 4231 y(for)k(calculating)g(appro)m(ximate)h (standard)e(errors)h(and)g(con\014dence)g(in)m(terv)-5 b(als)0 4472 y Fg(3.2)112 b(The)38 b(Sampling)e(Distribution)f(of)j (the)f(MLE)136 4643 y Fp(\017)46 b Fq(F)-8 b(reqen)m(tist)24 b(inference)e(is)g(concerned)h(with)f(the)h(p)s(erformance)f(of)h(an)g (estimator)g(o)m(v)m(er)h(rep)s(eated)f(sampling)136 4825 y Fp(\017)46 b Fq(In)28 b(other)h(w)m(ords,)g(if)f(w)m(e)h(w)m (ere)g(to)h(dra)m(w)e(a)h(large)g(n)m(um)m(b)s(er)e(of)i(random)f (samples)g(from)g(a)h(p)s(opulation)e(and)227 4938 y(calculate)k(the)g (MLE)f(for)g(eac)m(h)i(sample,)d(what)i(w)m(ould)e(the)h(distribution)d (of)k(these)f(MLEs)h(lo)s(ok)f(lik)m(e?)136 5119 y Fp(\017)46 b Fq(Often)30 b(not)h(p)s(ossible)d(to)j(calculate)g(this)e(sampling)f (distribution)f(of)k(the)f(MLE)h(analytically)136 5300 y Fp(\017)46 b Fq(Ho)m(w)m(ev)m(er,)33 b(as)e(the)f(sample)g(size)g (gets)h(large)g(w)m(e)g(can)f(rely)g(on)g(asymptotic)h(appro)m (ximations)136 5481 y Fp(\017)46 b Fq(F)-8 b(or)37 b(large)f(sample)f (sizes)g(the)h(sampling)e(distribution)e(of)k(the)h(MLE)e(is)g(appro)m (ximately)g(normal)g(with)227 5594 y(mean)43 b Fk(\022)j Fq(and)d(v)-5 b(ariance)43 b(equal)f(to)i(-1)g(times)e(the)h(in)m(v)m (erse)g(Hessian)g(of)g(the)g(log-lik)m(eliho)s(o)s(d)d(function)227 5707 y(ev)-5 b(aluated)31 b(at)749 5683 y(^)745 5707 y Fk(\022)s Fq(.)1927 5956 y(5)p eop %%Page: 6 6 6 5 bop 0 399 a Fg(3.3)112 b(Example)37 b(3:)50 b(Lik)m(eliho)s(o)s(d) 36 b(Inference)i(for)f(a)h(Binomial)d(Prop)s(ortion)136 570 y Fp(\017)46 b Fq(Consider)27 b(the)j(situation)e(in)f(whic)m(h)h (a)h(\(p)s(ossibly)e(biased\))h(coin)h(is)f(tossed)h(100)h(times)f(and)
f(43)i(heads)f(are)227 683 y(observ)m(ed.)41 b(Let)31 b Fl(n)f Fq(denote)h(the)f(sample)g(size)g(and)g(let)g Fl(y)j Fq(denote)e(the)g(n)m(um)m(b)s(er)e(of)h(heads)g(in)g Fl(n)f Fq(tosses)136 862 y Fp(\017)46 b Fq(W)-8 b(e)32 b(w)m(ould)d(lik)m(e)h(to)h(estimate)g(the)f(true)h(underlying)c (probabilit)m(y)h(of)i(the)h(coin)f(coming)g(up)f(heads)136 1041 y Fp(\017)46 b Fq(W)-8 b(e)32 b(assume)e(that)h(eac)m(h)g(toss)g (is)f(a)g(Bernoulli)e(trial)i(and)f(that)i(all)f(tosses)h(are)g(indep)s (enden)m(t)136 1221 y Fp(\017)46 b Fq(This)29 b(implies)e(that)k(the)g (sampling)d(densit)m(y)i(for)g(this)f(problem)g(is)h(a)g(binomial)e (sampling)g(densit)m(y)1500 1445 y Fl(p)p Fq(\()p Fl(y)s Fp(j)p Fl(\031)s Fq(\))e(=)1866 1316 y Fe(\022)1933 1383 y Fl(n)1936 1507 y(y)1987 1316 y Fe(\023)2054 1445 y Fl(\031)2109 1407 y Ff(y)2151 1445 y Fq(\(1)21 b Fp(\000)f Fl(\031)s Fq(\))2433 1407 y Fn(\()p Ff(n)p Fd(\000)p Ff(y)r Fn(\))136 1707 y Fp(\017)46 b Fq(Remem)m(b)s(ering)32 b(that)i(the)f(lik)m(eliho)s(o)s(d)d(function)h(is)h(an)m(y)h(function) f(prop)s(ortional)f(to)j Fl(p)p Fq(\()p Fl(y)s Fp(j)p Fl(\031)s Fq(\))f(considered)227 1819 y(as)e(a)g(function)e(of)h Fl(\031)k Fq(for)c(\014xed)f Fl(y)s Fq(,)i(w)m(e)g(can)f(write)1586 1998 y Fl(L)p Fq(\()p Fl(\031)s Fp(j)p Fl(y)s Fq(\))c(=)f Fl(\031)2023 1961 y Ff(y)2065 1998 y Fq(\(1)c Fp(\000)f Fl(\031)s Fq(\))2347 1961 y Fn(\()p Ff(n)p Fd(\000)p Ff(y)r Fn(\))136 2211 y Fp(\017)46 b Fq(the)31 b(log-lik)m(eliho)s(o)s (d)c(function)j(is)1368 2390 y Fl(`)p Fq(\()p Fl(\031)s Fp(j)p Fl(y)s Fq(\))c(=)f Fl(y)18 b Fq(ln)o(\()p Fl(\031)s Fq(\))j(+)f(\()p Fl(n)g Fp(\000)g Fl(y)s Fq(\))15 b(ln)o(\(1)21 b Fp(\000)f Fl(\031)s Fq(\))136 2602 y Fp(\017)46 b Fq(after)38 b(a)f(bit)f(a)i(calculus)d(w)m(e)j(can)f(see)h(that)f(the)h(the)f(v)-5 b(alue)36 b(of)i Fl(\031)i Fq(that)d(maximizes)f(the)i(log-lik)m(eliho) s(o)s(d)227 2715 y(function)29 b(is)35 b(^)-50 b Fl(\031)28 b Fq(=)864 2674 y Ff(y)p 861 2694 43 4 v 861 2746 a(n)939 2715 y Fq(=)1063 2679 y Fn(43)p 1045 2694 106 4 v 1045 2746 a(100)1186 2715 y Fq(=)d(0)p Fl(:)p Fq(43)136 2894 y Fp(\017)46 b Fq(to)30 b(\014nd)d(the)i(asymptotic)g(v)-5 b(ariance)29 b(of)g(the)g(MLE)f(w)m(e)i(need)e(to)i(calculate)f(the)g (second)g(deriv)-5 b(ativ)m(e)28 b(of)h(the)227 3007 y(log-lik)m(eliho)s(o)s(d)36 b(function)h(with)g(resp)s(ect)h(to)h Fl(\031)s Fq(,)i(ev)-5 b(aluate)39 b(this)e(at)i(the)g(MLE,)f(tak)m(e)i (the)f(recipro)s(cal)e(of)227 3120 y(this,)30 b(and)g(then)g(m)m (ultiply)d(this)j(b)m(y)g(-1.)330 3320 y Fo({)45 b Fq(the)31 b(second)f(deriv)-5 b(ativ)m(e)30 b(of)h(the)f(log-lik)m(eliho)s(o)s(d) e(function)h(with)g(resp)s(ect)h(to)i Fl(\031)h Fq(is)1644 3541 y Fl(`)1682 3504 y Fd(00)1725 3541 y Fq(\()p Fl(\031)s Fp(j)p Fl(y)s Fq(\))26 b(=)f Fp(\000)2149 3480 y Fl(y)p 2126 3520 95 4 v 2126 3604 a(\031)2181 3577 y Fn(2)2250 3541 y Fp(\000)2370 3480 y Fq(\()p Fl(n)20 b Fp(\000)g Fl(y)s Fq(\))p 2351 3520 322 4 v 2351 3604 a(\(1)h Fp(\000)f Fl(\031)s Fq(\))2633 3577 y Fn(2)330 3776 y Fo({)45 b Fq(the)31 b(in)m(v)m(erse)f(of)h(this)e(times)h(-1)h(ev)-5 b(aluated)30 b(at)i(the)e(MLE)g(is)1667 4040 y(\()p Fp(\000)p Fl(`)1811 4002 y Fd(00)1853 4040 y Fq(\()p Fl(\031)s Fp(j)p Fl(y)s Fq(\)\))2086 4002 y Fd(\000)p Fn(1)2207 4040 y Fq(=)2318 3978 y(^)-50 b Fl(\031)s Fq(\(1)21 b Fp(\000)k Fq(^)-50 b Fl(\031)s Fq(\))p 2313 4019 338 4 v 2454 4102 a Fl(n)427 4215 y Fq(whic)m(h)29 b(is)h(the)g (large-sample)g(v)-5 b(ariance)31 b(of)k(^)-50 b Fl(\031)330 4352 y Fo({)45 b Fq(to)31 b(get)h(the)e(standard)g(error)g(of)35 b(^)-49 b Fl(\031)33 b Fq(w)m(e)e(tak)m(e)h(the)e(square)g(ro)s(ot)1229
4641 y Fl(S)5 b(E)g Fq(\()g(^)-50 b Fl(\031)t Fq(\))26 b(=)1610 4493 y Fe(r)p 1701 4493 358 4 v 1716 4579 a Fq(^)-50 b Fl(\031)s Fq(\(1)21 b Fp(\000)j Fq(^)-50 b Fl(\031)t Fq(\))p 1711 4620 338 4 v 1852 4703 a Fl(n)2083 4641 y Fq(=)2179 4498 y Fe(r)p 2270 4498 455 4 v 2280 4579 a Fq(0)p Fl(:)p Fq(43)22 b Fp(\002)e Fq(0)p Fl(:)p Fq(57)p 2280 4620 435 4 v 2429 4703 a(100)2750 4641 y Fp(\031)k Fq(0)p Fl(:)p Fq(0495)330 4843 y Fo({)45 b Fq(to)27 b(calculate)g(a)f(\(1)12 b Fp(\000)g Fl(\013)p Fq(\))g Fp(\002)g Fq(100\045)26 b(con\014dence)g(in)m(terv)-5 b(al)26 b(for)f Fl(\031)k Fq(w)m(e)e(subtract)f(and)f(add)h Fl(z)3427 4861 y Ff(\013=)p Fn(2)3559 4843 y Fp(\002)12 b Fl(S)5 b(E)g Fq(\()g(^)-50 b Fl(\031)s Fq(\))427 4956 y(from)35 b(^)-50 b Fl(\031)s Fq(,)31 b(where)f Fl(z)1058 4974 y Ff(\013=)p Fn(2)1208 4956 y Fq(is)g(the)g(\(1)21 b Fp(\000)f Fl(\013=)p Fq(2\))32 b(quan)m(tile)e(of)g(the)h(standard)e (normal)h(distribution)330 5093 y Fo({)45 b Fq(in)29 b(this)h(case)h(the)g(95\045)g(con\014dence)f(in)m(terv)-5 b(al)29 b(for)i Fl(\031)i Fq(is)1366 5398 y(\()5 b(^)-50 b Fl(\031)23 b Fp(\000)d Fl(z)1609 5412 y Ff(:)p Fn(975)1759 5398 y Fp(\002)g Fl(S)5 b(E)g Fq(\()g(^)-50 b Fl(\031)t Fq(\))p Fl(;)81 b Fq(^)-50 b Fl(\031)24 b Fq(+)c Fl(z)2419 5412 y Ff(:)p Fn(975)2569 5398 y Fp(\002)g Fl(S)5 b(E)g Fq(\()g(^)-50 b Fl(\031)s Fq(\)\))26 b(=)1277 5535 y(\(0)p Fl(:)p Fq(43)c Fp(\000)e Fq(1)p Fl(:)p Fq(96)i Fp(\002)e Fl(:)p Fq(0495)p Fl(;)78 b Fq(0)p Fl(:)p Fq(43)22 b(+)e(1)p Fl(:)p Fq(96)i Fp(\002)e Fl(:)p Fq(0495\))27 b(=)2464 5673 y(\(0)p Fl(:)p Fq(333)p Fl(;)79 b Fq(0)p Fl(:)p Fq(527\))1927 5956 y(6)p eop %%Page: 7 7 7 6 bop 330 399 a Fo({)45 b Fq(It)29 b(is)g(not)g(correct)h(to)g(in)m (terpret)f(this)f(con\014dence)h(in)m(terv)-5 b(al)28 b(to)i(mean)f(that)h(there)f(is)f(a)h(95\045)h(proba-)427 511 y(bilit)m(y)f(that)i(the)f(true)g(parameter)h(v)-5 b(alue)30 b Fl(\031)j Fq(lies)d(in)f(this)g(in)m(terv)-5 b(al)330 658 y Fo({)45 b Fq(the)31 b(probabilit)m(y)d(of)i(this)g(is)f (either)h(0)h(or)f(1)h(but)e(unkno)m(wn)330 804 y Fo({)45 b Fq(Instead,)j(w)m(e)c(exp)s(ect)h(that)g(if)e(a)h(large)g(n)m(um)m(b) s(er)f(of)i(samples)e(of)h(size)g Fl(n)f Fq(are)i(dra)m(wn)e(from)h (the)427 917 y(p)s(opulation)35 b(and)h(MLEs)g(and)h(con\014dence)f(in) m(terv)-5 b(als)36 b(are)h(constructed)g(then,)i(on)d(a)m(v)m(erage,)42 b(o)m(v)m(er)427 1029 y(these)36 b(n)m(umerous)f(samples)f(w)m(e)i(w)m (ould)e(exp)s(ect)i(that)g(the)g(prop)s(ortion)d(of)j(times)f(that)h Fl(\031)i Fq(will)33 b(fall)427 1142 y(within)28 b(the)j(random,)f (sample-dep)s(enden)m(t)e(con\014dence)j(in)m(terv)-5 b(als)29 b(will)f(b)s(e)i(ab)s(out)g(1)20 b Fp(\000)g Fl(\013)p Fq(.)1927 5956 y(7)p eop %%Page: 8 8 8 7 bop 0 399 a Fr(4)135 b(Ba)l(y)l(esian)46 b(Inference)136 601 y Fp(\017)g Fq(The)29 b(goal)h(of)f(Ba)m(y)m(esian)i(inference)d (is)g(to)i(mak)m(e)g(probabilit)m(y)d(statemen)m(ts)k(ab)s(out)e(mo)s (del)f(parameters)i Fk(\022)s Fq(,)227 714 y(or)h(functions)e(of)h(mo)s (del)f(parameters)i Fl(g)s Fq(\()p Fk(\022)t Fq(\),)g(giv)m(en)f(a)h (probabilit)m(y)d(mo)s(del)h(and)h(observ)m(ed)g(data)3577 681 y Fn(2)3618 714 y Fq(.)136 902 y Fp(\017)46 b Fq(In)29 b(other)g(w)m(ords,)g(w)m(e)h(w)m(ould)d(lik)m(e)i(to)h(kno)m(w)f(the)g (conditional)e(densit)m(y)i(of)g Fk(\022)j Fq(giv)m(en)d(our)f(observ)m (ed)i(v)-5 b(alues)227 1015 y(of)31 b Fo(y)q Fq(.)136 1203 y Fp(\017)46 b Fq(W)-8 b(e)32 b(can)f(denote)f(this)g(conditional)
e(densit)m(y)i(as)h Fl(p)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\).)136 1390 y Fp(\017)46 b Fq(Ho)m(w)m(ev)m(er,)39 b(note)d(that)g(our)f(probabilit)m(y)e(mo)s(del)i(as)g(de\014ned)f(b)m (y)i(the)f(sampling)f(densit)m(y)g(treats)j Fo(y)g Fq(as)f(a)227 1503 y(function)29 b(of)i(\014xed)f(v)-5 b(alues)29 b(of)i Fk(\022)i Fq(whic)m(h)c(is)h(not)g(quite)g(what)g(w)m(e)h(w)m(an)m(t.) 136 1691 y Fp(\017)46 b Fq(Ho)m(w)31 b(do)g(w)m(e)f(get)i(from)e Fl(p)p Fq(\()p Fo(y)q Fp(j)p Fk(\022)t Fq(\))g(to)i Fl(p)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))f(?)136 1878 y Fp(\017)46 b Fq(Recall)25 b(from)g(elemen)m(tary)h(probabilit)m(y)d (that)j(a)f(conditional)f(probabilit)m(y)f(is)h(simply)f(the)j(ratio)f (of)g(a)h(join)m(t)227 1991 y(probabilit)m(y)i(to)j(a)g(marginal)e (probabilit)m(y)-8 b(.)39 b(This)28 b(giv)m(es)j(us:)1733 2279 y Fl(p)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))c(=)2117 2218 y Fl(p)p Fq(\()p Fk(\022)s Fl(;)15 b Fo(y)q Fq(\))p 2117 2259 268 4 v 2164 2342 a Fl(p)p Fq(\()p Fo(y)q Fq(\))3784 2279 y(\(1\))136 2527 y Fp(\017)46 b Fq(Realizing)30 b(that)i(a)f(join)m(t)g(densit)m(y)f(can)i(b)s(e)e(written)g(as)h(the)h (pro)s(duct)d(of)j(a)f(conditional)e(and)i(a)g(marginal)227 2640 y(densit)m(y)-8 b(,)23 b(and)c(that)i(a)g(marginal)e(densit)m(y)g (can)i(b)s(e)f(formed)g(from)f(a)i(join)m(t)f(densit)m(y)g(b)m(y)g(in)m (tegrating)g(v)-5 b(ariables)227 2753 y(out)21 b(of)g(the)g(join)m(t)g (densit)m(y)f(equation)g(1)h(can)g(b)s(e)f(reexpressed)h(in)e(terms)i (of)g(our)f(original)f(probabilit)m(y)f(mo)s(del)227 2866 y(as:)1534 3246 y Fl(p)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))26 b(=)1917 3184 y Fl(p)p Fq(\()p Fo(y)q Fp(j)p Fk(\022)t Fq(\))p Fl(p)p Fq(\()p Fk(\022)t Fq(\))p 1917 3225 423 4 v 2042 3308 a Fl(p)p Fq(\()p Fo(y)q Fq(\))1811 3494 y(=)2026 3432 y Fl(p)p Fq(\()p Fo(y)q Fp(j)p Fk(\022)t Fq(\))p Fl(p)p Fq(\()p Fk(\022)t Fq(\))p 1917 3473 642 4 v 1917 3484 a Fe(R)1960 3589 y Fn(\002)2034 3557 y Fl(p)p Fq(\()p Fo(y)q Fp(j)p Fk(\022)t Fq(\))p Fl(p)p Fq(\()p Fk(\022)t Fq(\))p Fl(d)p Fk(\022)2568 3494 y Fl(:)3784 3376 y Fq(\(2\))136 3794 y Fp(\017)46 b Fl(p)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))32 b(is)d(kno)m(wn)h(as)h(the)f Fm(p)-5 b(osterior)37 b(density)k Fq(of)30 b Fk(\022)s Fq(.)136 3982 y Fp(\017)46 b Fq(Equation)26 b(2)i(is)e(kno)m(wn)g(as)h Fm(Bayes)35 b(rule)g Fq(and)26 b(is)g(the)h(cornerstone)g(of)g(Ba)m(y)m (esian)h(inference.)39 b(It)27 b(pro)m(vides)227 4094 y(an)34 b(explicit)e(form)m(ula)g(that)j(one)e(can)h(use)g(to)g(mak)m (e)g(probabilit)m(y)d(statemen)m(ts)36 b(ab)s(out)d(unkno)m(wns)f(giv)m (en)227 4207 y(observ)m(ed)f(data.)330 4420 y Fo({)45 b Fq(Note)37 b(that)f(kno)m(wledge)f(of)g(the)h(marginal)e (distribution)d(of)k Fk(\022)k Fq(is)34 b(required)f(to)j(mak)m(e)g (probabilit)m(y)427 4533 y(statemen)m(ts)c(ab)s(out)e Fk(\022)k Fq(giv)m(en)c(the)h(data)136 4745 y Fp(\017)46 b Fq(Since)28 b Fl(p)p Fq(\()p Fo(y)q Fq(\))i(do)s(es)f(not)g(dep)s (end)e(on)i Fk(\022)j Fq(and)c(can)i(b)s(e)e(regarded)h(as)g(a)g (constan)m(t)h(for)f(\014xed)f(v)-5 b(alues)29 b(of)g Fo(y)h Fq(it)f(is)227 4858 y(often)i(con)m(v)m(enien)m(t)h(to)f (reexpress)f(equation)g(2)h(as:)1653 5121 y Fl(p)p Fq(\()p Fk(\022)s Fp(j)p Fo(y)q Fq(\))26 b Fp(/)f Fl(p)p Fq(\()p Fo(y)q Fp(j)p Fk(\022)t Fq(\))p Fl(p)p Fq(\()p Fk(\022)t Fq(\))p Fl(:)1310 b Fq(\(3\))136 5326 y Fp(\017)46 b Fq(In)26 b(w)m(ords,)h(the)f(p)s(osterior)f(densit)m(y)h(of)g Fk(\022)j Fq(giv)m(en)d Fo(y)i Fq(is)e(prop)s(ortional)e(to)j(the)f (sampling)e(densit)m(y)i(of)g Fo(y)i Fq(giv)m(en)227 5439 y Fk(\022)34 b Fq(times)29 b(the)i(prior)e(densit)m(y)g(of)i Fk(\022)s Fq(.)p 0 5518 1560 4 v 104 5571 a Fi(2)138
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forall
98.67 177.87 l 99.12 177.87 l 99.57 177.87 l 100.02 177.87 l 100.47 177.87 l 100.93 177.87 l 101.38 177.87 l 101.83 177.87 l 102.28 177.87 l 102.73 177.87 l 103.18 177.87 l 103.63 177.87 l 104.08 177.87 l 104.53 177.87 l 104.98 177.87 l 105.43 177.87 l 105.88 177.87 l 106.33 177.87 l 106.78 177.87 l 107.23 177.87 l 107.69 177.87 l 108.14 177.87 l 108.59 177.87 l 109.04 177.87 l 109.49 177.87 l 109.94 177.87 l 110.39 177.87 l 110.84 177.87 l 111.29 177.87 l 111.74 177.87 l 112.19 177.87 l 112.64 177.87 l 113.09 177.87 l 113.54 177.87 l 113.99 177.87 l 114.45 177.87 l 114.90 177.87 l 115.35 177.87 l 115.80 177.87 l 116.25 177.87 l 116.70 177.87 l 117.15 177.87 l 117.60 177.87 l 118.05 177.87 l 118.50 177.87 l 118.95 177.87 l 119.40 177.87 l 119.85 177.87 l 120.30 177.87 l 120.75 177.87 l 121.21 177.87 l 121.66 177.87 l 122.11 177.87 l 122.56 177.87 l 123.01 177.87 l 123.46 177.87 l 123.91 177.87 l 124.36 177.87 l 124.81 177.87 l
125.26 125.71 126.16 126.61 127.06 127.51 127.97 128.42 128.87 129.32 129.77 130.22 130.67 131.12 131.57 132.02 132.47 132.92 133.37 133.82 134.27 134.73 135.18 135.63 136.08 136.53 136.98 137.43 137.88 138.33 138.78 139.23 139.68 140.13 140.58 141.03 141.49 141.94 142.39 142.84 143.29 143.74 144.19 144.64 145.09 145.54 145.99 146.44 146.89 147.34 147.79 148.25 148.70 149.15 149.60 150.05 150.50 150.95 151.40
177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
151.85 152.30 152.75 153.20 153.65 154.10 154.55 155.01 155.46 155.91 156.36 156.81 157.26 157.71 158.16 158.61 159.06 159.51 159.96 160.41 160.86 161.31 161.77 162.22 162.67 163.12 163.57 164.02 164.47 164.92 165.37 165.82 166.27 166.72 167.17 167.62 168.07 168.53 168.98 169.43 169.88 170.33 170.78 171.23 171.68 172.13 172.58 173.03 173.48 173.93 174.38 174.83 175.29 175.74 176.19 176.64 177.09 177.54 177.99
177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
178.44 178.89 179.34 179.79 180.24 180.69 181.14 181.59 182.05 182.50 182.95 183.40 183.85 184.30 184.75 185.20 185.65 186.10 186.55 187.00 187.45 187.90 188.35 188.81 189.26 189.71 190.16 190.61 191.06 191.51 191.96 192.41 192.86 193.31 193.76 194.21 194.66 195.11 195.57 196.02 196.47 196.92 197.37 197.82 198.27 198.72 199.17 199.62 200.07 200.52 200.97 201.42 201.87 202.33 202.78 203.23 203.68 204.13 204.58
177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
205.03 205.48 205.93 206.38 206.83 207.28 207.73 208.18 208.63 209.09 209.54 209.99 210.44 210.89 211.34 211.79 212.24 212.69 213.14 213.59 214.04 214.49 214.94 215.39 215.85 216.30 216.75 217.20 217.65 218.10 218.55 219.00 219.45 219.90 220.35 220.80 221.25 221.70 222.15 222.61 223.06 223.51 223.96 224.41 224.86 225.31 225.76 226.21 226.66 227.11 227.56 228.01 228.46 228.91 229.37 229.82 230.27 230.72 231.17
177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.88 177.88 177.88 177.88 177.88 177.88 177.88 177.88 177.89 177.89 177.89 177.89 177.90 177.90 177.90 177.91 177.91 177.92 177.92 177.93 177.93 177.94 177.95 177.95 177.96 177.97
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
231.62 232.07 232.52 232.97 233.42 233.87 234.32 234.77 235.22 235.67 236.13 236.58 237.03 237.48 237.93 238.38 238.83 239.28 239.73 240.18 240.63 241.08 241.53 241.98 242.43 242.89 243.34 243.79 244.24 244.69 245.14 245.59 246.04 246.49 246.94 247.39 247.84 248.29 248.74 249.19 249.65 250.10 250.55 251.00 251.45 251.90 252.35 252.80 253.25 253.70 254.15 254.60 255.05 255.50 255.95 256.41 256.86 257.31 257.76
177.98 177.99 178.01 178.02 178.03 178.05 178.06 178.08 178.10 178.12 178.15 178.17 178.20 178.23 178.26 178.29 178.33 178.37 178.41 178.46 178.50 178.56 178.61 178.67 178.74 178.81 178.88 178.96 179.05 179.14 179.24 179.34 179.45 179.57 179.70 179.83 179.98 180.13 180.29 180.47 180.65 180.85 181.06 181.28 181.51 181.76 182.02 182.30 182.60 182.91 183.24 183.59 183.95 184.34 184.75 185.18 185.64 186.12 186.62
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
258.21 258.66 259.11 259.56 260.01 260.46 260.91 261.36 261.81 262.26 262.71 263.17 263.62 264.07 264.52 264.97 265.42 265.87 266.32 266.77 267.22 267.67 268.12 268.57 269.02 269.47 269.93 270.38 270.83 271.28 271.73 272.18 272.63 273.08 273.53 273.98 274.43 274.88 275.33 275.78 276.23 276.69 277.14 277.59 278.04 278.49 278.94 279.39 279.84 280.29 280.74 281.19 281.64 282.09 282.54 282.99 283.45 283.90 284.35
187.15 187.71 188.29 188.91 189.55 190.23 190.94 191.69 192.47 193.28 194.14 195.03 195.97 196.94 197.96 199.02 200.13 201.29 202.49 203.75 205.05 206.41 207.82 209.29 210.81 212.40 214.04 215.74 217.51 219.33 221.23 223.19 225.21 227.31 229.47 231.70 234.01 236.39 238.84 241.37 243.97 246.65 249.41 252.24 255.15 258.15 261.22 264.37 267.60 270.91 274.30 277.78 281.33 284.97 288.68 292.48 296.35 300.30 304.34
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
284.80 285.25 285.70 286.15 286.60 287.05 287.50 287.95 288.40 288.85 289.30 289.75 290.21 290.66 291.11 291.56 292.01 292.46 292.91 293.36 293.81 294.26 294.71 295.16 295.61 296.06 296.51 296.97 297.42 297.87 298.32 298.77 299.22 299.67 300.12 300.57 301.02 301.47 301.92 302.37 302.82 303.27 303.73 304.18 304.63 305.08 305.53 305.98 306.43 306.88 307.33 307.78 308.23 308.68 309.13 309.58 310.03 310.49 310.94
308.44 312.63 316.89 321.23 325.63 330.11 334.66 339.28 343.97 348.72 353.53 358.40 363.34 368.32 373.36 378.46 383.60 388.78 394.01 399.27 404.57 409.91 415.27 420.65 426.06 431.49 436.92 442.37 447.82 453.28 458.73 464.17 469.60 475.01 480.41 485.77 491.11 496.41 501.67 506.89 512.05 517.17 522.22 527.21 532.13 536.98 541.74 546.43 551.03 555.53 559.94 564.25 568.45 572.54 576.51 580.37 584.10 587.71 591.18
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
311.39 311.84 312.29 312.74 313.19 313.64 314.09 314.54 314.99 315.44 315.89 316.34 316.79 317.25 317.70 318.15 318.60 319.05 319.50 319.95 320.40 320.85 321.30 321.75 322.20 322.65 323.10 323.55 324.01 324.46 324.91 325.36 325.81 326.26 326.71 327.16 327.61 328.06 328.51 328.96 329.41 329.86 330.31 330.77 331.22 331.67 332.12 332.57 333.02 333.47 333.92 334.37 334.82 335.27 335.72 336.17 336.62 337.07 337.53
594.52 597.73 600.79 603.70 606.47 609.08 611.54 613.85 615.99 617.97 619.78 621.43 622.91 624.23 625.36 626.33 627.12 627.74 628.18 628.45 628.53 628.45 628.18 627.74 627.12 626.33 625.36 624.23 622.91 621.43 619.78 617.97 615.99 613.85 611.54 609.08 606.47 603.70 600.79 597.73 594.52 591.18 587.71 584.10 580.37 576.51 572.54 568.45 564.25 559.94 555.53 551.03 546.43 541.74 536.98 532.13 527.21 522.22 517.17
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
337.98 338.43 338.88 339.33 339.78 340.23 340.68 341.13 341.58 342.03 342.48 342.93 343.38 343.83 344.29 344.74 345.19 345.64 346.09 346.54 346.99 347.44 347.89 348.34 348.79 349.24 349.69 350.14 350.59 351.05 351.50 351.95 352.40 352.85 353.30 353.75 354.20 354.65 355.10 355.55 356.00 356.45 356.90 357.35 357.81 358.26 358.71 359.16 359.61 360.06 360.51 360.96 361.41 361.86 362.31 362.76 363.21 363.66 364.11
512.05 506.89 501.67 496.41 491.11 485.77 480.41 475.01 469.60 464.17 458.73 453.28 447.82 442.37 436.92 431.49 426.06 420.65 415.27 409.91 404.57 399.27 394.01 388.78 383.60 378.46 373.36 368.32 363.34 358.40 353.53 348.72 343.97 339.28 334.66 330.11 325.63 321.23 316.89 312.63 308.44 304.34 300.30 296.35 292.48 288.68 284.97 281.33 277.78 274.30 270.91 267.60 264.37 261.22 258.15 255.15 252.24 249.41 246.65
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
364.57 365.02 365.47 365.92 366.37 366.82 367.27 367.72 368.17 368.62 369.07 369.52 369.97 370.42 370.87 371.33 371.78 372.23 372.68 373.13 373.58 374.03 374.48 374.93 375.38 375.83 376.28 376.73 377.18 377.63 378.09 378.54 378.99 379.44 379.89 380.34 380.79 381.24 381.69 382.14 382.59 383.04 383.49 383.94 384.39 384.85 385.30 385.75 386.20 386.65 387.10 387.55 388.00 388.45 388.90 389.35 389.80 390.25 390.70
243.97 241.37 238.84 236.39 234.01 231.70 229.47 227.31 225.21 223.19 221.23 219.33 217.51 215.74 214.04 212.40 210.81 209.29 207.82 206.41 205.05 203.75 202.49 201.29 200.13 199.02 197.96 196.94 195.97 195.03 194.14 193.28 192.47 191.69 190.94 190.23 189.55 188.91 188.29 187.71 187.15 186.62 186.12 185.64 185.18 184.75 184.34 183.95 183.59 183.24 182.91 182.60 182.30 182.02 181.76 181.51 181.28 181.06 180.85
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
391.15 391.61 392.06 392.51 392.96 393.41 393.86 394.31 394.76 395.21 395.66 396.11 396.56 397.01 397.46 397.91 398.37 398.82 399.27 399.72 400.17 400.62 401.07 401.52 401.97 402.42 402.87 403.32 403.77 404.22 404.67 405.13 405.58 406.03 406.48 406.93 407.38 407.83 408.28 408.73 409.18 409.63 410.08 410.53 410.98 411.43 411.89 412.34 412.79 413.24 413.69 414.14 414.59 415.04 415.49 415.94 416.39 416.84 417.29
180.65 180.47 180.29 180.13 179.98 179.83 179.70 179.57 179.45 179.34 179.24 179.14 179.05 178.96 178.88 178.81 178.74 178.67 178.61 178.56 178.50 178.46 178.41 178.37 178.33 178.29 178.26 178.23 178.20 178.17 178.15 178.12 178.10 178.08 178.06 178.05 178.03 178.02 178.01 177.99 177.98 177.97 177.96 177.95 177.95 177.94 177.93 177.93 177.92 177.92 177.91 177.91 177.90 177.90 177.90 177.89 177.89 177.89 177.89
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
417.74 418.19 418.65 419.10 419.55 420.00 420.45 420.90 421.35 421.80 422.25 422.70 423.15 423.60 424.05 424.50 424.95 425.41 425.86 426.31 426.76 427.21 427.66 428.11 428.56 429.01 429.46 429.91 430.36 430.81 431.26 431.71 432.17 432.62 433.07 433.52 433.97 434.42 434.87 435.32 435.77 436.22 436.67 437.12 437.57 438.02 438.47 438.93 439.38 439.83 440.28 440.73 441.18 441.63 442.08 442.53 442.98 443.43 443.88
177.88 177.88 177.88 177.88 177.88 177.88 177.88 177.88 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
444.33 444.78 445.23 445.69 446.14 446.59 447.04 447.49 447.94 448.39 448.84 449.29 449.74 450.19 450.64 451.09 451.54 451.99 452.45 452.90 453.35 453.80 454.25 454.70 455.15 455.60 456.05 456.50 456.95 457.40 457.85 458.30 458.75 459.21 459.66 460.11 460.56 461.01 461.46 461.91 462.36 462.81 463.26 463.71 464.16 464.61 465.06 465.51 465.97 466.42 466.87 467.32 467.77 468.22 468.67 469.12 469.57 470.02 470.47
177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
470.92 471.37 471.82 472.27 472.73 473.18 473.63 474.08 474.53 474.98 475.43 475.88 476.33 476.78 477.23 477.68 478.13 478.58 479.03 479.49 479.94 480.39 480.84 481.29 481.74 482.19 482.64 483.09 483.54 483.99 484.44 484.89 485.34 485.79 486.25 486.70 487.15 487.60 488.05 488.50 488.95 489.40 489.85 490.30 490.75 491.20 491.65 492.10 492.55 493.01 493.46 493.91 494.36 494.81 495.26 495.71 496.16 496.61 497.06
177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87 177.87
l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
497.51 497.96 498.41 498.86 499.31 499.77 500.22 500.67 501.12 501.57 502.02 502.47 502.92 503.37 503.82 504.27 504.72 505.17 505.62 506.07 506.53 506.98 507.43 507.88 508.33 508.78 509.23 509.68 510.13 510.58 511.03 511.48 511.93 512.38 512.83 513.29 513.74 514.19 514.64 515.09 ...