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(ucleotide)g(p)r(olymorphisms)g(\(SNPs\))h(comprise)f(the)523 1053 y(largest)g(class)h(of)h(individual)g(di\013erences)f(in)h(DNA)h (and)e(ha)n(v)n(e)g(b)r(ecome)g(a)h(fo)r(cus)f(of)h(researc)n(h)523 1153 y(in)n(terest|millions)k(of)g(SNPs)h(ha)n(v)n(e)e(th)n(us)i(far)f (b)r(een)g(detected)h(out)g(of)f(an)h(estimated)f(total)523 1253 y(of)k(ten)g(million)f(common)g(SNPs)h([20)o(,)g(23)o(].)648 1354 y(The)c(list)h(of)g(alleles)f(in)h(con)n(tiguous)f(sites)h(in)g(a) f(lo)r(cal)g(region)g(of)g(a)h(single)f(c)n(hromosome)523 1453 y(is)h(called)f(a)h Fo(haplotyp)l(e)p Fx(.)i(F)-7 b(or)24 b(diploid)h(organisms,)e(t)n(w)n(o)h(haplot)n(yp)r(es)g(go)g (together)g(to)h(mak)n(e)523 1553 y(up)d(a)f Fo(genotyp)l(e)p Fx(,)h(whic)n(h)f(is)h(the)f(list)h(of)f(unordered)g(pairs)f(of)h (alleles)g(in)h(the)g(region.)e(That)h(is,)523 1652 y(a)31 b(genot)n(yp)r(e)g(is)g(obtained)g(from)g(a)g(pair)g(of)g(haplot)n(yp)r (es)g(b)n(y)g(omitting)h(the)f(sp)r(eci\014cation)523 1752 y(of)k(the)h(asso)r(ciation)e(of)h(eac)n(h)f(allele)h(with)h(one)f (of)g(the)h(t)n(w)n(o)e(c)n(hromosomes|its)f Fo(phase)p Fx(.)523 1852 y(Common)19 b(t)n(yping)g(metho)r(ds)g(yield)g(the)h (genot)n(yp)r(es)e(of)h(a)f(set)i(of)f(individuals,)g(and)g(t)n (ypically)523 1951 y(do)38 b(not)h(pro)n(vide)e(phase)h(information;)g (the)h(latter)f(information)g(can)g(b)r(e)h(obtained)f(at)523 2051 y(a)j(considerably)f(higher)h(cost)g([18)o(].)h(It)f(is)h (therefore)e(desirable)h(to)g(dev)n(elop)g(metho)r(ds)523 2151 y(for)34 b(inferring)f(haplot)n(yp)r(es)h(from)g(genot)n(yp)r(es)f (and)h(p)r(ossibly)g(other)g(data)f(sources)g(\(e.g.,)523 2250 y(p)r(edigrees\).)648 2351 y(F)-7 b(rom)32 b(the)i(p)r(oin)n(t)f (of)g(view)g(of)g(p)r(opulation)g(genetics,)g(the)h(basic)e(mo)r(del)i (underlying)523 2451 y(the)h(haplot)n(yp)r(e)g(inference)g(problem)f (is)h(a)f(\014nite)i(mixture)f(mo)r(del.)g(That)g(is,)g(letting)g Fn(H)523 2551 y Fx(denote)26 b(the)h(set)g(of)f(all)g(p)r(ossible)g (haplot)n(yp)r(es)g(asso)r(ciated)f(with)i(a)f(giv)n(en)g(region)f(\(a) i(set)f(of)523 2650 y(cardinalit)n(y)32 b(2)986 2620 y Fw(k)1060 2650 y Fx(in)i(the)f(case)g(of)g(binary)g(p)r (olymorphisms,)g(where)f Fm(k)37 b Fx(is)c(the)h(n)n(um)n(b)r(er)f(of) 523 2750 y(heterozygous)26 b(sites\),)h(the)h(probabilit)n(y)f(of)g(a)h (genot)n(yp)r(e)e(is)i(giv)n(en)f(b)n(y:)1262 2952 y Fm(p)p Fx(\()p Fm(g)s Fx(\))c(=)1593 2873 y Fl(X)1521 3052 y Fw(h)1560 3060 y Fk(1)1592 3052 y Fw(;h)1651 3060 y Fk(2)1683 3052 y Fj(2H)1799 2952 y Fm(p)p Fx(\()p Fm(h)1921 2964 y Fv(1)1958 2952 y Fm(;)14 b(h)2043 2964 y Fv(2)2080 2952 y Fx(\)1\()p Fm(h)2234 2964 y Fv(1)2290 2952 y Fn(\010)k Fm(h)2421 2964 y Fv(2)2481 2952 y Fx(=)k Fm(g)s Fx(\))p Fm(;)633 b Fx(\(1\))523 3228 y(where)20 b(1\()p Fm(h)878 3240 y Fv(1)918 3228 y Fn(\010)s Fm(h)1034 3240 y Fv(2)1094 3228 y Fx(=)j Fm(g)s Fx(\))d(is)g(the)g(indicator)f(function)i(of)f (the)h(ev)n(en)n(t)e(that)i(haplot)n(yp)r(es)e Fm(h)3214 3240 y Fv(1)3271 3228 y Fx(and)523 3327 y Fm(h)571 3339 y Fv(2)636 3327 y Fx(are)26 b(consisten)n(t)h(with)h Fm(g)s Fx(.)f(Under)g(the)h(assumption)f(of)g(Hardy-W)-7 b(ein)n(b)r(erg)26 b(equilibrium)523 3427 y(\(HWE\),)j(an)e(assumption) g(that)h(is)g(standard)f(in)h(the)g(literature)f(and)g(will)h(also)f(b) r(e)h(made)523 3527 y(here,)f(the)h(mixing)g(prop)r(ortion)e Fm(p)p Fx(\()p Fm(h)1679 3539 y Fv(1)1716 3527 y Fm(;)14 b(h)1801 3539 y Fv(2)1838 3527 y Fx(\))28 b(is)f(assumed)g(to)h(factor) f(as)g Fm(p)p Fx(\()p Fm(h)2878 3539 y Fv(1)2915 3527 y Fx(\))p Fm(p)p Fx(\()p Fm(h)3069 3539 y Fv(2)3106 3527 y Fx(\).)648 3628 y(Giv)n(en)35 b(this)i(basic)e(statistical)h (structure,)f(the)h(simplest)h(metho)r(dology)d(for)i(haplo-)523 3727 y(t)n(yp)r(e)25 b(inference)h(is)f(maxim)n(um)g(lik)n(eliho)r(o)r (d)g(via)f(the)i(EM)f(algorithm,)f(treating)g(the)i(haplo-)523 3827 y(t)n(yp)r(e)31 b(iden)n(tities)f(as)g(laten)n(t)g(v)-5 b(ariables)29 b(and)i(estimating)f(the)h(parameters)d Fm(p)p Fx(\()p Fm(h)p Fx(\))j([7)o(].)g(This)523 3927 y(metho)r(dology)19 b(has)h(rather)g(sev)n(ere)f(computational)h (requiremen)n(ts,)f(in)i(that)g(a)f(probabilit)n(y)523 4026 y(distribution)32 b(m)n(ust)h(b)r(e)f(main)n(tained)g(on)g(the)h (\(large\))e(set)i(of)f(p)r(ossible)g(haplot)n(yp)r(es,)f(but)523 4126 y(ev)n(en)c(more)g(fundamen)n(tally)g(it)h(fails)g(to)f(capture)g (the)h(notion)f(that)h(small)f(sets)g(of)h(haplo-)523 4226 y(t)n(yp)r(es)k(should)h(b)r(e)f(preferred.)g(This)g(notion)g (deriv)n(es)f(from)i(an)f(underlying)g(assumption)523 4325 y(that)27 b(for)f(relativ)n(ely)g(short)g(regions)f(there)i(is)f (limited)i(div)n(ersit)n(y)d(in)i(a)g(p)r(opulation)f(due)h(to)523 4425 y(p)r(opulation)g(b)r(ottlenec)n(ks)h(and)f(relativ)n(ely)f(lo)n (w)h(rates)g(of)h(recom)n(bination)e(and)h(m)n(utation.)648 4526 y(One)36 b(approac)n(h)e(to)j(dealing)f(with)h(this)g(issue)f(is)g (to)h(form)n(ulate)f(a)g(notion)g(of)h(\\par-)523 4625 y(simon)n(y)-7 b(,")36 b(and)g(to)h(dev)n(elop)f(algorithms)f(that)i (directly)g(attempt)g(to)g(maximize)f(parsi-)523 4725 y(mon)n(y)-7 b(.)24 b(Sev)n(eral)f(imp)r(ortan)n(t)h(pap)r(ers)g(ha)n (v)n(e)f(tak)n(en)h(this)g(approac)n(h)f([3)o(,)i(12)o(,)f(6,)g(22)o(]) h(and)f(ha)n(v)n(e)523 4825 y(yielded)i(new)g(insigh)n(ts)g(and)g (practical)f(algorithms.)f(Another)i(approac)n(h)e(is)i(to)g(elab)r (orate)523 4924 y(the)e(probabilistic)f(mo)r(del,)i(in)f(particular)e (b)n(y)i(incorp)r(orating)e(priors)h(on)g(the)i(parameters.)p eop %%Page: 3 3 3 2 bop 3367 232 a Ft(3)523 448 y Fx(Di\013eren)n(t)34 b(priors)e(ha)n(v)n(e)g(b)r(een)i(discussed)f(b)n(y)g(di\013eren)n(t)h (authors,)e(ranging)g(from)h(simple)523 548 y(Diric)n(hlet)27 b(priors)f([17)o(])h(to)g(priors)f(based)g(on)h(the)g(coalescen)n(t)f (pro)r(cess)g([21)o(])h(to)g(priors)f(that)523 648 y(capture)h(asp)r (ects)g(of)g(recom)n(bination)f([11)o(].)i(These)f(mo)r(dels)h(pro)n (vide)e(implicit)i(notions)f(of)523 747 y(parsimon)n(y)-7 b(,)26 b(via)h(the)h(implicit)h(\\Oc)n(kham)d(factor")g(of)h(the)h(Ba)n (y)n(esian)e(formalism)h([2)o(].)648 862 y(Both)36 b(parsimon)n (y-based)e(and)j(statistical)f(approac)n(hes)e(are)i(useful)h(in)g(the) g(case)f(of)523 962 y(ph)n(ylogenetic)25 b(inference)h([8],)g(and)g(w)n (e)f(feel)i(that)f(it)g(is)g(lik)n(ely)g(that)g(b)r(oth)g(will)h(con)n (tin)n(ue)e(to)523 1061 y(pla)n(y)f(a)g(role)g(in)g(haplot)n(yp)r(e)g (inference)h(as)f(w)n(ell.)g(The)h(approac)n(h)d(that)j(w)n(e)f(tak)n (e)g(in)h(the)g(cur-)523 1161 y(ren)n(t)d(pap)r(er)f(is)h(statistical,) g(but)g(w)n(e)g(attempt)h(to)e(pro)n(vide)g(more)g(explicit)i(con)n (trol)d(o)n(v)n(er)h(the)523 1261 y(n)n(um)n(b)r(er)h(of)f(inferred)h (haplot)n(yp)r(es)f(than)h(has)f(b)r(een)h(pro)n(vided)f(b)n(y)g(the)h (statistical)g(metho)r(ds)523 1360 y(prop)r(osed)h(th)n(us)i(far,)f (and)g(the)h(resulting)f(inference)g(algorithm)f(has)h(commonalities)f (with)523 1460 y(the)28 b(parsimon)n(y-based)d(sc)n(hemes.)648 1575 y(Our)f(approac)n(h)g(is)h(based)g(on)g(a)g(nonparametric)e(prior) i(kno)n(wn)f(as)h(the)h Fo(Dirichlet)j(pr)l(o-)523 1675 y(c)l(ess)k Fx([9,)27 b(1)o(].)g(In)f(the)h(setting)f(of)h(\014nite)g (mixture)f(mo)r(dels,)h(the)f(Diric)n(hlet)h(pro)r(cess|not)e(to)523 1774 y(b)r(e)d(confused)f(with)g(the)h(Diric)n(hlet)f(distribution|is)g (able)g(to)g(capture)g(uncertain)n(t)n(y)f(ab)r(out)523 1874 y(the)k(n)n(um)n(b)r(er)g(of)g(mixture)g(comp)r(onen)n(ts)g ([e.g.,)f(5].)h(The)g(basic)g(setup)g(can)g(b)r(e)g(explained)g(in)523 1974 y(terms)30 b(of)g(an)g(urn)h(mo)r(del,)f(and)g(a)g(pro)r(cess)f (that)i(pro)r(ceeds)e(through)h(data)g(sequen)n(tially)-7 b(.)523 2073 y(Consider)32 b(an)h(urn)f(whic)n(h)h(at)g(the)g(outset)g (con)n(tains)f(a)h(ball)f(of)h(a)g(single)f(color.)g(A)n(t)h(eac)n(h) 523 2173 y(step)i(w)n(e)g(either)g(dra)n(w)f(a)h(ball)g(from)g(the)h (urn,)f(and)g(replace)f(it)i(with)f(t)n(w)n(o)g(balls)f(of)i(the)523 2272 y(same)24 b(color,)g(or)g(w)n(e)g(are)g(giv)n(en)g(a)h(ball)g(of)f (a)h(new)g(color)e(whic)n(h)i(w)n(e)g(place)f(in)h(the)h(urn,)e(with) 523 2372 y(a)i(parameter)e(de\014ning)i(the)g(probabilities)g(of)g (these)g(t)n(w)n(o)f(p)r(ossibilities.)h(The)g(asso)r(ciation)523 2472 y(of)38 b(data)f(p)r(oin)n(ts)g(to)h(colors)e(de\014nes)h(a)g (\\clustering")f(of)i(the)g(data.)f(T)-7 b(o)37 b(mak)n(e)g(the)h(link) 523 2571 y(with)e(Ba)n(y)n(esian)d(mixture)j(mo)r(dels,)f(w)n(e)g(asso) r(ciate)f(with)i(eac)n(h)f(color)f(a)h(dra)n(w)f(from)i(the)523 2671 y(distribution)28 b(de\014ning)f(the)h(parameters)e(of)i(the)g (mixture)f(comp)r(onen)n(ts.)648 2786 y(This)f(pro)r(cess)g(de\014nes)g (a)g Fo(prior)k(distribution)k Fx(for)26 b(a)g(mixture)h(mo)r(del)g (with)g(a)f(random)523 2886 y(n)n(um)n(b)r(er)38 b(of)f(comp)r(onen)n (ts.)h(Multiplying)g(this)g(prior)f(b)n(y)g(a)g(lik)n(eliho)r(o)r(d)h (yields)f(a)h Fo(p)l(oste-)523 2985 y(rior)29 b(distribution)p Fx(.)e(Mark)n(o)n(v)d(c)n(hain)i(Mon)n(te)g(Carlo)e(algorithms)h(ha)n (v)n(e)g(b)r(een)h(dev)n(elop)r(ed)g(to)523 3085 y(sample)37 b(from)f(the)i(p)r(osterior)d(distributions)i(asso)r(ciated)f(with)i (Diric)n(hlet)f(pro)r(cess)f(pri-)523 3184 y(ors)26 b([5,)i(16)o(].)648 3300 y(The)21 b(usefulness)f(of)h(this)g(framew)n(ork)e(for)i(the)g (haplot)n(yp)r(e)g(problem)f(should)h(b)r(e)g(clear|)523 3399 y(using)j(a)f(Diric)n(hlet)h(pro)r(cess)f(prior)f(w)n(e)i(in)g (essence)f(main)n(tain)g(a)h(p)r(o)r(ol)f(of)h(haplot)n(yp)r(e)g (candi-)523 3499 y(dates)j(that)h(gro)n(ws)d(as)h(observ)n(ed)g(genot)n (yp)r(es)g(are)h(pro)r(cessed.)f(The)h(gro)n(wth)f(is)h(con)n(trolled) 523 3598 y(via)d(a)f(parameter)g(in)h(the)g(prior)f(distribution)h (that)h(corresp)r(onds)d(to)i(the)g(c)n(hoice)f(of)h(a)g(new)523 3698 y(color)k(in)i(the)g(urn)f(mo)r(del,)h(and)f(via)g(the)h(lik)n (eliho)r(o)r(d,)f(whic)n(h)g(assesses)f(the)i(matc)n(h)f(of)h(the)523 3798 y(new)e(genot)n(yp)r(e)f(to)g(the)h(a)n(v)-5 b(ailable)26 b(haplot)n(yp)r(es.)648 3913 y(T)-7 b(o)29 b(expand)h(on)f(this)h (latter)g(p)r(oin)n(t,)g(an)f(adv)-5 b(an)n(tage)29 b(of)h(the)g (probabilistic)f(formalism)523 4012 y(is)37 b(its)g(abilit)n(y)g(to)g (elab)r(orate)f(the)h(observ)-5 b(ation)36 b(mo)r(del)h(for)g(the)h (genot)n(yp)r(es)e(to)h(include)523 4112 y(the)30 b(p)r(ossibilit)n(y)g (of)g(errors.)d(In)j(particular,)f(the)h(indicator)f(function)h(1\()p Fm(h)2886 4124 y Fv(1)2943 4112 y Fn(\010)20 b Fm(h)3076 4124 y Fv(2)3140 4112 y Fx(=)26 b Fm(g)s Fx(\))k(in)523 4212 y(Eq.)24 b(\(1\))h(is)f(susp)r(ect|there)g(are)g(man)n(y)g (reasons)e(wh)n(y)j(an)f(individual)g(genot)n(yp)r(e)g(ma)n(y)g(not)523 4311 y(matc)n(h)k(with)h(a)f(curren)n(t)g(p)r(o)r(ol)g(of)g(haplot)n (yp)r(es,)g(suc)n(h)g(as)g(the)g(p)r(ossibilit)n(y)g(of)h(m)n(utation)f (or)523 4411 y(recom)n(bination)23 b(in)h(the)g(meiosis)g(for)f(that)i (individual,)f(and)g(errors)d(in)k(the)f(genot)n(yping)f(or)523 4510 y(data)28 b(recording)e(pro)r(cess.)i(Suc)n(h)g(sources)f(of)h (small)g(di\013erences)g(should)g(not)g(lead)g(to)h(the)523 4610 y(inference)f(pro)r(cedure)e(spa)n(wning)h(new)g(haplot)n(yp)r (es.)648 4725 y(In)32 b(the)g(curren)n(t)g(pap)r(er)f(w)n(e)h(presen)n (t)f(a)h(statistical)g(mo)r(del)g(for)g(haplot)n(yp)r(e)f(inference)523 4825 y(based)i(on)g(a)g(Diric)n(hlet)g(pro)r(cess)f(prior)g(and)h(a)g (lik)n(eliho)r(o)r(d)g(that)g(includes)h(error)d(mo)r(dels)523 4924 y(for)22 b(genot)n(yp)r(es.)g(W)-7 b(e)23 b(describ)r(e)f(a)g (Mark)n(o)n(v)f(c)n(hain)h(Mon)n(te)g(Carlo)g(pro)r(cedure,)f(in)i (particular)p eop %%Page: 4 4 4 3 bop 523 232 a Ft(4)523 448 y Fx(a)28 b(pro)r(cedure)f(that)h(mak)n (es)f(use)h(of)f(b)r(oth)i(Gibbs)f(and)g(Metrop)r(olis-Hasting)e(up)r (dates,)j(for)523 548 y(p)r(osterior)23 b(inference.)h(W)-7 b(e)24 b(presen)n(t)g(results)f(of)h(applying)g(our)f(metho)r(d)i(to)f (the)g(analysis)f(of)523 648 y(b)r(oth)c(sim)n(ulated)g(and)g(real)e (genot)n(yp)r(e)h(data,)h(comparing)e(to)i(the)g(state-of-the-art)e (PHASE)523 747 y(algorithm)29 b([21)o(].)i(On)f(the)h(sim)n(ulated)f (data)g(our)g(predictions)g(are)f(comparable)g(to)h(those)523 847 y(obtained)g(b)n(y)g(PHASE.)g(On)g(a)g(real)f(dataset)g(of)i([4)o (])f(our)g(results)f(are)h(again)f(comparable)523 946 y(to)g(those)f(of)h(PHASE,)g(and)g(w)n(e)f(outp)r(erform)g(t)n(w)n(o)h (other)f(algorithms:)f(HAP)i([13,)g(6)o(])g(and)523 1046 y(HAPLOTYPER)35 b([17)o(].)i(On)f(data)g(from)g([10)o(],)h(whic)n(h)f (is)g(a)g(di\016cult)h(test)g(case)f(due)g(to)523 1146 y(the)k(small)e(n)n(um)n(b)r(er)h(of)g(individuals)g(in)g(the)h (sample,)f(w)n(e)f(outp)r(erform)h(PHASE)g(b)n(y)f(a)523 1245 y(signi\014can)n(t)27 b(margin.)523 1516 y Fp(2)112 b(The)38 b(Statistical)d(Mo)s(del)523 1720 y Fx(The)20 b(input)i(to)e(a)g(phasing)g(algorithm)f(can)h(b)r(e)h(represen)n(ted)e (as)h(a)g Fo(genotyp)l(e)j(matrix)e Fm(G)f Fx(with)523 1819 y(columns)29 b(corresp)r(onding)f(to)h(SNPs)h(in)g(their)f(order)f (along)h(the)h(c)n(hromosome)d(and)i(ro)n(ws)523 1919 y(corresp)r(onding)f(to)h(genot)n(yp)r(ed)g(individuals.)g Fm(G)2075 1931 y Fw(i;j)2183 1919 y Fx(represen)n(ts)f(the)i (information)e(on)i(the)523 2018 y(t)n(w)n(o)22 b(alleles)f(of)i(the)g Fm(i)p Fx(-th)f(individual)g(in)h(SNP)g Fm(j)5 b Fx(.)22 b(W)-7 b(e)23 b(denote)f(the)h(t)n(w)n(o)f(alleles)f(of)i(a)f(SNP)g(b)n (y)523 2118 y(0)h(and)g(1,)g(and)g Fm(G)1055 2130 y Fw(i;j)1157 2118 y Fx(can)g(tak)n(e)g(on)g(one)g(of)g(four)g(v)-5 b(alues:)23 b(0)g(or)g(1,)g(indicating)g(a)g(homozygous)523 2218 y(site;)d(2,)f(indicating)g(a)g(heterozygous)f(site;)i(and)f(')9 b(?',)20 b(indicating)f(missing)h(data.)f(\(Although)523 2317 y(w)n(e)35 b(fo)r(cus)g(on)g(binary)f(data)h(here,)g(it)g(is)g(w)n (orth)g(noting)f(that)i(our)e(metho)r(ds)i(generalize)523 2417 y(immediately)28 b(to)f(non-binary)g(data,)g(and)g(accommo)r(date) g(missing)g(data\).)648 2517 y(W)-7 b(e)21 b(will)h(describ)r(e)f(our)g (mo)r(del)g(in)h(terms)f(of)h(a)f(p)r(o)r(ol)g(of)g(ancestral)f(haplot) n(yp)r(es,)h(or)g Fo(tem-)523 2617 y(plates)p Fx(,)29 b(from)e(whic)n(h)h(eac)n(h)f(p)r(opulation)h(haplot)n(yp)r(e)f (originates)f([cf.)j(11)o(].)f(The)f(haplot)n(yp)r(e)523 2717 y(itself)i(ma)n(y)f(undergo)f(p)r(oin)n(t)i(m)n(utation)f(with)h (resp)r(ect)f(to)g(its)h(template.)g(The)f(size)g(of)h(the)523 2816 y(p)r(o)r(ol)22 b(and)g(its)g(comp)r(osition)g(are)f(b)r(oth)h (unkno)n(wn,)g(and)g(are)f(treated)h(as)f(random)g(v)-5 b(ariables)523 2916 y(under)25 b(a)g(Diric)n(hlet)h(pro)r(cess)e (prior.)g(W)-7 b(e)26 b(b)r(egin)g(b)n(y)f(pro)n(viding)f(a)h(brief)g (description)g(of)h(the)523 3016 y(Diric)n(hlet)32 b(pro)r(cess)e(and)i (subsequen)n(tly)f(sho)n(w)g(ho)n(w)g(this)h(pro)r(cess)f(can)g(b)r(e)i (incorp)r(orated)523 3115 y(in)n(to)27 b(a)h(mo)r(del)f(for)g(haplot)n (yp)r(e)h(inference.)523 3369 y Fi(2.1)95 b(The)32 b(Diric)m(hlet)f (Pro)s(cess)523 3540 y Fx(Rather)39 b(than)g(presen)n(t)g(the)g(Diric)n (hlet)h(pro)r(cess)e(in)h(full)h(generalit)n(y)-7 b(,)38 b(w)n(e)g(fo)r(cus)i(on)f(the)523 3639 y(sp)r(eci\014c)32 b(setting)f(of)h(a)f(mixture)g(mo)r(del,)h(and)f(mak)n(e)g(use)g(of)g (the)h(urn)g(mo)r(del)f(to)h(presen)n(t)523 3739 y(the)27 b(essen)n(tial)f(features)g(of)h(the)g(pro)r(cess.)f(F)-7 b(or)26 b(a)g(fuller)h(presen)n(tation,)f(see,)g([e.g.,)h(14)o(].)g(W) -7 b(e)523 3839 y(assume)24 b(that)h(data)f Fm(X)32 b Fx(arise)23 b(from)i(a)f(mixture)h(distribution)g(with)g(mixture)g (comp)r(onen)n(ts)523 3938 y Fm(p)p Fx(\()p Fm(x)p Fn(j)p Fm(\036)p Fx(\).)36 b(W)-7 b(e)35 b(assume)f(the)h(existence)f(of)h(a)f Fo(b)l(ase)j(me)l(asur)l(e)k Fm(G)p Fx(\()p Fm(\036)p Fx(\),)36 b(whic)n(h)e(is)h(one)f(of)h(the)523 4038 y(t)n(w)n(o)c (parameters)g(of)g(the)i(Diric)n(hlet)f(pro)r(cess.)f(\(The)h(other)g (is)f(the)i(parameter)d Fm(\034)9 b Fx(,)33 b(whic)n(h)523 4137 y(w)n(e)h(presen)n(t)g(b)r(elo)n(w\).)g(The)h(parameter)d Fm(G)p Fx(\()p Fm(\036)p Fx(\))k(is)e(not)h(the)g(prior)e(for)h Fm(\036)p Fx(,)h(but)g(is)f(used)g(to)523 4237 y(generate)26 b(a)i(prior)e(for)h Fm(\036)p Fx(,)h(in)g(the)g(manner)f(that)h(w)n(e)f (no)n(w)g(discuss.)648 4338 y(Consider)i(the)i(follo)n(wing)e(pro)r (cess)g(for)h(generating)f(samples)g Fn(f)p Fm(x)2728 4350 y Fv(1)2766 4338 y Fm(;)14 b(x)2850 4350 y Fv(2)2887 4338 y Fm(;)g(:)g(:)g(:)g(;)g(x)3119 4350 y Fw(n)3164 4338 y Fn(g)30 b Fx(from)523 4437 y(a)25 b(mixture)g(mo)r(del)g (consisting)f(of)h(an)f(unsp)r(eci\014ed)i(n)n(um)n(b)r(er)f(of)f (mixture)h(comp)r(onen)n(ts,)g(or)523 4537 y Fo(e)l(quivalenc)l(e)31 b(classes)p Fx(:)581 4724 y({)41 b(The)34 b(\014rst)f(sample)g Fm(x)1348 4736 y Fv(1)1420 4724 y Fx(is)g(sampled)g(from)h(a)f (distribution)g Fm(p)p Fx(\()p Fm(x)p Fn(j)p Fm(\036)2767 4736 y Fv(1)2806 4724 y Fx(\),)h(where)f(the)h(pa-)664 4824 y(rameter)27 b Fm(\036)1023 4836 y Fv(1)1088 4824 y Fx(is)h(sampled)f(from)g(the)h(base)f(measure)g Fm(G)p Fx(\()p Fm(\036)p Fx(\).)581 4924 y({)41 b(The)28 b Fm(i)p Fx(th)g(sample,)f Fm(x)1317 4936 y Fw(i)1345 4924 y Fx(,)h(is)f (sampled)g(from)h(the)g(distribution)f Fm(p)p Fx(\()p Fm(x)p Fn(j)p Fm(\036)2787 4936 y Fw(c)2817 4944 y Fh(i)2849 4924 y Fx(\),)h(where:)p eop %%Page: 5 5 5 4 bop 3367 232 a Ft(5)722 448 y Fn(\017)41 b Fx(The)27 b(equiv)-5 b(alence)26 b(class)g(of)g(sample)h Fm(i)p Fx(,)f Fm(c)2090 460 y Fw(i)2118 448 y Fx(,)g(is)h(dra)n(wn)f(from)g (the)h(follo)n(wing)e(distri-)805 548 y(bution:)1208 730 y Fm(p)p Fx(\()p Fm(c)1318 742 y Fw(i)1369 730 y Fx(=)d Fm(c)1492 742 y Fw(j)1555 730 y Fx(for)27 b(some)g Fm(j)h(<)22 b(i)p Fn(j)p Fm(c)2127 742 y Fv(1)2164 730 y Fm(;)14 b(:)g(:)g(:)g(;)g(c)2385 742 y Fw(i)p Fj(\000)p Fv(1)2497 730 y Fx(\))24 b(=)2753 669 y Fm(n)2803 681 y Fw(c)2833 689 y Fh(j)p 2651 711 319 4 v 2651 787 a Fm(i)18 b Fn(\000)g Fx(1)g(+)g Fm(\034)3299 730 y Fx(\(2\))1301 919 y Fm(p)p Fx(\()p Fm(c)1411 931 y Fw(i)1461 919 y Fn(6)p Fx(=)23 b Fm(c)1585 931 y Fw(j)1647 919 y Fx(for)28 b(all)f Fm(j)h(<)22 b(i)p Fn(j)p Fm(c)2127 931 y Fv(1)2164 919 y Fm(;)14 b(:)g(:)g(:)g(;)g(c)2385 931 y Fw(i)p Fj(\000)p Fv(1)2497 919 y Fx(\))24 b(=)2787 863 y Fm(\034)p 2651 900 V 2651 976 a(i)18 b Fn(\000)g Fx(1)g(+)g Fm(\034)2979 919 y(;)297 b Fx(\(3\))805 1132 y(where)30 b Fm(n)1098 1144 y Fw(c)1128 1152 y Fh(i)1189 1132 y Fx(is)g(the)h Fo(o)l(c)l(cup)l(ancy)j(numb)l(er)c Fx(of)g(class)g Fm(c)2443 1144 y Fw(i)2470 1132 y Fx(|the)h(n)n(um)n(b)r(er)g(of)f(previous)805 1231 y(samples)d(that)h(b)r(elong)f(to)h(class)e Fm(c)1892 1243 y Fw(i)1920 1231 y Fx(.)722 1331 y Fn(\017)41 b Fx(The)22 b(parameter)f Fm(\036)1411 1343 y Fw(c)1441 1351 y Fh(i)1493 1331 y Fx(asso)r(ciated)g(with)h(the)g(mixture)g(comp) r(onen)n(t)g Fm(c)2964 1343 y Fw(i)3014 1331 y Fx(is)f(obtained)805 1431 y(as)27 b(follo)n(ws:)805 1613 y Fm(\036)854 1625 y Fw(c)884 1633 y Fh(i)938 1613 y Fx(=)c Fm(\036)1075 1625 y Fw(c)1105 1633 y Fh(j)1223 1613 y Fx(if)28 b Fm(c)1335 1625 y Fw(i)1386 1613 y Fx(=)23 b Fm(c)1510 1625 y Fw(j)1572 1613 y Fx(for)k(some)g Fm(j)h(<)23 b(i)k Fx(\(i.e.,)h Fm(c)2338 1625 y Fw(i)2394 1613 y Fx(is)f(a)g(p)r(opulated)h(equiv)-5 b(alence)27 b(class\))805 1737 y Fm(\036)854 1749 y Fw(c)884 1757 y Fh(i)938 1737 y Fn(\030)c Fm(G)p Fx(\()p Fm(\036)p Fx(\))84 b(if)28 b Fm(c)1400 1749 y Fw(i)1451 1737 y Fn(6)p Fx(=)23 b Fm(c)1575 1749 y Fw(j)1637 1737 y Fx(for)k(all)h Fm(j)g(<)22 b(i)28 b Fx(\(i.e.,)g Fm(c)2311 1749 y Fw(i)2366 1737 y Fx(is)f(a)h(new)f(equiv)-5 b(alence)27 b(class\))p Fm(:)648 1920 y Fx(Eqs.)38 b(\(2\))h(and)f(\(3\))h(de\014ne)g(a)g (conditional)f(prior)f(for)i(the)g(equiv)-5 b(alence)38 b(class)g(indi-)523 2019 y(cator)h Fm(c)784 2031 y Fw(i)851 2019 y Fx(of)h(eac)n(h)f(sample)h(during)f(a)g(sequen)n(tial)h (sampling)f(pro)r(cess.)g(They)g(imply)h(a)523 2119 y(self-reinforcing) 22 b(prop)r(ert)n(y)g(for)h(the)h(c)n(hoice)f(of)g(equiv)-5 b(alence)23 b(class)g(of)g(eac)n(h)g(new)g(sample|)523 2218 y(previously)j(p)r(opulated)i(classes)e(are)h(more)g(lik)n(ely)g (to)g(b)r(e)h(c)n(hosen.)648 2318 y(It)36 b(is)f(imp)r(ortan)n(t)h(to)f (emphasize)g(that)h(the)g(pro)r(cess)f(that)h(w)n(e)f(ha)n(v)n(e)g (discussed)g(will)523 2418 y(b)r(e)e(used)g(as)f(a)h Fo(prior)j(distribution)p Fx(.)d(W)-7 b(e)34 b(no)n(w)e(em)n(b)r(ed)h (this)g(prior)f(in)h(a)f(full)i(mo)r(del)f(that)523 2517 y(includes)c(a)g(lik)n(eliho)r(o)r(d)g(for)g(the)h(observ)n(ed)e(data.) g(In)i(App)r(endix)g(A)g(w)n(e)f(dev)n(elop)f(Mark)n(o)n(v)523 2617 y(c)n(hain)f(Mon)n(te)g(Carlo)g(\(MCMC\))h(inference)f(pro)r (cedures)g(for)g(this)h(mo)r(del.)523 2866 y Fi(2.2)95 b(The)32 b(Mo)s(del)523 3031 y Fx(W)-7 b(e)21 b(presen)n(t)f(a)g (probabilistic)f(mo)r(del)i(for)f(the)g(generation)f(of)i(haplot)n(yp)r (es)e(in)i(a)f(p)r(opulation)523 3131 y(and)39 b(for)h(the)g (generation)e(of)h(genot)n(yp)r(es)g(from)g(these)h(haplot)n(yp)r(es.)f (W)-7 b(e)40 b(assume)f(that)523 3231 y(eac)n(h)e(individual's)h(genot) n(yp)r(e)g(is)g(formed)g(b)n(y)f(dra)n(wing)g(t)n(w)n(o)h(random)f Fo(templates)h Fx(from)523 3330 y(an)24 b(ancestral)e(p)r(o)r(ol,)i (and)g(that)g(these)g(templates)g(are)f(sub)5 b(ject)24 b(to)g(random)e(p)r(erturbation.)523 3430 y(The)h(mo)r(del)g(is)g (displa)n(y)n(ed)f(as)g(a)h(graphical)e(mo)r(del)i(\(also)f(kno)n(wn)g (as)h(a)f(Ba)n(y)n(esian)f(net)n(w)n(ork\))523 3530 y(in)28 b(Figure)f(1.)648 3629 y(Let)i Fm(J)38 b Fx(b)r(e)30 b(an)f(ordered)f(list)i(of)f(lo)r(ci)g(of)h(in)n(terest.)f(F)-7 b(or)29 b(eac)n(h)f(individual)i Fm(i)p Fx(,)f(w)n(e)g(denote)523 3729 y(his/her)e(paternal)g(haplot)n(yp)r(e)h(b)n(y)f Fm(H)1708 3741 y Fw(i)1731 3749 y Fk(0)1792 3729 y Fx(:=)c([)p Fm(H)1995 3741 y Fw(i)2018 3749 y Fk(0)2051 3741 y Fw(;)p Fv(1)2108 3729 y Fm(;)14 b(:)g(:)g(:)g(;)g(H)2362 3741 y Fw(i)2385 3749 y Fk(0)2417 3741 y Fw(;J)2484 3729 y Fx(])28 b(and)f(maternal)g(haplot)n(yp)r(e)523 3828 y(b)n(y)38 b Fm(H)718 3840 y Fw(i)741 3848 y Fk(1)818 3828 y Fx(:=)i([)p Fm(H)1038 3840 y Fw(i)1061 3848 y Fk(1)1094 3840 y Fw(;)p Fv(1)1151 3828 y Fm(;)14 b(:)g(:)g(:)f(;)h(H)1404 3840 y Fw(i)1427 3848 y Fk(1)1460 3840 y Fw(;J)1526 3828 y Fx(].)38 b(W)-7 b(e)38 b(denote)g(a)g(set)g(of)f(ancestral)g(templates) h(as)f Fi(A)j Fx(=)523 3928 y Fn(f)p Fm(A)627 3940 y Fv(1)664 3928 y Fm(;)14 b(A)763 3940 y Fv(2)800 3928 y Fm(;)g(:)g(:)g(:)p Fn(g)p Fx(,)34 b(where)h Fm(A)1343 3940 y Fw(k)1418 3928 y Fx(:=)g([)p Fm(A)1626 3940 y Fw(k)q(;)p Fv(1)1720 3928 y Fm(;)14 b(:)g(:)g(:)f(;)h(A)1966 3940 y Fw(k)q(;J)2069 3928 y Fx(])35 b(is)g(a)f(particular)f(mem)n(b)r (er)h(of)h(this)g(set.)523 4028 y(The)23 b(set)h Fi(A)f Fx(is)h(a)f(random)f(v)-5 b(ariable)22 b(whose)h(cardinalit)n(y)f(and)h (comp)r(osition)g(are)g(not)g(\014xed,)523 4127 y(but)41 b(rather)d(v)-5 b(ary)39 b(with)i(realizations)d(of)i(the)g(Diric)n (hlet)g(pro)r(cess)f(and)g(v)-5 b(ary)39 b(with)i(the)523 4227 y(observ)n(ed)26 b(data.)648 4327 y(In)f(our)f(framew)n(ork,)f (the)i(probabilit)n(y)f(distribution)h(of)f(the)i(haplot)n(yp)r(e)e(v) -5 b(ariable)24 b Fm(H)3327 4339 y Fw(i)3350 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-1 sc (...) col0 sh gr /Times-Italic ff 240.00 scf sf 966 1854 m gs 1 -1 sc (G) col0 sh gr /Times-Italic ff 150.00 scf sf 1104 1899 m gs 1 -1 sc (i,1) col0 sh gr /Symbol ff 240.00 scf sf 1035 0 m gs 1 -1 sc 350.0 rot (q) col0 sh gr /Times-Italic ff 150.00 scf sf 1170 45 m gs 1 -1 sc (k) col0 sh gr /Times-Italic ff 240.00 scf sf 405 -855 m gs 1 -1 sc (A) col0 sh gr /Times-Italic ff 150.00 scf sf 540 -810 m gs 1 -1 sc (k) col0 sh gr /Times-Italic ff 240.00 scf sf -855 -428 m gs 1 -1 sc 350.0 rot (G) col0 sh gr /Times-Italic ff 135.00 scf sf -675 -383 m gs 1 -1 sc (0) col0 sh gr /Times-Italic ff 240.00 scf sf 0 135 m gs 1 -1 sc (K) col0 sh gr F2psBegin10setmiterlimit0.062990.06299scF2psEnd rs %%EndDocument @endspecial 523 1793 a Fr(Fig.)15 b(1.)i Ft(The)h(graphical)h(mo)r (del)f(represen)n(tation)g(of)h(the)e(haplot)n(yp)r(e)g(mo)r(del)h (with)g(a)g(Diric)n(hlet)g(pro-)523 1884 y(cess)j(prior.)g(Circles)h (represen)n(t)e(the)g(state)h(v)l(ariables,)g(o)n(v)l(als)g(represen)n (t)f(the)g(parameter)g(v)l(ariables,)523 1975 y(and)30 b(diamonds)f(represen)n(t)h(\014xed)e(parameters.)j(The)f(dashed)f(b)r (o)n(xes)h(denote)g(sets)g(of)h(v)l(ariables)523 2066 y(corresp)r(onding)e(to)f(the)g(same)f(ancestral)j(template,)d(haplot)n (yp)r(e,)h(and)g(genot)n(yp)r(e,)g(resp)r(ectiv)n(ely)-6 b(.)523 2158 y(The)33 b(solid)g(b)r(o)n(xes)g(corresp)r(onds)g(to)g (i.i.d.)h(replicates)f(of)h(a)e(set)h(of)h(v)l(ariables,)f(eac)n(h)g (asso)r(ciated)523 2249 y(with)j(a)g(particular)g(individual)g(\()p Fg(I)41 b Ft(copies\),)c(or)f(ancestral)h(template)e(\(an)g(un)n(b)r (ounded)f(n)n(um-)523 2340 y(b)r(er)f(of)h(copies\),)h(resp)r(ectiv)n (ely)-6 b(.)33 b(Arro)n(ws)h(b)r(et)n(w)n(een)g(v)l(ariables)g(or)g(b)r (o)n(xes)f(denote)g(dep)r(endencies)523 2432 y(b)r(et)n(w)n(een)26 b(v)l(ariables)g(or)g(sets)g(of)h(v)l(ariables.)523 2624 y Fx(to)32 b(the)h(equiv)-5 b(alence)32 b(classes)e(of)j(the)f(Diric)n (hlet)h(pro)r(cess.)e(The)h(base)g(measure)f Fm(G)h Fx(of)h(the)523 2723 y(Diric)n(hlet)38 b(pro)r(cess)f(is)h(a)f(join)n(t)h(measure)f(on) h(ancestral)f(haplot)n(yp)r(es)g Fm(A)h Fx(and)g(m)n(utation)523 2823 y(parameters)25 b Fm(\022)r Fx(,)i(where)g(the)g(latter)f (captures)g(the)h(probabilit)n(y)f(that)h(an)g(allele)f(at)h(a)f(lo)r (cus)523 2922 y(is)36 b(iden)n(tical)g(to)f(the)i(ancestor)d(at)i(this) g(lo)r(cus.)g(W)-7 b(e)36 b(let)h Fm(G)p Fx(\()p Fm(A;)14 b(\022)r Fx(\))38 b(=)e Fm(p)p Fx(\()p Fm(A)p Fx(\))p Fm(p)p Fx(\()p Fm(\022)r Fx(\),)i(and)d(w)n(e)523 3022 y(assume)40 b(that)g Fm(p)p Fx(\()p Fm(A)p Fx(\))h(is)g(a)f(uniform)g (distribution)g(o)n(v)n(er)f(all)h(p)r(ossible)g(haplot)n(yp)r(es.)g(W) -7 b(e)523 3122 y(let)36 b Fm(p)p Fx(\()p Fm(\022)r Fx(\))g(b)r(e)f(a)g (b)r(eta)h(distribution)f Fm(B)t(eta)p Fx(\()p Fm(\013)1951 3134 y Fw(h)1994 3122 y Fm(;)14 b(\014)2078 3134 y Fw(h)2121 3122 y Fx(\),)36 b(and)f(w)n(e)g(c)n(ho)r(ose)f(a)g(small)h(v)-5 b(alue)36 b(for)523 3221 y Fm(\014)570 3233 y Fw(h)613 3221 y Fm(=)p Fx(\()p Fm(\013)740 3233 y Fw(h)801 3221 y Fx(+)18 b Fm(\014)931 3233 y Fw(h)974 3221 y Fx(\),)28 b(corresp)r(onding)e(to)i(a)f(prior)f(exp)r(ectation)i(of)f(a)g(lo)n(w) g(m)n(utation)h(rate.)648 3321 y(Giv)n(en)c Fm(C)944 3333 y Fw(i)967 3341 y Fh(t)1025 3321 y Fx(and)h(a)g(set)g(of)g (ancestors,)e(w)n(e)i(de\014ne)h(the)f(conditional)f(probabilit)n(y)h (of)g(the)523 3421 y(corresp)r(onding)h(haplot)n(yp)r(e)h(instance)g Fm(h)c Fx(:=)g([)p Fm(h)2018 3433 y Fv(1)2055 3421 y Fm(;)14 b(:)g(:)g(:)f(;)h(h)2287 3433 y Fw(J)2334 3421 y Fx(])27 b(to)h(b)r(e:)921 3592 y Fm(p)p Fx(\()p Fm(H)1064 3604 y Fw(i)1087 3612 y Fh(t)1143 3592 y Fx(=)22 b Fm(h)p Fn(j)p Fm(C)1360 3604 y Fw(i)1383 3612 y Fh(t)1439 3592 y Fx(=)g Fm(k)s(;)14 b Fi(A)23 b Fx(=)g Fi(a)p Fm(;)14 b Ff(\022)s Fx(\))23 b(=)g Fm(p)p Fx(\()p Fm(H)2211 3604 y Fw(i)2234 3612 y Fh(t)2289 3592 y Fx(=)g Fm(h)p Fn(j)p Fm(A)2510 3604 y Fw(k)2574 3592 y Fx(=)g Fm(a;)14 b(\022)2782 3604 y Fw(k)2845 3592 y Fx(=)23 b Fm(\022)r Fx(\))1980 3733 y(=)2068 3654 y Fl(Y)2106 3831 y Fw(j)2188 3733 y Fm(p)p Fx(\()p Fm(h)2310 3745 y Fw(j)2345 3733 y Fn(j)p Fm(a)2412 3745 y Fw(j)2447 3733 y Fm(;)14 b(\022)r Fx(\))p Fm(;)719 b Fx(\(4\))523 3992 y(where)18 b Fm(p)p Fx(\()p Fm(h)876 4004 y Fw(j)911 3992 y Fn(j)p Fm(a)978 4004 y Fw(j)1013 3992 y Fm(;)c(\022)r Fx(\))19 b(is)g(the)g(probabilit)n(y)e (of)i(ha)n(ving)e(allele)i Fm(h)2362 4004 y Fw(j)2415 3992 y Fx(at)g(lo)r(cus)f Fm(j)24 b Fx(giv)n(en)18 b(its)g(ancestor.) 523 4092 y(Eq.)30 b(\(4\))g(assumes)g(that)h(eac)n(h)e(lo)r(cus)h(is)h (m)n(utated)f(indep)r(enden)n(tly)h(with)g(the)g(same)f(error)523 4191 y(rate.)40 b(F)-7 b(or)40 b(haplot)n(yp)r(es,)g Fm(H)1415 4203 y Fw(i)1438 4211 y Fh(t)1466 4203 y Fw(;j)1561 4191 y Fx(tak)n(es)g(v)-5 b(alues)40 b(from)g(a)g(set)h Fm(B)k Fx(of)40 b(alleles.)g(W)-7 b(e)41 b(use)g(the)523 4291 y(follo)n(wing)27 b Fo(single-lo)l(cus)j(mutation)f(mo)l(del)9 b Fx(:)1239 4505 y Fm(p)p Fx(\()p Fm(h)1361 4517 y Fw(j)1396 4505 y Fn(j)p Fm(a)1463 4517 y Fw(j)1497 4505 y Fm(;)14 b(\022)r Fx(\))24 b(=)f Fm(\022)1760 4471 y Fv(1\()p Fw(h)1858 4479 y Fh(j)1889 4471 y Fv(=)p Fw(a)1976 4479 y Fh(j)2007 4471 y Fv(\))2037 4413 y Fl(\020)2132 4449 y Fx(1)18 b Fn(\000)g Fm(\022)p 2096 4486 257 4 v 2096 4562 a Fn(j)p Fm(B)t Fn(j)h(\000)f Fx(1)2363 4413 y Fl(\021)2412 4431 y Fv(1\()p Fw(h)2510 4439 y Fh(j)2541 4431 y Fj(6)p Fv(=)p Fw(a)2628 4439 y Fh(j)2659 4431 y Fv(\))3299 4505 y Fx(\(5\))523 4725 y(where)27 b(1\()p Fn(\001)p Fx(\))h(is)f(the)h (indicator)f(function.)648 4825 y(The)35 b(join)n(t)g(conditional)g (distribution)h(of)f(haplot)n(yp)r(e)g(instances)g Fi(h)h Fx(=)f Fn(f)p Fm(h)3063 4837 y Fw(i)3086 4845 y Fh(t)3189 4825 y Fx(:)72 b Fm(t)36 b Fn(2)523 4924 y(f)p Fx(0)p Fm(;)14 b Fx(1)p Fn(g)p Fm(;)g(i)21 b Fn(2)i(f)p Fx(1)p Fm(;)14 b Fx(2)p Fm(;)g(:)g(:)g(:)e(;)i(I)7 b Fn(gg)22 b Fx(and)g(parameter)f(instances)h Ff(\022)k Fx(=)c Fn(f)p Fm(\022)2528 4936 y Fv(1)2565 4924 y Fm(;)14 b(:)g(:)g(:)g(;)g(\022) 2789 4936 y Fw(K)2853 4924 y Fn(g)p Fx(,)22 b(giv)n(en)g(the)h(an-)p eop %%Page: 7 7 7 6 bop 3367 232 a Ft(7)523 448 y Fx(cestor)17 b(indicator)h Fi(c)h Fx(of)f(haplot)n(yp)r(e)g(instances)g(and)g(the)h(set)f(of)h (ancestors)d Fi(a)24 b Fx(=)e Fn(f)p Fm(a)3051 460 y Fv(1)3088 448 y Fm(;)14 b(:)g(:)g(:)g(;)g(a)3317 460 y Fw(K)3380 448 y Fn(g)p Fx(,)523 548 y(can)27 b(b)r(e)h(written)g (explicitly)g(as:)1006 779 y Fm(p)p Fx(\()p Fi(h)p Fm(;)14 b Ff(\022)r Fn(j)p Fi(c)p Fm(;)g Fi(a)p Fx(\))25 b Fn(/)1512 700 y Fl(Y)1546 879 y Fw(k)1632 779 y Fm(\022)1673 742 y Fw(m)1732 751 y Fh(k)1768 742 y Fv(+)p Fw(\013)1862 751 y Fh(h)1901 742 y Fj(\000)p Fv(1)1671 804 y Fw(k)1990 687 y Fl(\020)2066 723 y Fx(1)18 b Fn(\000)g Fm(\022)2248 735 y Fw(k)p 2049 760 257 4 v 2049 836 a Fn(j)p Fm(B)t Fn(j)h(\000)f Fx(1)2316 687 y Fl(\021)2365 704 y Fw(m)2424 679 y Fe(0)2424 721 y Fh(k)2465 712 y Fl(\002)2499 779 y Fx(1)g Fn(\000)g Fm(\022)2681 791 y Fw(k)2722 712 y Fl(\003)2756 729 y Fw(\014)2794 738 y Fh(h)2833 729 y Fj(\000)p Fv(1)3299 779 y Fx(\(6\))523 1042 y(where)34 b Fm(m)843 1054 y Fw(k)918 1042 y Fx(=)1017 979 y Fl(P)1104 1067 y Fw(j)1153 979 y Fl(P)1241 1067 y Fw(i)1282 979 y Fl(P)1370 1067 y Fw(t)1413 1042 y Fx(1\()p Fm(h)1535 1054 y Fw(i)1558 1062 y Fh(t)1585 1054 y Fw(;j)1674 1042 y Fx(=)g Fm(a)1817 1054 y Fw(k)q(;j)1908 1042 y Fx(\)1\()p Fm(c)2050 1054 y Fw(i)2073 1062 y Fh(t)2140 1042 y Fx(=)g Fm(k)s Fx(\))g(is)g(the)h(n)n(um)n(b)r(er)f(of)g(alleles)g(that)523 1141 y(w)n(ere)17 b(not)i(m)n(utated)f(with)h(resp)r(ect)f(to)h(the)f (ancestral)f(allele,)h(and)h Fm(m)2639 1111 y Fj(0)2639 1165 y Fw(k)2702 1141 y Fx(=)2790 1079 y Fl(P)2878 1166 y Fw(j)2927 1079 y Fl(P)3014 1166 y Fw(i)3056 1079 y Fl(P)3143 1166 y Fw(t)3186 1141 y Fx(1\()p Fm(h)3308 1153 y Fw(i)3331 1161 y Fh(t)3359 1153 y Fw(;j)3437 1141 y Fn(6)p Fx(=)523 1241 y Fm(a)567 1253 y Fw(k)q(;j)658 1241 y Fx(\)1\()p Fm(c)800 1253 y Fw(i)823 1261 y Fh(t)884 1241 y Fx(=)27 b Fm(k)s Fx(\))k(is)g(the)g(n)n(um)n(b)r(er)f(of)h(m)n (utated)g(alleles.)f(The)g(coun)n(t)h Fi(m)2811 1253 y Fw(k)2878 1241 y Fx(=)d Fn(f)p Fm(m)3086 1253 y Fw(k)3126 1241 y Fm(;)14 b(m)3236 1211 y Fj(0)3236 1264 y Fw(k)3277 1241 y Fn(g)30 b Fx(is)523 1341 y(a)36 b(su\016cien)n(t)h(statistic)g (for)f(the)h(parameter)e Fm(\022)2016 1353 y Fw(k)2093 1341 y Fx(and)i(the)g(coun)n(t)f Fi(m)h Fx(=)g Fn(f)p Fi(m)2996 1353 y Fw(k)3035 1341 y Fm(;)14 b Fi(m)3153 1310 y Fj(0)3153 1364 y Fw(k)3193 1341 y Fn(g)36 b Fx(is)g(a)523 1440 y(su\016cien)n(t)26 b(statistic)h(for)e(the)i(parameter)e Ff(\022)r Fx(.)h(The)g(marginal)f(conditional)h(distribution)g(of)523 1540 y(haplot)n(yp)r(e)h(instances)g(can)g(b)r(e)h(obtained)g(b)n(y)f (in)n(tegrating)f(out)i Fm(\022)i Fx(in)e(Eq.)f(\(6\):)824 1776 y Fm(p)p Fx(\()p Fi(h)p Fn(j)p Fi(c)p Fm(;)14 b Fi(a)p Fx(\))25 b(=)1244 1697 y Fl(Y)1278 1876 y Fw(k)1364 1776 y Fm(R)q Fx(\()p Fm(\013)1513 1788 y Fw(h)1556 1776 y Fm(;)14 b(\014)1640 1788 y Fw(h)1683 1776 y Fx(\))1725 1720 y Fm(\000)e Fx(\()p Fm(\013)1873 1732 y Fw(h)1934 1720 y Fx(+)18 b Fm(m)2090 1732 y Fw(k)2131 1720 y Fx(\))p Fm(\000)12 b Fx(\()p Fm(\014)2305 1732 y Fw(h)2367 1720 y Fx(+)18 b Fm(m)2523 1689 y Fj(0)2523 1743 y Fw(k)2563 1720 y Fx(\))p 1725 1757 871 4 v 1738 1833 a Fm(\000)12 b Fx(\()p Fm(\013)1886 1845 y Fw(h)1947 1833 y Fx(+)18 b Fm(\014)2077 1845 y Fw(h)2139 1833 y Fx(+)g Fm(m)2295 1845 y Fw(k)2354 1833 y Fx(+)g Fm(m)2510 1804 y Fj(0)2510 1858 y Fw(k)2551 1833 y Fx(\))2606 1684 y Fl(\020)2773 1720 y Fx(1)p 2665 1757 257 4 v 2665 1833 a Fn(j)p Fm(B)t Fn(j)h(\000)f Fx(1)2931 1684 y Fl(\021)2981 1701 y Fw(m)3040 1676 y Fe(0)3040 1717 y Fh(k)3080 1776 y Fm(;)196 b Fx(\(7\))523 2060 y(where)23 b Fm(\000)12 b Fx(\()p Fn(\001)p Fx(\))24 b(is)g(the)g Fo(gamma)h Fx(function,)f(and)g Fm(R)q Fx(\()p Fm(\013)2086 2072 y Fw(h)2129 2060 y Fm(;)14 b(\014)2213 2072 y Fw(h)2256 2060 y Fx(\))23 b(=)2434 2020 y Fw(\000)9 b Fv(\()p Fw(\013)2552 2029 y Fh(h)2591 2020 y Fv(+)p Fw(\014)2680 2029 y Fh(h)2718 2020 y Fv(\))p 2409 2041 361 4 v 2409 2088 a Fw(\000)g Fv(\()p Fw(\013)2527 2097 y Fh(h)2566 2088 y Fv(\))p Fw(\000)g Fv(\()p Fw(\014)2705 2097 y Fh(h)2743 2088 y Fv(\))2803 2060 y Fx(is)24 b(the)g(normaliza-) 523 2174 y(tion)j(constan)n(t)f(asso)r(ciated)f(with)j Fm(B)t(eta)p Fx(\()p Fm(\013)1875 2186 y Fw(h)1918 2174 y Fm(;)14 b(\014)2002 2186 y Fw(h)2045 2174 y Fx(\).)27 b(\(F)-7 b(or)26 b(simplicit)n(y)-7 b(,)27 b(w)n(e)f(use)h(the)g (abbrevi-)523 2273 y(ation)g Fm(R)798 2285 y Fw(h)869 2273 y Fx(for)g Fm(R)q Fx(\()p Fm(\013)1145 2285 y Fw(h)1188 2273 y Fm(;)14 b(\014)1272 2285 y Fw(h)1315 2273 y Fx(\))28 b(in)g(the)g(sequel\).)648 2373 y(W)-7 b(e)26 b(no)n(w)f(in)n(tro)r (duce)g(a)g Fo(noisy)k(observation)h(mo)l(del)35 b Fx(for)25 b(the)h(genot)n(yp)r(es.)f(W)-7 b(e)26 b(let)g Fm(G)3289 2385 y Fw(i)3340 2373 y Fx(=)523 2473 y([)p Fm(G)611 2485 y Fw(i;)p Fv(1)692 2473 y Fm(;)14 b(:)g(:)g(:)f(;)h(G)941 2485 y Fw(i;J)1031 2473 y Fx(])27 b(denote)g(the)g Fo(joint)j(genotyp)l (e)e Fx(of)e(individual)h Fm(i)g Fx(at)g(lo)r(ci)f([1)p Fm(;)14 b(:)g(:)g(:)g(;)g(J)8 b Fx(],)27 b(where)523 2572 y(eac)n(h)21 b Fm(G)769 2584 y Fw(j;i)866 2572 y Fx(denotes)g(the)i Fo(genotyp)l(e)f Fx(at)g(lo)r(cus)g Fm(j)5 b Fx(.)22 b(W)-7 b(e)22 b(assume)f(that)h(the)g(observ)n(ed)f (genot)n(yp)r(e)523 2672 y(at)35 b(a)f(lo)r(cus)g(is)h(determined)g(b)n (y)f(the)h(paternal)f(and)h(maternal)f(alleles)g(of)g(this)h(lo)r(cus)g (as)523 2772 y(follo)n(ws:)565 2985 y Fm(p)p Fx(\()p Fm(g)679 2997 y Fw(i;j)756 2985 y Fn(j)p Fm(h)827 2997 y Fw(i)850 3005 y Fk(0)883 2997 y Fw(;j)938 2985 y Fm(;)14 b(h)1023 2997 y Fw(i)1046 3005 y Fk(1)1079 2997 y Fw(;j)1133 2985 y Fm(;)g(\015)5 b Fx(\))23 b(=)g Fm(\015)1409 2951 y Fv(1\()p Fw(h)1507 2959 y Fh(i;j)1578 2951 y Fv(=)p Fw(g)1661 2959 y Fh(i;j)1734 2951 y Fv(\))1764 2985 y Fx([)p Fm(\026)1837 2997 y Fv(1)1874 2985 y Fx(\(1)14 b Fn(\000)f Fm(\015)5 b Fx(\)])2143 2951 y Fv(1\()p Fw(h)2241 2959 y Fh(i;j)2325 2899 y Fk(1)2314 2951 y Fj(6)p Fv(=)o Fw(g)2396 2959 y Fh(i;j)2469 2951 y Fv(\))2499 2985 y Fx([)p Fm(\026)2572 2997 y Fv(2)2609 2985 y Fx(\(1)14 b Fn(\000)g Fm(\015)5 b Fx(\)])2879 2951 y Fv(1\()p Fw(h)2977 2959 y Fh(i;j)3060 2899 y Fk(2)3049 2951 y Fj(6)p Fv(=)p Fw(g)3132 2959 y Fh(i;j)3204 2951 y Fv(\))3234 2985 y Fm(;)42 b Fx(\(8\))523 3177 y(where)32 b Fm(h)816 3189 y Fw(i;j)926 3177 y Fd(,)e Fm(h)1070 3189 y Fw(i)1093 3197 y Fk(0)1126 3189 y Fw(;j)1202 3177 y Fn(\010)22 b Fm(h)1337 3189 y Fw(i)1360 3197 y Fk(1)1392 3189 y Fw(;j)1480 3177 y Fx(denotes)32 b(the)h(unordered)f(pair)g(of)h(t)n(w)n (o)f(actual)g(SNP)g(allele)523 3318 y(instances)38 b(at)g(lo)r(cus)f Fm(j)5 b Fx(;)39 b(\\)1382 3244 y Fk(1)1365 3318 y Fn(6)p Fx(=)o(")e(denotes)h(set)g(di\013erence)g(b)n(y)g(exactly)g(one)f (elemen)n(t)h(\(i.e.,)523 3457 y(the)28 b(observ)n(ed)e(genot)n(yp)r(e) g(is)i(heterozygous,)d(while)j(the)g(true)f(one)g(is)h(homozygous\);)d (\\)3317 3382 y Fk(2)3299 3457 y Fn(6)p Fx(=)o(")523 3556 y(denotes)35 b(set)g(di\013erence)g(of)g(b)r(oth)g(elemen)n(ts)g (\(i.e.,)g(the)h(observ)n(ed)d(and)i(true)g(genot)n(yp)r(es)523 3656 y(are)e(di\013eren)n(t)h(and)g(b)r(oth)h(are)e(homozygous\);)g (and)h Fm(\026)2267 3668 y Fv(1)2338 3656 y Fx(and)g Fm(\026)2556 3668 y Fv(2)2628 3656 y Fx(are)f(appropriately)f(de-)523 3756 y(\014ned)h(normalizing)d(constan)n(ts.)i(W)-7 b(e)32 b(place)g(a)g(b)r(eta)g(prior)g Fm(B)t(eta)p Fx(\()p Fm(\013)2686 3768 y Fw(g)2724 3756 y Fm(;)14 b(\014)2808 3768 y Fw(g)2847 3756 y Fx(\))32 b(on)g Fm(\015)5 b Fx(.)32 b(Assum-)523 3855 y(ing)c(indep)r(enden)n(t)h(and)g(iden)n(tical)f (error)e(mo)r(dels)j(for)f(eac)n(h)f(lo)r(cus,)h(the)h(join)n(t)g (conditional)523 3955 y(probabilit)n(y)g(of)g(the)h(en)n(tire)f(genot)n (yp)r(e)g(observ)-5 b(ation)28 b Fi(g)f Fx(=)f Fn(f)p Fm(g)2472 3967 y Fw(i)2554 3955 y Fx(:)56 b Fm(i)26 b Fn(2)h(f)p Fx(1)p Fm(;)14 b Fx(2)p Fm(;)g(:)g(:)g(:)e(;)i(I)7 b Fn(gg)28 b Fx(and)523 4054 y(parameter)e Fm(\015)5 b Fx(,)27 b(giv)n(en)g(all)h(haplot)n(yp)r(e)f(instances)g(is:)1129 4250 y Fm(p)p Fx(\()p Fi(g)q Fm(;)14 b(\015)5 b Fn(j)p Fi(h)p Fx(\))23 b(=)1556 4171 y Fl(Y)1597 4348 y Fw(i)1676 4250 y Fm(p)p Fx(\()p Fm(g)1790 4262 y Fw(i)1817 4250 y Fm(;)14 b(\015)5 b Fn(j)p Fm(h)1973 4262 y Fw(i)1996 4270 y Fk(0)2033 4250 y Fm(;)14 b(h)2118 4262 y Fw(i)2141 4270 y Fk(1)2177 4250 y Fx(\))1468 4488 y(=)23 b Fm(\015)1604 4453 y Fw(\013)1647 4461 y Fh(g)1682 4453 y Fv(+)p Fw(u)p Fj(\000)p Fv(1)1861 4420 y Fl(\002)1895 4488 y Fx(1)18 b Fn(\000)g Fm(\015)2086 4420 y Fl(\003)2121 4437 y Fw(\014)2159 4445 y Fh(g)2193 4437 y Fv(+)p Fw(u)2283 4412 y Fe(0)2306 4437 y Fv(+)p Fw(u)2396 4412 y Fe(00)2437 4437 y Fj(\000)p Fv(1)2526 4488 y Fm(\026)2576 4453 y Fw(u)2615 4428 y Fe(0)2576 4508 y Fv(1)2642 4488 y Fm(\026)2692 4453 y Fw(u)2731 4428 y Fe(00)2692 4508 y Fv(2)2776 4488 y Fm(;)500 b Fx(\(9\))523 4671 y(where)27 b(the)g(su\016cien)n(t)h(statistics)f Fi(u)c Fx(=)g Fn(f)p Fm(u;)14 b(u)1940 4641 y Fj(0)1961 4671 y Fm(;)g(u)2046 4641 y Fj(00)2088 4671 y Fn(g)27 b Fx(are)f(computed)i(as)f Fm(u)22 b Fx(=)2938 4609 y Fl(P)3025 4696 y Fw(i;j)3117 4671 y Fx(1\()p Fm(h)3239 4683 y Fw(i;j)3340 4671 y Fx(=)523 4825 y Fm(g)563 4837 y Fw(i;j)641 4825 y Fx(\),)34 b Fm(u)778 4795 y Fj(0)835 4825 y Fx(=)934 4762 y Fl(P)1022 4850 y Fw(i;j)1114 4825 y Fx(1\()p Fm(h)1236 4837 y Fw(i;j)1366 4750 y Fk(1)1347 4825 y Fn(6)p Fx(=)g Fm(g)1486 4837 y Fw(i;j)1564 4825 y Fx(\),)g(and)h Fm(u)1870 4795 y Fj(00)1946 4825 y Fx(=)2044 4762 y Fl(P)2132 4850 y Fw(i;j)2224 4825 y Fx(1\()p Fm(h)2346 4837 y Fw(j;i)2472 4750 y Fk(2)2454 4825 y Fn(6)p Fx(=)f Fm(g)2593 4837 y Fw(j;i)2667 4825 y Fx(\),)h(resp)r(ectiv)n(ely)-7 b(.)33 b(Note)523 4924 y(that)i Fm(u)22 b Fx(+)h Fm(u)916 4894 y Fj(0)962 4924 y Fx(+)f Fm(u)1097 4894 y Fj(00)1173 4924 y Fx(=)34 b Fm(I)7 b(J)h Fx(.)35 b(T)-7 b(o)34 b(re\015ect)g(an)g (assumption)g(that)h(the)f(observ)-5 b(ational)33 b(error)p eop %%Page: 8 8 8 7 bop 523 232 a Ft(8)523 448 y Fx(rate)25 b(is)h(lo)n(w)g(w)n(e)f (set)h Fm(\014)1218 460 y Fw(g)1257 448 y Fm(=)p Fx(\()p Fm(\013)1384 460 y Fw(g)1438 448 y Fx(+)15 b Fm(\014)1565 460 y Fw(g)1603 448 y Fx(\))27 b(to)f(a)f(small)h(constan)n(t)f (\(0.001\).)g(Again,)h(the)g(marginal)523 548 y(conditional)h (distribution)h(of)f Fi(g)i Fx(is)e(computed)h(b)n(y)g(in)n(tegrating)e (out)h Fm(\015)5 b Fx(.)648 648 y(Ha)n(ving)39 b(describ)r(ed)i(our)f (Ba)n(y)n(esian)e(haplot)n(yp)r(e)j(mo)r(del,)g(the)g(problem)f(of)h (phasing)523 747 y(individual)c(haplot)n(yp)r(es)g(and)f(estimating)h (the)h(size)e(and)h(con\014guration)f(of)h(the)g(laten)n(t)523 847 y(ancestral)26 b(p)r(o)r(ol)h(can)g(b)r(e)h(solv)n(ed)e(via)h(p)r (osterior)f(inference)h(giv)n(en)g(the)g(genot)n(yp)r(e)g(data.)g(In) 523 946 y(the)f(App)r(endix,)g(w)n(e)f(describ)r(e)g(a)g(Mark)n(o)n(v)e (c)n(hain)h(Mon)n(te)h(Carlo)f(\(MCMC\))i(algorithm)f(for)523 1046 y(this)j(purp)r(ose.)523 1295 y Fp(3)112 b(Exp)s(erimen)m(tal)36 b(Results)523 1478 y Fx(W)-7 b(e)27 b(v)-5 b(alidated)27 b(our)f(algorithm)g(b)n(y)h(applying)f(it)h(to)g(sim)n(ulated)g(and)g (real)e(data)i(and)g(com-)523 1578 y(pared)f(its)g(p)r(erformance)f(to) h(that)h(of)f(the)h(state-of-the-art)e(PHASE)h(algorithm)f([21)o(])h (and)523 1677 y(other)34 b(curren)n(t)f(algorithms.)g(W)-7 b(e)34 b(rep)r(ort)g(on)f(the)i(results)f(of)g(b)r(oth)g(v)-5 b(arian)n(ts)33 b(of)h(our)g(al-)523 1777 y(gorithm:)e(The)g(Gibbs)h (sampler,)f(denoted)g(DP\(Gibbs\),)h(and)g(the)g(Metrop)r(olis-Hasting) 523 1877 y(sampler,)g(denoted)h(DP\(MH\).)g(Throughout)f(the)h(exp)r (erimen)n(ts,)f(w)n(e)g(set)h(the)g(h)n(yp)r(erpa-)523 1976 y(rameter)27 b Fm(\034)38 b Fx(in)28 b(the)h(Diric)n(hlet)f(pro)r (cess)f(to)h(b)r(e)h(roughly)d(1\045)i(of)g(the)h(p)r(opulation)f (size,)g(i.e.,)523 2076 y(for)33 b(a)g(data)f(set)i(of)f(100)f (individuals,)h Fm(\034)42 b Fx(=)32 b(1.)h(W)-7 b(e)34 b(used)f(a)g(burn-in)g(of)h(2000)d(iterations)523 2175 y(\(or)i(4000)f(for)h(datasets)f(with)i(more)f(than)h(50)f (individuals\),)g(and)h(used)f(the)h(next)g(6000)523 2275 y(iterations)27 b(for)g(estimation.)523 2508 y Fi(3.1)95 b(Sim)m(ulated)30 b(data)523 2657 y Fx(In)22 b(our)f(\014rst)g(set)g (of)h(exp)r(erimen)n(ts)f(w)n(e)g(applied)h(our)f(metho)r(d)h(to)f(sim) n(ulated)g(data)h(from)f([21)o(,)523 2757 y(\\short)41 b(sequence)g(data"].)g(This)g(data)h(con)n(tains)f(sets)g(of)h(2)p Fm(n)f Fx(haplot)n(yp)r(es,)g(randomly)523 2857 y(paired)24 b(to)g(form)g Fm(n)g Fx(genot)n(yp)r(es,)g(under)g(an)g (in\014nite-sites)g(mo)r(del)h(with)g(parameters)e Fm(\021)j Fx(=)c(4)523 2956 y(and)42 b Fm(R)47 b Fx(=)f(4)41 b(determining)h(the) g(m)n(utation)f(and)h(recom)n(bination)e(rates,)h(resp)r(ectiv)n(ely) 523 3056 y(\(see)27 b([21)o(])h(for)e(additional)h(details\).)h(W)-7 b(e)27 b(used)g(the)h(\014rst)f(40)f(datasets)h(for)g(eac)n(h)f(com)n (bina-)523 3155 y(tion)33 b(of)g(individuals)f(and)h(sites,)f(where)h (the)g(n)n(um)n(b)r(er)f(of)h(individuals)g(ranged)e(b)r(et)n(w)n(een) 523 3255 y(10)c(and)g(50,)g(and)g(the)h(n)n(um)n(b)r(er)g(of)f(sites)h (ranged)e(b)r(et)n(w)n(een)i(5)f(and)g(30.)p 957 3314 1967 4 v 955 3380 4 67 v 1451 3380 V 1601 3360 a Fv(DP\(MH\))p 2002 3380 V 311 w(PHASE)p 2554 3380 V 202 w(EM)p 2738 3380 V 51 w(Clark)p 2922 3380 V 957 3383 1967 4 v 955 3450 4 67 v 968 3430 a(#individuals)p 1451 3450 V 123 w Fw(er)r(r)1559 3438 y Fh(s)p 1635 3450 V 1648 3430 a Fw(er)r(r)1743 3438 y Fh(i)p 1818 3450 V 1832 3430 a Fw(d)1867 3438 y Fh(s)p 2002 3450 V 2016 3430 a Fw(er)r(r)2111 3438 y Fh(s)p 2186 3450 V 2199 3430 a Fw(er)r(r)2294 3438 y Fh(i)p 2370 3450 V 2383 3430 a Fw(d)2418 3438 y Fh(s)p 2554 3450 V 2567 3430 a Fw(er)r(r)2662 3438 y Fh(i)p 2738 3450 V 2751 3430 a Fw(er)r(r)2846 3438 y Fh(i)p 2922 3450 V 957 3453 1967 4 v 957 3470 V 955 3536 4 67 v 968 3516 a Fv(10)p 1451 3536 V 430 w(0.060)p 1635 3536 V 33 w(0.216)p 1818 3536 V 33 w(0.051)p 2002 3536 V 33 w(0.046)p 2186 3536 V 32 w(0.182)p 2370 3536 V 33 w(0.054)p 2554 3536 V 33 w(0.424)p 2738 3536 V 33 w(0.398)p 2922 3536 V 957 3539 1967 4 v 955 3606 4 67 v 968 3586 a(20)p 1451 3606 V 430 w(0.039)p 1635 3606 V 33 w(0.152)p 1818 3606 V 33 w(0.039)p 2002 3606 V 33 w(0.029)p 2186 3606 V 32 w(0.136)p 2370 3606 V 33 w(0.046)p 2554 3606 V 33 w(0.296)p 2738 3606 V 33 w(0.397)p 2922 3606 V 957 3609 1967 4 v 955 3676 4 67 v 968 3656 a(30)p 1451 3676 V 430 w(0.036)p 1635 3676 V 33 w(0.121)p 1818 3676 V 33 w(0.038)p 2002 3676 V 33 w(0.024)p 2186 3676 V 32 w(0.101)p 2370 3676 V 33 w(0.027)p 2554 3676 V 33 w(0.231)p 2738 3676 V 33 w(0.356)p 2922 3676 V 957 3679 1967 4 v 955 3745 4 67 v 968 3725 a(40)p 1451 3745 V 430 w(0.030)p 1635 3745 V 33 w(0.094)p 1818 3745 V 33 w(0.029)p 2002 3745 V 33 w(0.019)p 2186 3745 V 32 w(0.071)p 2370 3745 V 33 w(0.026)p 2554 3745 V 33 w(0.195)p 2738 3745 V 33 w(0.355)p 2922 3745 V 957 3749 1967 4 v 955 3815 4 67 v 968 3795 a(50)p 1451 3815 V 430 w(0.028)p 1635 3815 V 33 w(0.082)p 1818 3815 V 33 w(0.024)p 2002 3815 V 33 w(0.019)p 2186 3815 V 32 w(0.072)p 2370 3815 V 33 w(0.025)p 2554 3815 V 33 w(0.167)p 2738 3815 V 33 w(0.343)p 2922 3815 V 957 3818 1967 4 v 1087 3879 a Fr(T)-7 b(able)29 b(1.)c Fv(P)n(erformance)20 b(results)i(on)g(sim)n(ulated)f (data)h(from)f([21].)648 4028 y Fx(T)-7 b(o)32 b(ev)-5 b(aluate)33 b(the)h(p)r(erformance)e(of)h(the)h(algorithms)d(w)n(e)i (used)g(the)h(follo)n(wing)e(error)523 4127 y(measures:)22 b Fm(er)r(r)1013 4139 y Fw(s)1050 4127 y Fx(,)h(the)h(ratio)f(of)g (incorrectly)g(phased)g(SNP)g(sites)h(o)n(v)n(er)d(all)j(non-trivial)e (het-)523 4227 y(erozygous)31 b(SNPs)h(\(excluding)h(individuals)g (with)h(a)f(single)f(heterozygous)f(SNP\);)i Fm(er)r(r)3353 4239 y Fw(i)3382 4227 y Fx(,)523 4327 y(the)26 b(the)g(ratio)f(of)g (incorrectly)g(phased)g(individuals)g(o)n(v)n(er)f(all)h(non-trivial)g (heterogeneous)523 4426 y(individuals;)31 b(and)g Fm(d)1180 4438 y Fw(s)1216 4426 y Fx(,)g(the)g Fo(switch)j(distanc)l(e)p Fx(,)d(whic)n(h)g(is)g(the)h(n)n(um)n(b)r(er)e(of)h(phase)g(\015ips)g (re-)523 4526 y(quired)40 b(to)g(correct)e(the)j(predicted)f(haplot)n (yp)r(es)f(o)n(v)n(er)f(all)i(non-trivial)f(heterogeneous)523 4625 y(SNPs.)22 b(The)h(results)f(are)f(summarized)h(in)h(T)-7 b(able)22 b(1.)g(Ov)n(erall,)f(w)n(e)h(p)r(erform)g(sligh)n(tly)g(w)n (orse)523 4725 y(than)33 b(PHASE)f(on)h(the)g(\014rst)g(t)n(w)n(o)f (measures,)f(and)i(sligh)n(tly)f(b)r(etter)h(on)g(the)g(switc)n(h)f (dis-)523 4825 y(tance)27 b(measure)f(\(whic)n(h)h(uses)g(100,000)d (sampling)j(steps\),)g(and)g(are)f(signi\014can)n(tly)h(b)r(etter)523 4924 y(than)h(EM)f(and)g(Clark.)p eop %%Page: 9 9 9 8 bop 3367 232 a Ft(9)523 448 y Fi(3.2)95 b(Real)31 b(data)523 645 y Fx(W)-7 b(e)20 b(also)f(applied)h(our)f(algorithm)g (to)h(t)n(w)n(o)f(real)g(datasets)g(and)h(compared)f(its)h(p)r (erformance)523 745 y(to)28 b(that)f(of)h(PHASE)f([21)o(])h(and)g (other)f(algorithms.)648 851 y(The)35 b(\014rst)g(dataset)g(con)n (tains)g(the)g(genot)n(yp)r(es)g(of)g(129)f(individuals)i(o)n(v)n(er)d (103)h(p)r(oly-)523 951 y(morphic)f(sites)h([4].)g(In)g(addition)g(it)g (con)n(tains)f(the)h(genot)n(yp)r(es)f(of)h(the)g(paren)n(ts)f(of)h (eac)n(h)523 1050 y(individual,)i(whic)n(h)h(allo)n(ws)d(the)j (inference)f(of)g(a)g(large)f(p)r(ortion)g(of)i(the)f(haplot)n(yp)r(es) g(as)523 1150 y(in)h([6].)g(The)h(p)r(erformance)e(results)g(are)h (summarized)f(in)h(T)-7 b(able)37 b(2.)g(F)-7 b(rom)37 b(T)-7 b(able)37 b(2,)g(it)523 1249 y(is)f(apparen)n(t)e(that)i(the)h (Metrop)r(olis-Hasting)d(sampling)h(algorithm)f(signi\014can)n(tly)h (out-)523 1349 y(p)r(erforms)29 b(the)i(Gibbs)f(sampler,)f(and)h(is)g (to)g(b)r(e)h(preferred)e(giv)n(en)g(the)h(relativ)n(ely)f(limited)523 1449 y(n)n(um)n(b)r(er)i(of)h(sampling)e(steps)i(\()p Fn(\030)d Fx(6000\).)h(The)h(o)n(v)n(erall)f(p)r(erformance)g(is)h (comparable)f(to)523 1548 y(that)e(of)f(PHASE)h(and)f(b)r(etter)h(than) g(b)r(oth)g(HAP)g([13)o(,)g(6)o(])g(and)f(HAPLOTYPER)g([17)o(].)p 533 1703 2862 4 v 531 1769 4 67 v 815 1769 V 994 1769 V 1011 1769 V 1123 1749 a Fv(DP\(Gibbs\))p 1548 1769 V 1564 1769 V 267 w(DP\(MH\))p 2102 1769 V 2118 1769 V 314 w(PHASE)p 2656 1769 V 2673 1769 V 207 w(HAP)p 2885 1769 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/epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore 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 dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} 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 /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /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 {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /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 /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix 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MP PP 0 sg -164 0 0 66 164 0 0 -66 2256 2172 5 MP stroke c8 0 53 165 0 0 -53 2461 2172 4 MP PP 0 sg -165 0 0 53 165 0 0 -53 2461 2172 5 MP stroke c8 0 43 164 0 0 -43 2667 2172 4 MP PP 0 sg -164 0 0 43 164 0 0 -43 2667 2172 5 MP stroke c8 0 36 164 0 0 -36 2873 2172 4 MP PP 0 sg -164 0 0 36 164 0 0 -36 2873 2172 5 MP stroke 0 783 1104 667 2 MP stroke 37 0 1086 667 2 MP stroke 37 0 1086 1450 2 MP stroke 0 369 1310 1562 2 MP stroke 37 0 1291 1562 2 MP stroke 37 0 1291 1931 2 MP stroke 0 215 1515 1826 2 MP stroke 37 0 1497 1826 2 MP stroke 37 0 1497 2041 2 MP stroke 0 136 1721 1947 2 MP stroke 37 0 1703 1947 2 MP stroke 37 0 1703 2083 2 MP stroke 0 95 1927 2011 2 MP stroke 37 0 1908 2011 2 MP stroke 37 0 1908 2106 2 MP stroke 0 71 2132 2051 2 MP stroke 37 0 2114 2051 2 MP stroke 37 0 2114 2122 2 MP stroke 0 56 2338 2078 2 MP stroke 37 0 2319 2078 2 MP stroke 37 0 2319 2134 2 MP stroke 0 45 2544 2097 2 MP stroke 37 0 2525 2097 2 MP stroke 37 0 2525 2142 2 MP stroke 0 36 2749 2111 2 MP stroke 37 0 2731 2111 2 MP stroke 37 0 2731 2147 2 MP stroke 0 31 2955 2121 2 MP stroke 37 0 2936 2121 2 MP stroke 37 0 2936 2152 2 MP stroke gr 24 W 1104 1058 PD 24 W 1310 1747 PD 24 W 1515 1934 PD 24 W 1721 2015 PD 24 W 1927 2059 PD 24 W 2132 2086 PD 24 W 2338 2106 PD 24 W 2544 2119 PD 24 W 2749 2129 PD 24 W 2955 2136 PD gs 899 389 2263 1784 MR c np gr %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 96 FMSR 2503 787 mt ( 112.3680) s 2556 897 mt ( 42.8975) s 2556 1007 mt ( 24.0215) s 2556 1117 mt ( 15.8180) s 2556 1227 mt ( 11.4060) s 2583 1337 mt ( 8.5905) s 2583 1447 mt ( 6.6175) s 2583 1557 mt ( 5.2755) s 2583 1667 mt ( 4.2945) s 2583 1777 mt ( 3.5750) s end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 1009 w @beginspecial 73 @llx 416 @lly 285 @urx 583 @ury 1037 @rwi @setspecial %%BeginDocument: figures/c1a.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /.automount/coeus.EECS.Berkeley.EDU/vol/vol0/eecs/epxing/matlab/haplo/daly/block_exp/sec1_new/exp4/c1a.eps %%CreationDate: 09/17/2003 23:27:52 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK %%Pages: 1 %%BoundingBox: 73 416 285 583 %%EndComments %%BeginProlog % 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 /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 /rc {rectclip} bdef /rf {rectfill} bdef % 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 /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore 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 dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} 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 /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /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 {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /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 /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile exch readhexstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 73 416 285 583 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 664 340 2547 2005 MR c np 92 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 6918 5185 PR 6 w 0 1784 2262 0 0 -1784 899 2172 4 MP PP -2262 0 0 1784 2262 0 0 -1784 899 2172 5 MP stroke 4 w DO SO 6 w 0 sg 899 2172 mt 3161 2172 L 899 388 mt 3161 388 L 899 2172 mt 899 388 L 3161 2172 mt 3161 388 L 899 2172 mt 3161 2172 L 899 2172 mt 899 388 L 1104 2172 mt 1104 2149 L 1104 389 mt 1104 411 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 120 FMSR 1071 2317 mt (1) s 1310 2172 mt 1310 2149 L 1310 389 mt 1310 411 L 1277 2317 mt (2) s 1515 2172 mt 1515 2149 L 1515 389 mt 1515 411 L 1482 2317 mt (3) s 1721 2172 mt 1721 2149 L 1721 389 mt 1721 411 L 1688 2317 mt (4) s 1927 2172 mt 1927 2149 L 1927 389 mt 1927 411 L 1894 2317 mt (5) s 2132 2172 mt 2132 2149 L 2132 389 mt 2132 411 L 2099 2317 mt (6) s 2338 2172 mt 2338 2149 L 2338 389 mt 2338 411 L 2305 2317 mt (7) s 2544 2172 mt 2544 2149 L 2544 389 mt 2544 411 L 2511 2317 mt (8) s 2749 2172 mt 2749 2149 L 2749 389 mt 2749 411 L 2716 2317 mt (9) s 2955 2172 mt 2955 2149 L 2955 389 mt 2955 411 L 2889 2317 mt (10) s 899 2172 mt 921 2172 L 3161 2172 mt 3138 2172 L 798 2216 mt (0) s 899 1676 mt 921 1676 L 3161 1676 mt 3138 1676 L 731 1720 mt (50) s 899 1181 mt 921 1181 L 3161 1181 mt 3138 1181 L 664 1225 mt (100) s 899 686 mt 921 686 L 3161 686 mt 3138 686 L 664 730 mt (150) s 899 2172 mt 3161 2172 L 899 388 mt 3161 388 L 899 2172 mt 899 388 L 3161 2172 mt 3161 388 L gs 899 389 2263 1784 MR c np /c8 { 0.000000 0.000000 0.562500 sr} bdef c8 0 1546 164 0 0 -1546 1022 2172 4 MP PP 0 sg -164 0 0 1546 164 0 0 -1546 1022 2172 5 MP stroke c8 0 606 164 0 0 -606 1228 2172 4 MP PP 0 sg -164 0 0 606 164 0 0 -606 1228 2172 5 MP stroke c8 0 335 165 0 0 -335 1433 2172 4 MP PP 0 sg -165 0 0 335 165 0 0 -335 1433 2172 5 MP stroke c8 0 49 164 0 0 -49 1639 2172 4 MP PP 0 sg -164 0 0 49 164 0 0 -49 1639 2172 5 MP stroke c8 0 21 165 0 0 -21 1844 2172 4 MP PP 0 sg -165 0 0 21 165 0 0 -21 1844 2172 5 MP stroke c8 0 2 165 0 0 -2 2050 2172 4 MP PP 0 sg -165 0 0 2 165 0 0 -2 2050 2172 5 MP stroke c8 0 1 164 0 0 -1 2256 2172 4 MP PP 0 sg -164 0 0 1 164 0 0 -1 2256 2172 5 MP stroke c8 0 1 165 0 0 -1 2461 2172 4 MP PP 0 sg -165 0 0 1 165 0 0 -1 2461 2172 5 MP stroke c8 0 0 164 0 0 0 2667 2172 4 MP PP 0 sg -164 0 0 0 164 0 0 0 2667 2172 5 MP stroke c8 0 0 164 0 0 0 2873 2172 4 MP PP 0 sg -164 0 0 0 164 0 0 0 2873 2172 5 MP stroke 0 461 1104 396 2 MP stroke 37 0 1086 396 2 MP stroke 37 0 1086 857 2 MP stroke 0 456 1310 1338 2 MP stroke 37 0 1291 1338 2 MP stroke 37 0 1291 1794 2 MP stroke 0 29 1515 1823 2 MP stroke 37 0 1497 1823 2 MP stroke 37 0 1497 1852 2 MP stroke 0 7 1721 2119 2 MP stroke 37 0 1703 2119 2 MP stroke 37 0 1703 2126 2 MP stroke 0 5 1927 2149 2 MP stroke 37 0 1908 2149 2 MP stroke 37 0 1908 2154 2 MP stroke 0 7 2132 2166 2 MP stroke 37 0 2114 2166 2 MP stroke 0 4 2338 2169 2 MP stroke 37 0 2319 2169 2 MP stroke 0 2 2544 2170 2 MP stroke 37 0 2525 2170 2 MP stroke 37 0 2525 2172 2 MP stroke 2749 2172 PD 37 0 2731 2172 2 MP stroke 37 0 2731 2172 2 MP stroke 2955 2172 PD 37 0 2936 2172 2 MP stroke 37 0 2936 2172 2 MP stroke gr 24 W 1104 626 PD 24 W 1310 1566 PD 24 W 1515 1837 PD 24 W 1721 2123 PD 24 W 1927 2151 PD 24 W 2132 2170 PD 24 W 2338 2171 PD 24 W 2544 2171 PD 24 W 2749 2172 PD 24 W 2955 2172 PD gs 899 389 2263 1784 MR c np c8 0 1546 164 0 0 -1546 1022 2172 4 MP PP 0 sg -164 0 0 1546 164 0 0 -1546 1022 2172 5 MP stroke c8 0 606 164 0 0 -606 1228 2172 4 MP PP 0 sg -164 0 0 606 164 0 0 -606 1228 2172 5 MP stroke c8 0 335 165 0 0 -335 1433 2172 4 MP PP 0 sg -165 0 0 335 165 0 0 -335 1433 2172 5 MP stroke c8 0 49 164 0 0 -49 1639 2172 4 MP PP 0 sg -164 0 0 49 164 0 0 -49 1639 2172 5 MP stroke c8 0 21 165 0 0 -21 1844 2172 4 MP PP 0 sg -165 0 0 21 165 0 0 -21 1844 2172 5 MP stroke c8 0 2 165 0 0 -2 2050 2172 4 MP PP 0 sg -165 0 0 2 165 0 0 -2 2050 2172 5 MP stroke c8 0 1 164 0 0 -1 2256 2172 4 MP PP 0 sg -164 0 0 1 164 0 0 -1 2256 2172 5 MP stroke c8 0 1 165 0 0 -1 2461 2172 4 MP PP 0 sg -165 0 0 1 165 0 0 -1 2461 2172 5 MP stroke c8 0 0 164 0 0 0 2667 2172 4 MP PP 0 sg -164 0 0 0 164 0 0 0 2667 2172 5 MP stroke c8 0 0 164 0 0 0 2873 2172 4 MP PP 0 sg -164 0 0 0 164 0 0 0 2873 2172 5 MP stroke 0 461 1104 396 2 MP stroke 37 0 1086 396 2 MP stroke 37 0 1086 857 2 MP stroke 0 456 1310 1338 2 MP stroke 37 0 1291 1338 2 MP stroke 37 0 1291 1794 2 MP stroke 0 29 1515 1823 2 MP stroke 37 0 1497 1823 2 MP stroke 37 0 1497 1852 2 MP stroke 0 7 1721 2119 2 MP stroke 37 0 1703 2119 2 MP stroke 37 0 1703 2126 2 MP stroke 0 5 1927 2149 2 MP stroke 37 0 1908 2149 2 MP stroke 37 0 1908 2154 2 MP stroke 0 7 2132 2166 2 MP stroke 37 0 2114 2166 2 MP stroke 0 4 2338 2169 2 MP stroke 37 0 2319 2169 2 MP stroke 0 2 2544 2170 2 MP stroke 37 0 2525 2170 2 MP stroke 37 0 2525 2172 2 MP stroke 2749 2172 PD 37 0 2731 2172 2 MP stroke 37 0 2731 2172 2 MP stroke 2955 2172 PD 37 0 2936 2172 2 MP stroke 37 0 2936 2172 2 MP stroke gr 24 W 1104 626 PD 24 W 1310 1566 PD 24 W 1515 1837 PD 24 W 1721 2123 PD 24 W 1927 2151 PD 24 W 2132 2170 PD 24 W 2338 2171 PD 24 W 2544 2171 PD 24 W 2749 2172 PD 24 W 2955 2172 PD gs 899 389 2263 1784 MR c np gr %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 96 FMSR 2516 733 mt ( 155.9960) s 2543 843 mt ( 61.1640) s 2543 953 mt ( 33.7340) s 2570 1063 mt ( 4.9460) s 2570 1173 mt ( 2.0220) s 2570 1283 mt ( 0.1100) s 2570 1393 mt ( 0.0220) s 2570 1503 mt ( 0.0060) s 2678 1613 mt ( 0) s 2626 1723 mt ( 0) s end eplot %%EndObject epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 1008 w @beginspecial 73 @llx 416 @lly 285 @urx 583 @ury 1037 @rwi @setspecial %%BeginDocument: figures/c2a.eps %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /.automount/coeus.EECS.Berkeley.EDU/vol/vol0/eecs/epxing/matlab/haplo/daly/block_exp/sec1_new/exp4/c2a.eps %%CreationDate: 09/17/2003 23:24:16 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Extensions: CMYK %%Pages: 1 %%BoundingBox: 73 416 285 583 %%EndComments %%BeginProlog % 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 /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 /rc {rectclip} bdef /rf {rectfill} bdef % 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 /rotateMode 2 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS {/FontSize xstore 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 dup landscapeMode eq {pop -90 rotate} {rotateMode eq {90 rotate} if} ifelse} 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 /PDlw -1 def /W {/PDlw currentlinewidth def setlinewidth} def /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 {currentlinewidth 2 div 0 360 arc fill PDlw -1 eq not {PDlw w /PDlw -1 def} if} def /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 /stri 6 array def /dtri 6 array def /smat 6 array def /dmat 6 array def /tmat1 6 array def /tmat2 6 array def /dif 3 array def /asub {/ind2 exch def /ind1 exch def dup dup ind1 get exch ind2 get sub exch } bdef /tri_to_matrix { 2 0 asub 3 1 asub 4 0 asub 5 1 asub dup 0 get exch 1 get 7 -1 roll astore } bdef /compute_transform { dmat dtri tri_to_matrix tmat1 invertmatrix smat stri tri_to_matrix tmat2 concatmatrix } bdef /ds {stri astore pop} bdef /dt {dtri astore pop} bdef /db {2 copy /cols xdef /rows xdef mul dup 3 mul string currentfile exch readhexstring pop dup 0 3 index getinterval /rbmap xdef dup 2 index dup getinterval /gbmap xdef 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef /it {gs np dtri aload pop moveto lineto lineto cp c cols rows 8 compute_transform rbmap gbmap bbmap true 3 colorimage gr}bdef /il {newpath moveto lineto stroke}bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 73 416 285 583 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 664 340 2547 2005 MR c np 92 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 6918 5185 PR 6 w 0 1784 2262 0 0 -1784 899 2172 4 MP PP -2262 0 0 1784 2262 0 0 -1784 899 2172 5 MP stroke 4 w DO SO 6 w 0 sg 899 2172 mt 3161 2172 L 899 388 mt 3161 388 L 899 2172 mt 899 388 L 3161 2172 mt 3161 388 L 899 2172 mt 3161 2172 L 899 2172 mt 899 388 L 1104 2172 mt 1104 2149 L 1104 389 mt 1104 411 L %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 120 FMSR 1071 2317 mt (1) s 1310 2172 mt 1310 2149 L 1310 389 mt 1310 411 L 1277 2317 mt (2) s 1515 2172 mt 1515 2149 L 1515 389 mt 1515 411 L 1482 2317 mt (3) s 1721 2172 mt 1721 2149 L 1721 389 mt 1721 411 L 1688 2317 mt (4) s 1927 2172 mt 1927 2149 L 1927 389 mt 1927 411 L 1894 2317 mt (5) s 2132 2172 mt 2132 2149 L 2132 389 mt 2132 411 L 2099 2317 mt (6) s 2338 2172 mt 2338 2149 L 2338 389 mt 2338 411 L 2305 2317 mt (7) s 2544 2172 mt 2544 2149 L 2544 389 mt 2544 411 L 2511 2317 mt (8) s 2749 2172 mt 2749 2149 L 2749 389 mt 2749 411 L 2716 2317 mt (9) s 2955 2172 mt 2955 2149 L 2955 389 mt 2955 411 L 2889 2317 mt (10) s 899 2172 mt 921 2172 L 3161 2172 mt 3138 2172 L 798 2216 mt (0) s 899 1676 mt 921 1676 L 3161 1676 mt 3138 1676 L 731 1720 mt (50) s 899 1181 mt 921 1181 L 3161 1181 mt 3138 1181 L 664 1225 mt (100) s 899 686 mt 921 686 L 3161 686 mt 3138 686 L 664 730 mt (150) s 899 2172 mt 3161 2172 L 899 388 mt 3161 388 L 899 2172 mt 899 388 L 3161 2172 mt 3161 388 L gs 899 389 2263 1784 MR c np /c8 { 0.000000 0.000000 0.562500 sr} bdef c8 0 1535 164 0 0 -1535 1022 2172 4 MP PP 0 sg -164 0 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b(of)h(our)523 4208 y(mo)r(del)j(as)f(a)g(building)h(blo)r(c)n(k)f(for) g(more)g(expressiv)n(e)f(mo)r(dels.)h(In)h(particular,)e(as)h(in)60 b([11)o(])523 4307 y(and)38 b([15)o(],)h(the)g(graphical)e(mo)r(del)i (formalism)e(naturally)h(accommo)r(dates)f(v)-5 b(arious)38 b(ex-)523 4407 y(tensions,)33 b(suc)n(h)h(as)f(segmen)n(tation)g(of)h (c)n(hromosomes)d(in)n(to)j(haplot)n(yp)r(e)f(blo)r(c)n(ks)g(and)h(the) 523 4506 y(inclusion)c(of)g(p)r(edigree)f(relationships.)g(The)h(Diric) n(hlet)g(pro)r(cess)f(parameterization)f(also)523 4606 y(pro)n(vides)h(a)g(natural)g(upgrade)g(path)h(for)g(the)g (consideration)e(of)i(ric)n(her)f(mo)r(dels;)h(in)g(par-)523 4706 y(ticular,)k(it)i(is)e(p)r(ossible)h(to)g(incorp)r(orate)e(more)h (elab)r(orate)f(base)h(measures)g Fm(G)h Fx(in)n(to)g(the)523 4805 y(Diric)n(hlet)h(pro)r(cess)f(framew)n(ork|the)f (coalescence-based)g(distribution)i(of)g([21)o(])g(w)n(ould)523 4905 y(b)r(e)28 b(an)f(in)n(teresting)g(c)n(hoice.)p eop %%Page: 11 11 11 10 bop 1587 448 a Fy(Bibliograph)l(y)565 880 y Fx([1])41 b(C.)48 b(E.)g(An)n(toniak.)98 b(Mixtures)48 b(of)g(Diric)n(hlet)g(pro) r(cesses)f(with)i(applications)e(to)694 980 y(Ba)n(y)n(esian)23 b(nonparametric)i(problems.)33 b Fo(A)n(nnals)27 b(of)i(Statistics)p Fx(,)d(2:1152{1174,)21 b(1973.)565 1083 y([2])41 b(J.M.)27 b(Bernardo)f(and)h(A.F.M.)i(Smith.)37 b Fo(Bayesian)32 b(The)l(ory)p Fx(.)38 b(Wiley)-7 b(,)28 b(1994.)565 1186 y([3])41 b(A.)26 b(Clark)f(et)h(al.)33 b(Haplot)n(yp)r(e)26 b(structure)f(and)h(p)r(opulation)f(genetic)h(inferences)f(from)694 1286 y(n)n(ucleotide-sequence)31 b(v)-5 b(ariation)31 b(in)i(h)n(uman)g(lip)r(oprotein)f(lipase.)51 b Fo(A)n(m)33 b(J.)i(Human)694 1385 y(Genetics)p Fx(,)28 b(63:595{612,)23 b(1998.)565 1488 y([4])41 b(M.J.)24 b(Daly)g(et)g(al.)31 b(High-resolution)23 b(haplot)n(yp)r(e)g(structure)h(in)g(the)h(h)n (uman)f(genome.)694 1588 y Fo(Natur)l(e)29 b(Genetics)p Fx(,)f(29\(2\):229{232,)23 b(2001.)565 1691 y([5])41 b(M.D.)f(Escobar)d(and)j(M.)g(W)-7 b(est.)73 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4419 y(mo)r(dels.)36 b Fo(J.)30 b(Computational)h(and)g(Gr)l(aphic)l(al)h(Statistics)p Fx(,)27 b(9\(2\):249{256,)d(2000.)523 4522 y([17])41 b(T.)22 b(Niu,)g(S.)h(Qin,)e(X.)i(Xu,)f(and)g(J.)g(Liu.)28 b(Ba)n(y)n(esian)20 b(haplot)n(yp)r(e)h(inference)h(for)f(m)n(ultiple) 694 4622 y(link)n(ed)32 b(single)f(n)n(ucleotide)h(p)r(olymorphisms.)49 b Fo(A)n(m.)34 b(J.)g(Hum.Genet.)p Fx(,)e(70:157{169,)694 4722 y(2002.)523 4825 y([18])41 b(N.)h(P)n(atil)g(et)g(al.)80 b(Blo)r(c)n(ks)41 b(of)h(limited)h(haplot)n(yp)r(e)f(div)n(ersit)n(y)f (rev)n(ealed)g(b)n(y)g(high-)694 4924 y(resolution)17 b(scanning)h(of)g(h)n(uman)h(c)n(hromosome)d(21.)21 b Fo(Scienc)l(e)p Fx(,)e(294:1719{1723,)13 b(2001.)p eop %%Page: 12 12 12 11 bop 523 232 a Ft(12)523 448 y Fx([19])41 b(N.)d(J.)f(Risc)n(h.)67 b(Searc)n(hing)37 b(for)g(genetic)h(determinan)n(ts)f(in)h(the)g(new)g (millennium.)694 548 y Fo(Natur)l(e)p Fx(,)27 b(405\(6788\):847{56,)22 b(2000.)523 641 y([20])41 b(R.)27 b(Sac)n(hidanandam)f(et)h(al.)35 b(A)28 b(map)e(of)h(h)n(uman)g(genome)f(sequence)h(v)-5 b(ariation)26 b(con-)694 741 y(taining)37 b(1.42)g(million)h(single)f (n)n(ucleotide)g(p)r(olymorphisms.)67 b Fo(Natur)l(e)p Fx(,)37 b(291:1298{)694 841 y(2302,)25 b(2001.)523 934 y([21])41 b(M.)24 b(Stephens,)h(N.)g(Smith,)h(and)e(P)-7 b(.)24 b(Donnelly)-7 b(.)32 b(A)25 b(new)g(statistical)f(metho)r(d)h (for)e(hap-)694 1034 y(lot)n(yp)r(e)31 b(reconstruction)e(from)i(p)r (opulation)g(data.)47 b Fo(A)n(meric)l(an)33 b(Journal)g(of)h(Human)694 1133 y(Genetics)p Fx(,)28 b(68:978{989,)23 b(2001.)523 1227 y([22])41 b(Bafna)22 b(V,)i(Halldorsson)d(BV,)j(Sc)n(h)n(w)n(artz) d(R,)j(Clark)e(A)n(G,)h(and)g(Istrail)g(S.)30 b(Haplot)n(yp)r(es)694 1326 y(and)f(informativ)n(e)h(snp)g(selection)f(algorithms:)g(Don't)h (blo)r(c)n(k)g(out)g(information.)43 b(In)694 1426 y Fo(Pr)l(o)l(c)l(e)l(e)l(dings)29 b(of)h(the)e(7th)h(International)g (Confer)l(enc)l(e)h(on)e(Computational)i(Mole)l(cu-)694 1526 y(lar)g(Biolo)l(gy)i(\(RECOMB)e(2003\))p Fx(,)g(pages)c(19{27,)g (2002.)523 1619 y([23])41 b(C.)25 b(V)-7 b(en)n(ter)26 b(et)g(al.)33 b(The)26 b(sequence)f(of)g(the)h(h)n(uman)g(genome.)33 b Fo(Scienc)l(e)p Fx(,)26 b(291:1304{51,)694 1719 y(2001.)523 2011 y Fp(App)s(endix)523 2203 y(A)112 b(Mark)m(o)m(v)38 b(c)m(hain)f(Mon)m(te)h(Carlo)e(for)i(Haplot)m(yp)s(e)f(Inference)523 2377 y Fx(In)26 b(this)h(section,)e(w)n(e)h(describ)r(e)g(a)f(Gibbs)h (sampling)g(algorithm)f(for)g(exploring)g(the)h(p)r(oste-)523 2477 y(rior)k(distribution)h(under)g(our)f(mo)r(del,)h(including)h(the) f(laten)n(t)g(ancestral)f(p)r(o)r(ol.)h(W)-7 b(e)31 b(also)523 2576 y(presen)n(t)40 b(a)g(Metrop)r(olis-Hastings)e(v)-5 b(arian)n(t)39 b(of)h(this)h(algorithm)e(that)i(app)r(ears)e(to)h(mix) 523 2676 y(b)r(etter)28 b(in)g(practice.)523 2900 y Fi(A.1)95 b(A)32 b(Gibbs)f(sampling)e(algorithm)523 3041 y Fx(The)36 b(Gibbs)h(sampler)e(dra)n(ws)g(samples)g(of)h(eac)n(h)g(random)f(v)-5 b(ariable)35 b(from)h(a)g(predictiv)n(e)523 3141 y(distribution)30 b(of)g(the)g(v)-5 b(ariable)29 b(to)h(b)r(e)h(sampled)e(giv)n(en)g (\(previously)g(sampled\))h(v)-5 b(alues)30 b(of)523 3241 y(all)d(the)h(remaining)e(v)-5 b(ariables)26 b(of)i(the)f(mo)r (del.)h(The)f(v)-5 b(ariables)26 b(needed)i(in)g(our)e(algorithm)523 3340 y(are:)19 b Fm(c)713 3352 y Fw(i)736 3360 y Fh(t)768 3340 y Fx(,)h(the)g(index)g(of)g(the)g(ancestral)e(template)i(of)g(a)f (haplot)n(yp)r(e)h(instance)f Fm(t)h Fx(of)g(individual)523 3440 y Fm(i)p Fx(;)29 b Fm(a)648 3452 y Fw(k)q(;j)739 3440 y Fx(,)h(the)g(allele)f(pattern)g(at)g(the)h Fm(j)5 b Fx(-th)29 b(lo)r(cus)g(of)g(the)h Fm(k)s Fx(-th)g(ancestral)e (template;)h Fm(h)3276 3452 y Fw(i)3299 3460 y Fh(t)3327 3452 y Fw(;j)3382 3440 y Fx(,)523 3539 y(the)c Fm(t)p Fx(-th)f(allele)f(of)h(the)h(SNP)f(at)g(the)g Fm(j)5 b Fx(-th)24 b(lo)r(cus)g(of)g(individual)h Fm(i)p Fx(;)e(and)h Fm(g)2815 3551 y Fw(i;j)2893 3539 y Fx(,)g(the)h(genot)n(yp)r(e)523 3639 y(at)31 b(lo)r(cus)g Fm(j)37 b Fx(of)31 b(individual)g Fm(i)g Fx(\(the)h(only)f(observ)n(ed)f(v)-5 b(ariables)30 b(in)h(the)h(mo)r(del\).)g(All)g(other)523 3739 y(v)-5 b(ariables)31 b(in)i(the)g(mo)r(del|)p Fm(\022)i Fx(and)d Fm(\015)5 b Fx(|are)32 b(in)n(tegrated)f(out.)i(The)f(Gibbs)h(sampler)f (th)n(us)523 3838 y(estimates)27 b(the)h(v)-5 b(alues)28 b(of)f Fm(c)1409 3850 y Fw(i)1432 3858 y Fh(t)1464 3838 y Fx(,)h Fm(a)1559 3850 y Fw(k)q(;j)1678 3838 y Fx(and)f Fm(h)1887 3850 y Fw(i)1910 3858 y Fh(t)1938 3850 y Fw(;j)1993 3838 y Fx(.)648 3938 y(Conceptually)-7 b(,)44 b(the)h(Gibbs)g(sampler)f (alternates)f(b)r(et)n(w)n(een)i(t)n(w)n(o)f(coupled)g(stages.)523 4038 y(First,)29 b(giv)n(en)f(the)h(curren)n(t)f(v)-5 b(alues)28 b(of)h(the)g(hidden)g(haplot)n(yp)r(es,)f(w)n(e)h(sample)f (the)h Fm(c)3187 4050 y Fw(i)3210 4058 y Fh(t)3271 4038 y Fx(and)523 4137 y(subsequen)n(tly)24 b Fm(a)1056 4149 y Fw(k)q(;j)1147 4137 y Fx(,)g(whic)n(h)g(are)f(asso)r(ciated)g(with)i (the)f(Diric)n(hlet)g(pro)r(cess)f(prior.)g(Second,)523 4237 y(giv)n(en)28 b(the)g(curren)n(t)g(state)g(of)h(the)f(ancestral)f (p)r(o)r(ol)i(and)f(the)h(ancestral)e(template)i(assign-)523 4336 y(men)n(t)34 b(for)g(eac)n(h)f(individual,)h(w)n(e)f(sample)h(the) g Fm(h)2094 4348 y Fw(j;i)2164 4356 y Fh(t)2230 4336 y Fx(v)-5 b(ariables)33 b(in)h(the)g(basic)f(haplot)n(yp)r(e)523 4436 y(mo)r(del.)648 4536 y(In)27 b(the)h(\014rst)g(stage,)e(the)i (predictiv)n(e)f(distribution)h(of)g Fm(c)2411 4548 y Fw(i)2434 4556 y Fh(t)2493 4536 y Fx(is:)531 4687 y Fm(p)p Fx(\()p Fm(c)641 4699 y Fw(i)664 4707 y Fh(t)719 4687 y Fx(=)23 b Fm(k)30 b Fn(j)p Fi(c)945 4702 y Fv([)p Fj(\000)p Fw(i)1039 4710 y Fh(t)1067 4702 y Fv(])1090 4687 y Fm(;)14 b Fi(h)p Fm(;)g Fi(a)p Fx(\))24 b Fn(/)f Fm(p)p Fx(\()p Fm(c)1517 4699 y Fw(i)1540 4707 y Fh(t)1595 4687 y Fx(=)g Fm(k)30 b Fn(j)p Fi(c)1821 4702 y Fv([)p Fj(\000)p Fw(i)1915 4710 y Fh(t)1944 4702 y Fv(])1966 4687 y Fx(\))p Fm(p)p Fx(\()p Fm(h)2120 4699 y Fw(i)2143 4707 y Fh(t)2176 4687 y Fn(j)p Fm(a)2243 4699 y Fw(k)2284 4687 y Fm(;)14 b Fi(c)p Fm(;)g Fi(h)2453 4702 y Fv([)p Fj(\000)p Fw(i)2547 4710 y Fh(t)2575 4702 y Fv(])2598 4687 y Fx(\))1319 4879 y(=)1407 4762 y Fl(\032)1491 4775 y Fw(n)1532 4786 y Fk([)p Fe(\000)p Fh(i)1616 4797 y(t)1643 4786 y Fk(])p Fh(;k)p 1491 4804 225 4 v 1496 4852 a Fw(n)p Fj(\000)p Fv(1+)p Fw(\034)1725 4823 y Fm(p)p Fx(\()p Fm(h)1847 4835 y Fw(i)1870 4843 y Fh(t)1902 4823 y Fn(j)p Fm(a)1969 4835 y Fw(k)2010 4823 y Fm(;)g Fi(m)2128 4838 y Fv([)p Fj(\000)p Fw(i)2222 4846 y Fh(t)2248 4838 y Fv(])p Fw(;k)2328 4823 y Fx(\))2394 4811 y Ft(if)26 b Fg(k)e Ft(=)d Fg(c)2642 4825 y Fq(i)2664 4808 y Fe(0)2664 4853 y Fh(t)2687 4841 y Fe(0)2744 4811 y Ft(for)26 b(some)g Fg(i)3080 4779 y Fc(0)3080 4829 y Fq(t)3104 4815 y Fe(0)3152 4811 y Fb(6)p Ft(=)21 b Fg(i)3259 4819 y Fq(t)1579 4898 y Fw(\034)p 1491 4912 215 4 v 1491 4959 a(n)p Fj(\000)p Fv(1+)p Fw(\034)1729 4868 y Fl(P)1817 4955 y Fw(a)1853 4939 y Fe(0)1893 4930 y Fm(p)p Fx(\()p Fm(h)2015 4942 y Fw(i)2038 4950 y Fh(t)2070 4930 y Fn(j)p Fm(a)2137 4900 y Fj(0)2160 4930 y Fx(\))p Fm(p)p Fx(\()p Fm(a)2310 4900 y Fj(0)2334 4930 y Fx(\))2394 4918 y Ft(if)26 b Fg(k)e Fb(6)p Ft(=)d Fg(c)2642 4932 y Fq(i)2664 4915 y Fe(0)2664 4960 y Fh(t)2687 4948 y Fe(0)2744 4918 y Ft(for)26 b(all)h Fg(i)2994 4886 y Fc(0)2994 4936 y Fq(t)3018 4922 y Fe(0)3066 4918 y Fb(6)p Ft(=)21 b Fg(i)3173 4926 y Fq(t)3350 4879 y Fm(;)-116 b Fx(\(10\))p eop %%Page: 13 13 13 12 bop 3328 232 a Ft(13)523 448 y Fx(where)31 b([)p Fn(\000)p Fm(i)884 460 y Fw(t)912 448 y Fx(])h(denotes)f(the)g(set)g (of)h(indices)f(excluding)g Fm(i)2329 460 y Fw(t)2358 448 y Fx(;)g Fm(n)2462 460 y Fk([)p Fe(\000)p Fh(i)2546 471 y(t)2573 460 y Fk(])p Fh(;k)2677 448 y Fx(represen)n(ts)e(the)j(n)n (um-)523 548 y(b)r(er)37 b(of)h Fm(c)819 562 y Fw(i)842 542 y Fe(0)842 589 y Fh(t)865 577 y Fe(0)934 548 y Fx(for)f Fm(i)1100 518 y Fj(0)1100 571 y Fw(t)1125 554 y Fe(0)1191 548 y Fn(6)p Fx(=)i Fm(i)1324 560 y Fw(t)1391 548 y Fx(that)f(are)e (equal)h(to)h Fm(k)s Fx(;)f Fm(n)h Fx(represen)n(ts)e(the)h(total)h(n)n (um)n(b)r(er)f(of)523 648 y(instances)30 b(sampled)h(so)f(far;)g(and)h Fi(m)1713 663 y Fv([)p Fj(\000)p Fw(i)1807 671 y Fh(t)1833 663 y Fv(])p Fw(;k)1943 648 y Fx(denote)g(the)g Fm(m)g Fx(su\016cien)n(t)g(statistics)f(asso)r(ci-)523 747 y(ated)h(with)g (all)f(haplot)n(yp)r(e)h(instances)f(originating)f(from)h(ancestor)g Fm(k)s Fx(,)g(except)h Fm(h)3134 759 y Fw(i)3157 767 y Fh(t)3189 747 y Fx(.)g(This)523 847 y(expression)f(is)i(simply)g(Ba)n (y)n(es)e(theorem)i(with)g Fm(p)p Fx(\()p Fm(h)2169 859 y Fw(i)2192 867 y Fh(t)2224 847 y Fn(j)p Fm(a)2291 859 y Fw(k)2332 847 y Fm(;)14 b Fi(c)p Fm(;)g Fi(h)2501 862 y Fv([)p Fj(\000)p Fw(i)2595 870 y Fh(t)2623 862 y Fv(])p Fw(;)2666 847 y Fx(\))32 b(pla)n(ying)f(the)h(role)f(of)523 946 y(the)d(lik)n(eliho)r(o)r(d)f(and)h Fm(p)p Fx(\()p Fm(c)1316 958 y Fw(i)1339 966 y Fh(t)1394 946 y Fx(=)22 b Fm(k)31 b Fn(j)p Fi(c)1620 961 y Fv([)p Fj(\000)p Fw(i)1714 969 y Fh(t)1742 961 y Fv(])1765 946 y Fx(\))d(pla)n(ying)e(the)i(role)f (of)g(the)h(prior.)f(The)g(lik)n(eliho)r(o)r(d)523 1046 y Fm(p)p Fx(\()p Fm(h)645 1058 y Fw(i)668 1066 y Fh(t)700 1046 y Fn(j)p Fm(a)767 1058 y Fw(k)808 1046 y Fm(;)14 b Fi(m)926 1061 y Fv([)p Fj(\000)p Fw(i)1020 1069 y Fh(t)1046 1061 y Fv(])p Fw(;k)1125 1046 y Fx(\))23 b(is)e(obtained)h(b)n(y)f(in)n (tegrating)f(o)n(v)n(er)g(the)j(parameter)d Fm(\022)2858 1058 y Fw(k)2899 1046 y Fx(,)i(as)f(in)h(Eq.)f(\(7\).)648 1153 y(The)28 b(conditional)f(probabilit)n(y)g(for)g(a)h(newly)g(prop)r (osed)f(equiv)-5 b(alence)27 b(class)g Fm(k)k Fx(that)d(is)523 1253 y(not)k(p)r(opulated)f(b)n(y)h(an)n(y)e(previous)h(samples)g (requires)f(a)h(summation)g(o)n(v)n(er)f(all)h(p)r(ossible)523 1352 y(ancestors:)f Fm(p)p Fx(\()p Fm(h)1033 1364 y Fw(i)1056 1372 y Fh(t)1088 1352 y Fx(\))h(=)1245 1290 y Fl(P)1333 1377 y Fw(a)1369 1361 y Fe(0)1410 1352 y Fm(p)p Fx(\()p Fm(h)1532 1364 y Fw(i)1555 1372 y Fh(t)1587 1352 y Fn(j)p Fm(a)1654 1322 y Fj(0)1677 1352 y Fx(\))p Fm(p)p Fx(\()p Fm(a)1827 1322 y Fj(0)1850 1352 y Fx(\).)i(Since)f(the)g(gamma)f (function)h(do)r(es)g(not)f(fac-)523 1452 y(torize)25 b(o)n(v)n(er)e(lo)r(ci,)i(computing)g(this)h(summation)f(tak)n(es)f (time)i(that)f(is)g(exp)r(onen)n(tial)g(in)h(the)523 1552 y(n)n(um)n(b)r(er)c(of)g(lo)r(ci.)g(T)-7 b(o)22 b(skirt)g(this)h(problem)e(w)n(e)h(endo)n(w)g(eac)n(h)f(lo)r(cus)h (with)h(its)g(o)n(wn)e(m)n(utation)523 1651 y(parameter)h Fm(\022)955 1663 y Fw(k)q(;j)1046 1651 y Fx(,)h(with)h(all)f (parameters)e(admitting)i(the)h(same)e(b)r(eta)i(prior)e Fm(B)t(eta)p Fx(\()p Fm(\013)3180 1663 y Fw(h)3223 1651 y Fm(;)14 b(\014)3307 1663 y Fw(h)3350 1651 y Fx(\).)523 1751 y(This)26 b(giv)n(es)e(rise)h(to)h(a)f(closed-form)g(form)n(ula)g (for)g(the)h(summation)f(and)h(also)f(for)g(the)h(nor-)523 1851 y(malization)j(constan)n(t)h(in)g(Eq.)g(\(10\).)g(It)g(is)g(also,) f(arguably)-7 b(,)29 b(a)g(more)h(accurate)f(re\015ection)523 1950 y(of)f(realit)n(y)-7 b(.)648 2057 y(No)n(w)40 b(w)n(e)g(need)h(to) g(sample)f(the)h(ancestor)f(template)h Fm(a)2498 2069 y Fw(k)2538 2057 y Fx(,)g(where)g Fm(k)i Fx(is)e(the)g(newly)523 2157 y(sampled)27 b(ancestor)f(index)h(for)f Fm(c)1559 2169 y Fw(i)1582 2177 y Fh(t)1615 2157 y Fx(.)h(When)g Fm(k)k Fx(is)c(not)g(equal)f(to)h(an)n(y)g(other)f(existing)h(index)523 2257 y Fm(c)559 2271 y Fw(i)582 2251 y Fe(0)582 2298 y Fh(t)605 2286 y Fe(0)637 2257 y Fx(,)j(a)f(v)-5 b(alue)30 b(for)f Fm(a)1151 2269 y Fw(k)1222 2257 y Fx(needs)g(to)h(b)r(e)g(c)n (hosen)f(from)h Fm(p)p Fx(\()p Fm(A)p Fn(j)p Fm(h)2343 2269 y Fw(i)2366 2277 y Fh(t)2398 2257 y Fx(\),)g(the)h(p)r(osterior)d (distribution)523 2368 y(of)33 b Fm(A)h Fx(based)f(on)f(the)i(prior)e Fm(p)p Fx(\()p Fm(A)p Fx(\))i(and)f(the)h(single)f(dep)r(enden)n(t)g (haplot)n(yp)r(e)g Fm(h)3034 2380 y Fw(i)3057 2388 y Fh(t)3089 2368 y Fx(.)h(On)f(the)523 2468 y(other)g(hand,)g(if)h Fm(k)j Fx(is)c(an)g(equiv)-5 b(alence)33 b(class)g(p)r(opulated)h(b)n (y)f(previous)f(samples)h(of)g Fm(c)3304 2482 y Fw(i)3327 2462 y Fe(0)3327 2509 y Fh(t)3350 2497 y Fe(0)3382 2468 y Fx(,)523 2579 y(w)n(e)c(dra)n(w)g(a)g(new)g(v)-5 b(alue)30 b(of)f Fm(a)1455 2591 y Fw(k)1525 2579 y Fx(from)h Fm(p)p Fx(\()p Fm(A)p Fn(j)p Fm(h)1931 2591 y Fw(i)1954 2599 y Fh(t)1986 2579 y Fm(;)43 b Fx(s.t.)30 b Fm(c)2229 2591 y Fw(i)2252 2599 y Fh(t)2311 2579 y Fx(=)25 b Fm(k)s Fx(\).)30 b(If)g(after)f(a)g(new)h(sample)f(of)523 2679 y Fm(c)559 2691 y Fw(i)582 2699 y Fh(t)614 2679 y Fx(,)h(a)f(template)h (is)g(no)f(longer)g(asso)r(ciated)f(with)j(an)n(y)e(haplot)n(yp)r(e)g (instance,)g(w)n(e)h(remo)n(v)n(e)523 2779 y(this)j(template)g(from)f (the)h(p)r(o)r(ol.)f(The)h(predictiv)n(e)f(distribution)h(for)f(this)h (Gibbs)f(step)h(is)523 2878 y(therefore:)570 3162 y Fg(p)p Ft(\()p Fg(a)680 3171 y Fq(k)q(;j)764 3162 y Fb(j)p Fg(h)829 3170 y Fq(i)851 3178 y Fh(t)879 3170 y Fq(;j)956 3162 y Ft(s.t.)27 b Fg(c)1118 3170 y Fq(i)1140 3178 y Fh(t)1193 3162 y Ft(=)21 b Fg(k)r Ft(\))h Fb(/)570 3219 y Fa(8)570 3286 y(>)570 3308 y(>)570 3331 y(>)570 3353 y(<)570 3488 y(>)570 3510 y(>)570 3533 y(>)570 3555 y(:)649 3311 y Fg(p)p Ft(\()p Fg(h)762 3319 y Fq(i)784 3327 y Fh(t)812 3319 y Fq(;j)863 3311 y Fb(j)p Fg(a)925 3320 y Fq(k)q(;j)1011 3311 y Ft(\))f(=)1143 3251 y Fa(\000)1248 3276 y Fq(\013)1288 3288 y Fh(h)p 1188 3294 199 4 v 1188 3336 a Fq(\013)1228 3348 y Fh(h)1267 3336 y Fu(+)p Fq(\014)1349 3348 y Fh(h)1396 3251 y Fa(\001)1432 3268 y Fu(1\()p Fq(h)1522 3279 y Fh(i)1544 3290 y(t)1572 3279 y(;j)1622 3268 y Fu(=)p Fq(a)1703 3280 y Fh(k)q(;j)1784 3268 y Fu(\))1813 3251 y Fa(\000)2051 3276 y Fq(\014)2086 3288 y Fh(h)p 1858 3294 460 4 v 1858 3336 a Fu(\()p Fc(j)p Fq(B)r Fc(j\000)p Fu(1\)\()p Fq(\013)2134 3348 y Fh(h)2173 3336 y Fu(+)p Fq(\014)2255 3348 y Fh(h)2293 3336 y Fu(\))2327 3251 y Fa(\001)2362 3268 y Fu(1\()p Fq(h)2452 3279 y Fh(i)2474 3290 y(t)2502 3279 y(;j)2552 3268 y Fc(6)p Fu(=)p Fq(a)2633 3280 y Fh(k)q(;j)2715 3268 y Fu(\))2820 3265 y Ft(if)37 b Fg(k)i Ft(is)e(not)g(previ-)2820 3357 y(ously)18 b(instan)n(tiated) 649 3556 y Fg(p)p Ft(\()p Fg(h)762 3564 y Fq(i)784 3572 y Fh(t)812 3564 y Fq(;j)889 3556 y Ft(s.t.)26 b Fg(c)1050 3564 y Fq(i)1072 3572 y Fh(t)1126 3556 y Ft(=)21 b Fg(k)r Fb(j)p Fg(a)1311 3565 y Fq(k)q(;j)1396 3556 y Ft(\))h(=)1577 3507 y Fq(\000)8 b Fu(\()p Fq(\013)1686 3519 y Fh(h)1725 3507 y Fu(+)p Fq(m)1826 3519 y Fh(k)q(;j)1908 3507 y Fu(\))p Fq(\000)g Fu(\()p Fq(\014)2036 3519 y Fh(h)2074 3507 y Fu(+)p Fq(m)2175 3486 y Fe(0)2175 3524 y Fh(k)q(;j)2258 3507 y Fu(\))p 1539 3539 782 4 v 1539 3615 a Fq(\000)g Fu(\()p Fq(\013)1648 3627 y Fh(h)1686 3615 y Fu(+)p Fq(\014)1768 3627 y Fh(h)1806 3615 y Fu(+)p Fq(n)1891 3627 y Fh(k)1927 3615 y Fu(\))p Fc(\001)p Fu(\()p Fc(j)p Fq(B)r Fc(j\000)p Fu(1\))2182 3579 y Fh(m)2233 3565 y Fe(0)2233 3602 y Fh(k)q(;j)2820 3517 y Ft(if)37 b Fg(k)i Ft(is)e(previously)2820 3609 y(instan)n(tiated)3487 3439 y Fg(;)3407 3717 y Ft(\(11\))561 3910 y Fx(where)h Fm(m)885 3922 y Fw(k)q(;j)1015 3910 y Fx(\(resp)r(ectiv)n(ely)-7 b(,)38 b Fm(m)1600 3879 y Fj(0)1600 3933 y Fw(k)q(;j)1691 3910 y Fx(\))h(is)f(the)g(n)n(um)n(b) r(er)g(of)h(allelic)f(instances)f(originated)523 4009 y(from)26 b(ancestor)e Fm(k)29 b Fx(at)d(lo)r(cus)f Fm(j)31 b Fx(that)c(are)e(iden)n(tical)g(to)h(\(resp)r(ectiv)n(ely)-7 b(,)26 b(di\013eren)n(t)g(from\))g(the)523 4109 y(ancestor,)g(when)i (the)g(ancestor)e(has)h(the)h(pattern)g Fm(a)2193 4121 y Fw(k)q(;j)2284 4109 y Fx(.)648 4208 y(W)-7 b(e)27 b(no)n(w)g(pro)r (ceed)g(to)g(the)h(second)f(sampling)f(stage,)h(in)g(whic)n(h)h(w)n(e)f (sample)g(the)g(hap-)523 4299 y(lot)n(yp)r(es)38 b Fm(h)866 4311 y Fw(i)889 4319 y Fh(t)921 4299 y Fx(.)g(W)-7 b(e)39 b(sample)f(eac)n(h)f Fm(h)1669 4311 y Fw(i)1692 4319 y Fh(t)1720 4311 y Fw(;j)1775 4299 y Fx(,)h(for)g(all)g Fm(j;)14 b(i;)g(t)p Fx(,)38 b(sequen)n(tially)f(according)g(to)h(the) 523 4390 y(follo)n(wing)27 b(predictiv)n(e)g(distribution:)539 4575 y Fg(p)p Ft(\()p Fg(h)652 4583 y Fq(i)674 4591 y Fh(t)701 4583 y Fq(;j)753 4575 y Fb(j)p Fr(h)823 4587 y Fu([)p Fc(\000)p Fu(\()p Fq(i;j)s Fu(\)])1028 4575 y Fg(;)13 b(h)1106 4583 y Fq(i)1130 4590 y Fk(\026)1128 4598 y Fh(t)1156 4583 y Fq(;j)1207 4575 y Fg(;)g Fr(c)p Fg(;)h Fr(a)p Fg(;)f Fr(g)q Ft(\))22 b Fb(/)f Fg(p)p Ft(\()p Fg(g)1676 4583 y Fq(i)1701 4575 y Fb(j)p Fg(h)1766 4583 y Fq(i)1788 4591 y Fh(t)1816 4583 y Fq(;j)1868 4575 y Fg(;)13 b(h)1946 4583 y Fq(i)1970 4590 y Fk(\026)1968 4598 y Fh(t)1996 4583 y Fq(;j)2047 4575 y Fg(;)g Fr(u)2130 4587 y Fu([)p Fc(\000)p Fu(\()p Fq(i;j)s Fu(\)])2335 4575 y Ft(\))p Fg(p)p Ft(\()p Fg(h)2478 4583 y Fq(i)2500 4591 y Fh(t)2527 4583 y Fq(;j)2579 4575 y Fb(j)p Fg(a)2641 4584 y Fq(k)q(;j)2726 4575 y Fg(;)g Fr(m)2834 4587 y Fu([)p Fc(\000)p Fu(\()p Fq(i)2946 4595 y Fh(t)2973 4587 y Fq(;j)s Fu(\)])p Fq(;k)3119 4575 y Ft(\))562 4746 y(=)23 b Fg(R)703 4754 y Fq(g)749 4698 y Fg(\000)11 b Ft(\()p Fg(\013)886 4706 y Fq(g)939 4698 y Ft(+)17 b Fg(u)p Ft(\))p Fg(\000)11 b Ft(\()p Fg(\014)1221 4706 y Fq(g)1274 4698 y Ft(+)17 b(\()p Fg(u)1425 4666 y Fc(0)1464 4698 y Ft(+)g Fg(u)1585 4666 y Fc(00)1626 4698 y Ft(\)\))p 749 4729 937 4 v 938 4797 a Fg(\000)11 b Ft(\()p Fg(\013)1075 4805 y Fq(g)1128 4797 y Ft(+)17 b Fg(\014)1248 4805 y Fq(g)1301 4797 y Ft(+)g Fg(I)6 b(J)h Ft(\))1696 4746 y([)p Fg(\026)1763 4754 y Fu(1)1798 4746 y Ft(])1819 4710 y Fq(u)1856 4690 y Fe(0)1882 4746 y Ft([)p Fg(\026)1949 4754 y Fu(2)1985 4746 y Ft(])2006 4710 y Fq(u)2043 4690 y Fe(00)2105 4746 y Fb(\002)16 b Fg(R)2239 4755 y Fq(h)2368 4692 y Fg(\000)11 b Ft(\()p Fg(\013)2505 4701 y Fq(h)2562 4692 y Ft(+)17 b Fg(m)2707 4701 y Fq(j;k)2789 4692 y Ft(\))p Fg(\000)11 b Ft(\()p Fg(\014)2950 4701 y Fq(h)3007 4692 y Ft(+)17 b Fg(m)3152 4660 y Fc(0)3152 4707 y Fq(k)q(;j)3237 4692 y Ft(\))p 2290 4729 1056 4 v 2290 4816 a Fg(\000)11 b Ft(\()p Fg(\013)2427 4825 y Fq(h)2484 4816 y Ft(+)17 b Fg(\014)2604 4825 y Fq(h)2661 4816 y Ft(+)g Fg(n)2784 4825 y Fq(k)2822 4816 y Ft(\))g Fb(\001)h Ft(\()p Fb(j)p Fg(B)t Fb(j)f(\000)g Ft(1\))3204 4779 y Fq(m)3258 4762 y Fe(0)3258 4800 y Fh(k)q(;j)3355 4746 y Fg(;)3268 4924 y Ft(\(12\))p eop %%Page: 14 14 14 13 bop 523 232 a Ft(14)541 448 y Fx(where)18 b([)p Fn(\000)p Fx(\()p Fm(i)921 460 y Fw(t)950 448 y Fm(;)c(j)5 b Fx(\)])19 b(denotes)f(the)h(set)f(of)g(indices)h(excluding)f(\()p Fm(i)2417 460 y Fw(t)2446 448 y Fm(;)c(j)5 b Fx(\))18 b(and)h Fm(m)2798 460 y Fw(k)q(;j)2912 448 y Fx(=)k Fm(m)3073 463 y Fv([)p Fj(\000)p Fv(\()p Fw(i)3193 471 y Fh(t)3220 463 y Fw(;j)s Fv(\)])p Fw(;k)3376 448 y Fx(+)523 548 y(1\()p Fm(h)645 560 y Fw(i)668 568 y Fh(t)696 560 y Fw(;j)773 548 y Fx(=)g Fm(a)905 560 y Fw(k)q(;j)996 548 y Fx(\))h(\(and)g(similarly)f(for)g(the)h(other)f(su\016cien)n(t)h (statistics\).)g(Note)g(that)f(during)523 648 y(eac)n(h)g(sampling)g (step,)g(w)n(e)g(do)g(not)h(ha)n(v)n(e)e(to)h(recompute)g(the)h Fm(\000)12 b Fx(\()p Fn(\001)p Fx(\),)24 b(b)r(ecause)f(the)h (su\016cien)n(t)523 747 y(statistics)d(are)f(either)h(not)g(going)e(to) i(c)n(hange)f(\(e.g.,)h(when)g(the)g(newly)g(sampled)g Fm(h)3086 759 y Fw(i)3109 767 y Fh(t)3137 759 y Fw(;j)3213 747 y Fx(is)g(the)523 847 y(same)29 b(as)g(the)h(old)g(sample\),)g(or)e (only)i(going)e(to)i(c)n(hange)f(b)n(y)g(one)g(\(e.g.,)h(when)g(the)g (newly)523 946 y(sampled)k Fm(h)901 958 y Fw(i)924 966 y Fk(1)956 958 y Fw(;j)1045 946 y Fx(results)f(in)h(a)g(c)n(hange)e(of) i(the)g(allele\).)g(In)g(suc)n(h)f(cases)g(the)h(new)g(gamma)523 1046 y(function)28 b(can)f(b)r(e)h(easily)f(up)r(dated)h(from)g(the)g (old)f(one.)523 1295 y Fi(A.2)95 b(Metrop)s(olis-Hasting)29 b(sampling)g(algorithm)523 1461 y Fx(Note)24 b(that)g(for)g(a)f(long)h (list)g(of)g(lo)r(ci,)g(a)f(uniform)h Fm(p)p Fx(\()p Fm(A)p Fx(\))h(of)f(all)f(p)r(ossible)h(ancestral)f(template)523 1561 y(patterns)e(will)h(render)f(the)h(probabilit)n(y)e(of)i(sampling) f(a)g(new)h(ancestor)e(in\014nitesimal,)i(due)523 1660 y(to)i(the)h(small)g(v)-5 b(alue)24 b(of)g(the)h(smo)r(othed)g (marginal)e(lik)n(eliho)r(o)r(d)h(of)g(an)n(y)g(haplot)n(yp)r(e)g (pattern)523 1760 y Fm(h)571 1772 y Fw(i)594 1780 y Fh(t)626 1760 y Fx(,)k(as)f(computed)h(from)f(Eq.)g(\(10\).)g(This)h(could)f (result)g(in)h(slo)n(w)f(mixing.)648 1860 y(An)f(alternativ)n(e)f (sampling)g(strategy)g(is)h(to)g(use)g(a)f(partial)h(Gibbs)g(sampling)f (strategy)523 1959 y(with)f(the)g(follo)n(wing)e(Metrop)r(olis-Hasting) g(up)r(dates.)i(F)-7 b(or)22 b(the)i(prop)r(osal)e(distribution)i(for) 523 2059 y(the)k(equiv)-5 b(alence)27 b(class)g(of)g Fm(h)1441 2071 y Fw(i)1464 2079 y Fh(t)1524 2059 y Fx(w)n(e)g(use:)846 2316 y Fm(q)s Fx(\()p Fm(c)954 2282 y Fj(\003)954 2337 y Fw(i)977 2345 y Fh(t)1032 2316 y Fx(=)c Fm(k)s Fn(j)p Fm(c)1225 2331 y Fv([)p Fj(\000)p Fw(i)1319 2339 y Fh(t)1346 2331 y Fv(])1369 2316 y Fx(\))h(=)1512 2174 y Fl(\()1601 2208 y Fw(n)1642 2219 y Fk([)p Fe(\000)p Fh(i)1726 2230 y(t)1753 2219 y Fk(])p Fh(;k)p 1601 2237 225 4 v 1606 2285 a Fw(n)p Fj(\000)p Fv(1+)p Fw(\034)1918 2256 y Fx(:)111 b(if)56 b Fm(k)26 b Fx(=)c Fm(c)2348 2270 y Fw(i)2371 2251 y Fe(0)2371 2297 y Fh(t)2394 2285 y Fe(0)2454 2256 y Fx(for)27 b(some)g Fm(i)2818 2226 y Fj(0)2818 2279 y Fw(t)2843 2262 y Fe(0)2892 2256 y Fn(6)p Fx(=)c Fm(i)3009 2226 y Fw(t)1700 2334 y(\034)p 1611 2348 215 4 v 1611 2396 a(n)p Fj(\000)p Fv(1+)p Fw(\034)1918 2367 y Fx(:)111 b(if)56 b Fm(k)26 b Fn(6)p Fx(=)c Fm(c)2348 2381 y Fw(i)2371 2361 y Fe(0)2371 2408 y Fh(t)2394 2396 y Fe(0)2454 2367 y Fx(for)27 b(all)g Fm(i)2725 2337 y Fj(0)2725 2390 y Fw(t)2750 2373 y Fe(0)2799 2367 y Fn(6)p Fx(=)c Fm(i)2916 2379 y Fw(t)3049 2316 y Fm(:)185 b Fx(\(13\))523 2574 y(Then)29 b(w)n(e)f(sample)g Fm(a)1186 2586 y Fh(c)1213 2569 y Fe(\003)1213 2605 y Fh(i)1235 2616 y(t)1300 2574 y Fx(sequen)n(tially)g(according)e(to)j(Eq.)f(\(11\).)g(F)-7 b(or)28 b(target)g(distribution)523 2687 y Fm(p)p Fx(\()p Fm(c)633 2699 y Fw(i)656 2707 y Fh(t)711 2687 y Fx(=)23 b Fm(k)s Fn(j)p Fi(c)910 2702 y Fv([)p Fj(\000)p Fw(i)1004 2710 y Fh(t)1032 2702 y Fv(])1055 2687 y Fm(;)14 b Fi(h)p Fm(;)g Fi(a)p Fx(\),)20 b(the)g(prop)r(osal)d(factor)i(cancels)f(when)i (computing)f(the)h(acceptance)523 2787 y(probabilit)n(y)27 b Fm(\030)t Fx(,)g(lea)n(ving:)1273 3031 y Fm(\030)t Fx(\()p Fm(c)1381 2997 y Fj(\003)1381 3052 y Fw(i)1404 3060 y Fh(t)1436 3031 y Fm(;)14 b(c)1509 3043 y Fw(i)1532 3051 y Fh(t)1564 3031 y Fx(\))24 b(=)e(min)1860 2939 y Fl(h)1899 3031 y Fx(1)p Fm(;)1987 2956 y(p)p Fx(\()p Fm(h)2109 2968 y Fw(i)2132 2976 y Fh(t)2164 2956 y Fn(j)p Fm(a)2231 2968 y Fh(c)2258 2952 y Fe(\003)2258 2988 y Fh(i)2280 2999 y(t)2316 2956 y Fm(;)14 b Fi(h)2406 2971 y Fv([)p Fj(\000)p Fw(i)2500 2979 y Fh(t)2528 2971 y Fv(])2551 2956 y Fx(\))p 1987 3012 596 4 v 1987 3088 a Fm(p)p Fx(\()p Fm(h)2109 3100 y Fw(i)2132 3108 y Fh(t)2164 3088 y Fn(j)p Fm(a)2231 3100 y Fh(c)2258 3113 y(i)2280 3124 y(t)2316 3088 y Fm(;)g Fi(h)2406 3103 y Fv([)p Fj(\000)p Fw(i)2500 3111 y Fh(t)2528 3103 y Fv(])2551 3088 y Fx(\))2593 2939 y Fl(i)2632 3031 y Fm(:)602 b Fx(\(14\))648 3274 y(In)38 b(practice,)g(w)n(e)g(found)g(that)h(the)f(ab)r(o)n(v)n(e)f(mo) r(di\014cation)h(to)g(the)h(Gibbs)g(sampling)523 3374 y(algorithm)23 b(leads)g(to)g(substan)n(tial)h(impro)n(v)n(emen)n(t)e (of)i(e\016ciency)f(for)h(long)f(haplot)n(yp)r(e)g(lists,)523 3474 y(whereas)f(for)g(short)g(lists,)h(the)g(Gibbs)g(sampler)f (remains)g(b)r(etter)i(due)f(to)g(the)g(high)g(\(100\045\))523 3573 y(acceptance)k(rate.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF