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y(distrib)n(utions)20 b(of)h(the)h(form)e(\(3\).)27 b(The)21 b(gradients)f(of)h(the)h(log)f (lik)o(elihood)e(bound)h(with)h(respect)g(to)h(the)450 4859 y(model)j(parameters)g Fm(J)1121 4871 y Fg(ij)1206 4859 y Fp(and)g Fm(b)1388 4871 y Fg(i)1442 4859 y Fp(can)h(then)f(be)h (e)n(v)n(aluated)f(for)g(this)i(pattern)e(and)g(used)h(to)g(adapt)g (the)450 4950 y(parameters)19 b(by)h(taking)f(a)i(step)f(in)h(the)f (gradient)f(direction.)p eop %%Page: 3 3 3 2 bop 450 83 a Fo(2)99 b(Mixtur)n(es)450 286 y Fp(Although)20 b(mean)h(\002eld)h(theory)f(leads)g(to)h(a)h(tractable)e(algorithm,)f (the)i(assumption)e(of)i(a)g(completely)450 377 y(f)o(actorized)e (distrib)n(ution)f(is)j(a)g(v)o(ery)d(strong)i(one.)26 b(In)21 b(particular)m(,)e(such)i(representations)e(can)i(only)f(ef-) 450 468 y(fecti)n(v)o(ely)g(model)g(posterior)g(distrib)n(utions)g (which)g(are)h(uni-modal.)26 b(Since)21 b(we)g(e)o(xpect)f(multi-modal) 450 560 y(distrib)n(utions)29 b(to)g(be)h(common,)f(we)h(therefore)e (seek)i(a)g(richer)e(class)j(of)e(approximating)d(distrib)n(u-)450 651 y(tions)16 b(which)f(ne)n(v)o(ertheless)g(remain)g(computationally) e(tractable.)23 b(One)15 b(approach)f(\(Saul)i(and)f(Jordan,)450 742 y(1996\))g(is)k(to)e(identify)f(a)i(tractable)e(substructure)g (within)h(the)g(model)f(\(for)h(e)o(xample)e(a)j(chain\))e(and)h(then) 450 834 y(to)k(use)g(mean)f(\002eld)h(techniques)f(to)g(approximate)f (the)i(remaining)e(interactions.)25 b(This)c(can)g(be)g(ef)n(fec-)450 925 y(ti)n(v)o(e)i(where)f(the)g(additional)g(interactions)g(are)g (weak)h(or)f(are)h(fe)n(w)g(in)f(number)m(,)f(b)n(ut)i(will)h(again)d (pro)o(v)o(e)450 1016 y(to)k(be)f(restricti)n(v)o(e)f(for)h(more)g (general,)g(densely)f(connected)g(netw)o(orks.)36 b(W)-7 b(e)26 b(therefore)c(consider)i(an)450 1107 y(alternati)n(v)o(e)19 b(approach)1125 1077 y Fd(1)1176 1107 y Fp(based)h(on)g(mixture)f (representations)f(of)i(the)g(form)1448 1352 y Fm(Q)1514 1364 y Ff(mix)1626 1352 y Fl(\()p Fm(H)7 b Fl(\))23 b(=)1914 1248 y Fg(M)1889 1273 y Fi(X)1877 1449 y Fg(m)p Ff(=1)2034 1352 y Fm(\013)2087 1364 y Fg(m)2150 1352 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))901 b Fp(\(5\))450 1593 y(in)23 b(which)g(each)f(of)h(the)g(components)e Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))23 b Fp(is)h(itself)g(gi)n(v)o(en) d(by)i(a)g(mean-\002eld)f(distrib)n(ution,)g(for)450 1685 y(e)o(xample)i(of)h(the)g(form)g(\(3\))f(in)i(the)f(case)h(of)f (sigmoid)f(belief)h(netw)o(orks.)40 b(Substituting)24 b(\(5\))g(into)i(the)450 1776 y(lo)n(wer)20 b(bound)e(\(1\))i(we)g (obtain)1218 1952 y Fk(F)8 b Fl([)p Fm(Q)1375 1964 y Ff(mix)1487 1952 y Fl(])23 b(=)1621 1874 y Fi(X)1652 2048 y Fg(m)1755 1952 y Fm(\013)1808 1964 y Fg(m)1871 1952 y Fk(F)8 b Fl([)p Fm(Q)p Fl(\()p Fm(H)f Fk(j)p Fm(m)p Fl(\)])18 b(+)g Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))672 b Fp(\(6\))450 2193 y(where)25 b Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))25 b Fp(is)i(the)e(mutual)f(information)f(between)h(the)i (component)c(label)j Fm(m)h Fp(and)f(the)g(set)h(of)450 2284 y(hidden)19 b(v)n(ariables)g Fm(H)7 b Fp(,)20 b(and)g(is)h(gi)n(v) o(en)e(by)1173 2495 y Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))23 b(=)1577 2416 y Fi(X)1607 2590 y Fg(m)1714 2416 y Fi(X)1710 2598 y Fh(f)p Fg(H)t Fh(g)1851 2495 y Fm(\013)1904 2507 y Fg(m)1967 2495 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))14 b(ln)2384 2439 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))p 2376 2476 319 4 v 2376 2552 a Fm(Q)2442 2564 y Ff(mix)2554 2552 y Fl(\()p Fm(H)g Fl(\))2704 2495 y Fm(:)626 b Fp(\(7\))450 2752 y(The)27 b(\002rst)i(term)e(in)h(\(6\))f(is)i(simply)e(a)h(con)m(v)o(e) o(x)d(combination)h(of)h(standard)g(mean-\002eld)f(bounds)g(and)450 2843 y(hence)h(is)i(no)f(greater)f(than)h(the)g(lar)o(gest)g(of)g (these)g(and)g(so)g(gi)n(v)o(es)g(no)g(useful)f(impro)o(v)o(ement)e(o)o (v)o(er)i(a)450 2934 y(single)i(mean-\002eld)f(distrib)n(ution.)51 b(It)29 b(is)h(the)f(second)g(term,)i(i.e.)e(the)g(mutual)f (information,)h(which)450 3026 y(characterises)i(the)h(gain)f(in)h (using)f(mixtures.)58 b(Since)32 b Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))45 b Fk(\025)f Fl(0)p Fp(,)34 b(the)e(mutual)f (information)450 3117 y(increases)15 b(the)h(v)n(alue)f(of)g(the)h (bound)e(and)h(hence)f(impro)o(v)o(es)g(the)i(approximation)c(to)k(the) f(true)h(posterior)-5 b(.)450 3334 y Fq(2.1)82 b(Smoothing)19 b(Distrib)n(utions)450 3509 y Fp(As)28 b(it)h(stands,)g(the)f(mutual)f (information)e(itself)j(in)m(v)n(olv)o(es)f(a)h(summation)e(o)o(v)o(er) g(the)i(con\002gurations)450 3600 y(of)f(hidden)f(v)n(ariables,)j(and)e (so)g(is)i(computationally)c(intractable.)46 b(In)27 b(order)f(to)i(be)f(able)h(to)f(treat)h(it)450 3692 y(ef)n(\002ciently) e(we)i(\002rst)f(introduce)f(a)h(set)h(of)f(`smoothing')d(distrib)n (utions)i Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))p Fp(,)29 b(and)e(re)n(write)g(the)450 3783 y(mutual)19 b(information)f(\(7\))i(in)g(the)g(form)844 3959 y Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))84 b(=)1368 3880 y Fi(X)1399 4055 y Fg(m)1505 3880 y Fi(X)1502 4062 y Fh(f)p Fg(H)t Fh(g)1643 3959 y Fm(\013)1696 3971 y Fg(m)1759 3959 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))14 b(ln)g Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))18 b Fk(\000)2559 3880 y Fi(X)2590 4055 y Fg(m)2693 3959 y Fm(\013)2746 3971 y Fg(m)2823 3959 y Fl(ln)c Fm(\013)2959 3971 y Fg(m)1368 4239 y Fk(\000)1447 4160 y Fi(X)1477 4334 y Fg(m)1584 4160 y Fi(X)1580 4342 y Fh(f)p Fg(H)t Fh(g)1721 4239 y Fm(\013)1774 4251 y Fg(m)1837 4239 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))14 b(ln)2236 4122 y Fi(\032)2308 4183 y Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))p 2308 4220 300 4 v 2400 4296 a Fm(\013)2453 4308 y Fg(m)2628 4183 y Fm(Q)2694 4195 y Ff(mix)2806 4183 y Fl(\()p Fm(H)g Fl(\))p 2628 4220 319 4 v 2636 4296 a Fm(Q)p Fl(\()p Fm(H)g Fk(j)p Fm(m)p Fl(\))2956 4122 y Fi(\033)3033 4239 y Fm(:)297 b Fp(\(8\))450 4501 y(It)20 b(is)h(easily)g(v)o(eri\002ed)e (that)h(\(8\))f(is)i(equi)n(v)n(alent)e(to)h(\(7\))f(for)h(arbitrary)e Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))p Fp(.)25 b(W)-7 b(e)22 b(ne)o(xt)d(mak)o(e)h(use)g(of)450 4593 y(the)g(follo)n(wing)f(inequality)1522 4696 y Fk(\000)14 b Fl(ln)f Fm(x)24 b Fk(\025)f(\000)p Fm(\025x)c Fl(+)f(ln)c Fm(\025)k Fl(+)g(1)975 b Fp(\(9\))p 450 4780 1196 4 v 555 4835 a Fc(1)584 4867 y Fb(Here)21 b(we)f(outline)h(the)f(k)o(e)o(y) i(steps.)27 b(A)20 b(more)h(detailed)g(discussion)g(can)g(be)g(found)h (in)e(Jaakk)o(ola)i(and)f(Jordan)450 4950 y(\(1997\).)p eop %%Page: 4 4 4 3 bop 450 83 a Fp(to)29 b(replace)e(the)i(logarithm)e(in)h(the)h (third)f(term)g(in)h(\(8\))e(with)i(a)g(linear)f(function)f (\(conditionally)f(on)450 174 y(the)h(component)d(label)i Fm(m)p Fp(\).)45 b(This)26 b(yields)h(a)g(lo)n(wer)f(bound)f(on)h(the)h (mutual)e(information)g(gi)n(v)o(en)g(by)450 266 y Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))23 b Fk(\025)g Fm(I)890 278 y Fg(\025)934 266 y Fl(\()p Fm(m;)14 b(H)7 b Fl(\))21 b Fp(where)758 407 y Fm(I)794 419 y Fg(\025)839 407 y Fl(\()p Fm(m;)14 b(H)7 b Fl(\))83 b(=)1319 328 y Fi(X)1350 503 y Fg(m)1456 328 y Fi(X)1453 510 y Fh(f)p Fg(H)t Fh(g)1594 407 y Fm(\013)1647 419 y Fg(m)1710 407 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))14 b(ln)g Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))18 b Fk(\000)2510 328 y Fi(X)2541 503 y Fg(m)2644 407 y Fm(\013)2697 419 y Fg(m)2774 407 y Fl(ln)c Fm(\013)2910 419 y Fg(m)1319 653 y Fk(\000)1398 574 y Fi(X)1428 749 y Fg(m)1531 653 y Fm(\025)1579 665 y Fg(m)1660 574 y Fi(X)1657 756 y Fh(f)p Fg(H)t Fh(g)1797 653 y Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))p Fm(Q)2163 665 y Ff(mix)2275 653 y Fl(\()p Fm(H)g Fl(\))19 b(+)2517 574 y Fi(X)2548 749 y Fg(m)2651 653 y Fm(\013)2704 665 y Fg(m)2781 653 y Fl(ln)14 b Fm(\025)2912 665 y Fg(m)2994 653 y Fl(+)k(1)p Fm(:)170 b Fp(\(10\))450 875 y(W)m(ith)23 b Fm(I)672 887 y Fg(\025)716 875 y Fl(\()p Fm(m;)14 b(H)7 b Fl(\))23 b Fp(substituted)e(for)h Fm(I)7 b Fl(\()p Fm(m;)14 b(H)7 b Fl(\))23 b Fp(in)g(\(6\))f(we)g(again)g(obtain)f(a)i (rigorous)e(lo)n(wer)g(bound)g(on)450 966 y(the)f(true)g(log)g(lik)o (elihood)f(gi)n(v)o(en)f(by)1101 1105 y Fk(F)1161 1117 y Fg(\025)1204 1105 y Fl([)p Fm(Q)1293 1117 y Ff(mix)1405 1105 y Fl(\()p Fm(H)7 b Fl(\)])23 b(=)1679 1026 y Fi(X)1709 1200 y Fg(m)1813 1105 y Fm(\013)1866 1117 y Fg(m)1929 1105 y Fk(F)8 b Fl([)p Fm(Q)p Fl(\()p Fm(H)f Fk(j)p Fm(m)p Fl(\)])18 b(+)g Fm(I)2482 1117 y Fg(\025)2526 1105 y Fl(\()p Fm(m;)c(H)7 b Fl(\))p Fm(:)513 b Fp(\(11\))450 1304 y(The)23 b(summations)f(o)o(v)o(er)g(hidden)f(con\002gurations)g Fk(f)p Fm(H)7 b Fk(g)22 b Fp(in)i(\(10\))e(can)h(be)g(performed)d (analytically)i(if)450 1396 y(we)k(assume)h(that)f(the)g(smoothing)e (distrib)n(utions)h Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))27 b Fp(f)o(actorize.)42 b(In)26 b(particular)m(,)f(we)i(ha)n(v)o (e)e(to)450 1487 y(consider)19 b(the)h(follo)n(wing)f(tw)o(o)h (summations)f(o)o(v)o(er)g(hidden)g(v)n(ariable)g(con\002gurations)776 1555 y Fi(X)772 1737 y Fh(f)p Fg(H)t Fh(g)913 1634 y Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))p Fm(Q)p Fl(\()p Fm(H)g Fk(j)p Fm(k)s Fl(\))83 b(=)1718 1555 y Fi(Y)1760 1732 y Fg(i)1838 1555 y Fi(X)1866 1734 y Fg(h)1905 1742 y Fe(i)1972 1634 y Fm(R)q Fl(\()p Fm(h)2116 1646 y Fg(i)2143 1634 y Fk(j)p Fm(m)p Fl(\))p Fm(Q)p Fl(\()p Fm(h)2417 1646 y Fg(i)2445 1634 y Fk(j)p Fm(k)s Fl(\))2569 1587 y Ff(def)2583 1634 y Fl(=)36 b Fm(\031)2731 1646 y Fg(R;Q)2858 1634 y Fl(\()p Fm(m;)14 b(k)s Fl(\))234 b Fp(\(12\))652 1810 y Fi(X)648 1992 y Fh(f)p Fg(H)t Fh(g)789 1889 y Fm(Q)p Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))14 b(ln)g Fm(R)q Fl(\()p Fm(H)7 b Fk(j)p Fm(m)p Fl(\))83 b(=)1718 1810 y Fi(X)1767 1987 y Fg(i)1852 1810 y Fi(X)1879 1989 y Fg(h)1918 1997 y Fe(i)1986 1889 y Fm(Q)p Fl(\()p Fm(h)2132 1901 y Fg(i)2159 1889 y Fk(j)p Fm(m)p Fl(\))14 b(ln)g Fm(R)q Fl(\()p Fm(h)2528 1901 y Fg(i)2556 1889 y Fk(j)p Fm(m)p Fl(\))2707 1842 y Ff(def)2720 1889 y Fl(=)37 b Fm(H)7 b Fl(\()p Fm(Q)p Fk(k)p Fm(R)q Fk(j)p Fm(m)p Fl(\))p Fm(:)59 b Fp(\(13\))450 2105 y(W)-7 b(e)25 b(note)e(that)h(the)g(left)g(hand)e(sides)j(of)e(\(12\))g(and)g 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1533 1548 5 MP stroke c8 0 7 141 0 0 -7 1675 1548 4 MP PP 0 sg -141 0 0 7 141 0 0 -7 1675 1548 5 MP stroke c8 0 2 141 0 0 -2 1816 1548 4 MP PP 0 sg -141 0 0 2 141 0 0 -2 1816 1548 5 MP stroke c8 0 0 141 0 0 0 1957 1548 4 MP PP 0 sg -141 0 0 0 141 0 0 0 1957 1548 5 MP stroke c8 0 0 141 0 0 0 2098 1548 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2098 1548 5 MP stroke c8 0 0 141 0 0 0 2239 1548 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2239 1548 5 MP stroke c8 0 0 142 0 0 0 2380 1548 4 MP PP 0 sg -142 0 0 0 142 0 0 0 2380 1548 5 MP stroke c8 0 0 141 0 0 0 2522 1548 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2522 1548 5 MP stroke gr 1360 292 mt (5 components, mean: 0.011394) s 1 sg 0 1160 2260 0 0 -1160 3994 1548 4 MP PP -2260 0 0 1160 2260 0 0 -1160 3994 1548 5 MP stroke DO 4 w SO 6 w 0 sg 3994 1548 mt 6254 1548 L 3994 388 mt 6254 388 L 6254 1548 mt 6254 388 L 3994 1548 mt 3994 388 L 3994 1548 mt 6254 1548 L 3994 1548 mt 3994 388 L 3994 1548 mt 3994 1525 L 3994 388 mt 3994 411 L 3961 1694 mt (0) s 4559 1548 mt 4559 1525 L 4559 388 mt 4559 411 L 4443 1694 mt (0.02) s 5124 1548 mt 5124 1525 L 5124 388 mt 5124 411 L 5008 1694 mt (0.04) s 5689 1548 mt 5689 1525 L 5689 388 mt 5689 411 L 5573 1694 mt (0.06) s 6254 1548 mt 6254 1525 L 6254 388 mt 6254 411 L 6138 1694 mt (0.08) s 3994 1548 mt 4017 1548 L 6254 1548 mt 6231 1548 L 3893 1592 mt (0) s 3994 1161 mt 4017 1161 L 6254 1161 mt 6231 1161 L 3693 1205 mt (1000) s 3994 775 mt 4017 775 L 6254 775 mt 6231 775 L 3693 819 mt (2000) s 3994 388 mt 4017 388 L 6254 388 mt 6231 388 L 3693 432 mt (3000) s 3994 388 mt 6254 388 L 3994 1548 mt 6254 1548 L 3994 1548 mt 3994 388 L 6254 1548 mt 6254 388 L gs 3994 388 2261 1161 MR c np c8 0 295 141 0 0 -295 4065 1548 4 MP PP 0 sg -141 0 0 295 141 0 0 -295 4065 1548 5 MP stroke c8 0 867 141 0 0 -867 4206 1548 4 MP PP 0 sg -141 0 0 867 141 0 0 -867 4206 1548 5 MP stroke c8 0 539 141 0 0 -539 4347 1548 4 MP PP 0 sg -141 0 0 539 141 0 0 -539 4347 1548 5 MP stroke c8 0 178 142 0 0 -178 4488 1548 4 MP PP 0 sg -142 0 0 178 142 0 0 -178 4488 1548 5 MP stroke c8 0 41 141 0 0 -41 4630 1548 4 MP PP 0 sg -141 0 0 41 141 0 0 -41 4630 1548 5 MP stroke c8 0 9 141 0 0 -9 4771 1548 4 MP PP 0 sg -141 0 0 9 141 0 0 -9 4771 1548 5 MP stroke c8 0 3 141 0 0 -3 4912 1548 4 MP PP 0 sg -141 0 0 3 141 0 0 -3 4912 1548 5 MP stroke c8 0 0 142 0 0 0 5053 1548 4 MP PP 0 sg -142 0 0 0 142 0 0 0 5053 1548 5 MP stroke c8 0 1 141 0 0 -1 5195 1548 4 MP PP 0 sg -141 0 0 1 141 0 0 -1 5195 1548 5 MP stroke c8 0 0 141 0 0 0 5336 1548 4 MP PP 0 sg -141 0 0 0 141 0 0 0 5336 1548 5 MP stroke c8 0 0 141 0 0 0 5477 1548 4 MP PP 0 sg -141 0 0 0 141 0 0 0 5477 1548 5 MP stroke c8 0 0 142 0 0 0 5618 1548 4 MP PP 0 sg -142 0 0 0 142 0 0 0 5618 1548 5 MP stroke gr 4457 292 mt (4 components, mean: 0.012024) s 1 sg 0 1159 2259 0 0 -1159 898 3080 4 MP PP -2259 0 0 1159 2259 0 0 -1159 898 3080 5 MP stroke DO 4 w SO 6 w 0 sg 898 3080 mt 3157 3080 L 898 1921 mt 3157 1921 L 3157 3080 mt 3157 1921 L 898 3080 mt 898 1921 L 898 3080 mt 3157 3080 L 898 3080 mt 898 1921 L 898 3080 mt 898 3057 L 898 1921 mt 898 1944 L 865 3226 mt (0) s 1463 3080 mt 1463 3057 L 1463 1921 mt 1463 1944 L 1347 3226 mt (0.02) s 2028 3080 mt 2028 3057 L 2028 1921 mt 2028 1944 L 1912 3226 mt (0.04) s 2592 3080 mt 2592 3057 L 2592 1921 mt 2592 1944 L 2476 3226 mt (0.06) s 3157 3080 mt 3157 3057 L 3157 1921 mt 3157 1944 L 3041 3226 mt (0.08) s 898 3080 mt 921 3080 L 3157 3080 mt 3134 3080 L 797 3124 mt (0) s 898 2694 mt 921 2694 L 3157 2694 mt 3134 2694 L 597 2738 mt (1000) s 898 2307 mt 921 2307 L 3157 2307 mt 3134 2307 L 597 2351 mt (2000) s 898 1921 mt 921 1921 L 3157 1921 mt 3134 1921 L 597 1965 mt (3000) s 898 1921 mt 3157 1921 L 898 3080 mt 3157 3080 L 898 3080 mt 898 1921 L 3157 3080 mt 3157 1921 L gs 898 1921 2260 1160 MR c np c8 0 245 141 0 0 -245 969 3080 4 MP PP 0 sg -141 0 0 245 141 0 0 -245 969 3080 5 MP stroke c8 0 780 141 0 0 -780 1110 3080 4 MP PP 0 sg -141 0 0 780 141 0 0 -780 1110 3080 5 MP stroke c8 0 588 141 0 0 -588 1251 3080 4 MP PP 0 sg -141 0 0 588 141 0 0 -588 1251 3080 5 MP stroke c8 0 229 141 0 0 -229 1392 3080 4 MP PP 0 sg -141 0 0 229 141 0 0 -229 1392 3080 5 MP stroke c8 0 65 142 0 0 -65 1533 3080 4 MP PP 0 sg -142 0 0 65 142 0 0 -65 1533 3080 5 MP stroke c8 0 19 141 0 0 -19 1675 3080 4 MP PP 0 sg -141 0 0 19 141 0 0 -19 1675 3080 5 MP stroke c8 0 3 141 0 0 -3 1816 3080 4 MP PP 0 sg -141 0 0 3 141 0 0 -3 1816 3080 5 MP stroke c8 0 2 141 0 0 -2 1957 3080 4 MP PP 0 sg -141 0 0 2 141 0 0 -2 1957 3080 5 MP stroke c8 0 0 141 0 0 0 2098 3080 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2098 3080 5 MP stroke c8 0 0 141 0 0 0 2239 3080 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2239 3080 5 MP stroke c8 0 0 142 0 0 0 2380 3080 4 MP PP 0 sg -142 0 0 0 142 0 0 0 2380 3080 5 MP stroke c8 0 0 141 0 0 0 2522 3080 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2522 3080 5 MP stroke gr 1393 1825 mt (3 components, mean: 0.01288) s 1 sg 0 1159 2260 0 0 -1159 3994 3080 4 MP PP -2260 0 0 1159 2260 0 0 -1159 3994 3080 5 MP stroke DO 4 w SO 6 w 0 sg 3994 3080 mt 6254 3080 L 3994 1921 mt 6254 1921 L 6254 3080 mt 6254 1921 L 3994 3080 mt 3994 1921 L 3994 3080 mt 6254 3080 L 3994 3080 mt 3994 1921 L 3994 3080 mt 3994 3057 L 3994 1921 mt 3994 1944 L 3961 3226 mt (0) s 4559 3080 mt 4559 3057 L 4559 1921 mt 4559 1944 L 4443 3226 mt (0.02) s 5124 3080 mt 5124 3057 L 5124 1921 mt 5124 1944 L 5008 3226 mt (0.04) s 5689 3080 mt 5689 3057 L 5689 1921 mt 5689 1944 L 5573 3226 mt (0.06) s 6254 3080 mt 6254 3057 L 6254 1921 mt 6254 1944 L 6138 3226 mt (0.08) s 3994 3080 mt 4017 3080 L 6254 3080 mt 6231 3080 L 3893 3124 mt (0) s 3994 2694 mt 4017 2694 L 6254 2694 mt 6231 2694 L 3693 2738 mt (1000) s 3994 2307 mt 4017 2307 L 6254 2307 mt 6231 2307 L 3693 2351 mt (2000) s 3994 1921 mt 4017 1921 L 6254 1921 mt 6231 1921 L 3693 1965 mt (3000) s 3994 1921 mt 6254 1921 L 3994 3080 mt 6254 3080 L 3994 3080 mt 3994 1921 L 6254 3080 mt 6254 1921 L gs 3994 1921 2261 1160 MR c np c8 0 192 141 0 0 -192 4065 3080 4 MP PP 0 sg -141 0 0 192 141 0 0 -192 4065 3080 5 MP stroke c8 0 675 141 0 0 -675 4206 3080 4 MP PP 0 sg -141 0 0 675 141 0 0 -675 4206 3080 5 MP stroke c8 0 623 141 0 0 -623 4347 3080 4 MP PP 0 sg -141 0 0 623 141 0 0 -623 4347 3080 5 MP stroke c8 0 298 142 0 0 -298 4488 3080 4 MP PP 0 sg -142 0 0 298 142 0 0 -298 4488 3080 5 MP stroke c8 0 99 141 0 0 -99 4630 3080 4 MP PP 0 sg -141 0 0 99 141 0 0 -99 4630 3080 5 MP stroke c8 0 28 141 0 0 -28 4771 3080 4 MP PP 0 sg -141 0 0 28 141 0 0 -28 4771 3080 5 MP stroke c8 0 12 141 0 0 -12 4912 3080 4 MP PP 0 sg -141 0 0 12 141 0 0 -12 4912 3080 5 MP stroke c8 0 3 142 0 0 -3 5053 3080 4 MP PP 0 sg -142 0 0 3 142 0 0 -3 5053 3080 5 MP stroke c8 0 1 141 0 0 -1 5195 3080 4 MP PP 0 sg -141 0 0 1 141 0 0 -1 5195 3080 5 MP stroke c8 0 0 141 0 0 0 5336 3080 4 MP PP 0 sg -141 0 0 0 141 0 0 0 5336 3080 5 MP stroke c8 0 0 141 0 0 0 5477 3080 4 MP PP 0 sg -141 0 0 0 141 0 0 0 5477 3080 5 MP stroke c8 0 0 142 0 0 0 5618 3080 4 MP PP 0 sg -142 0 0 0 142 0 0 0 5618 3080 5 MP stroke gr 4457 1825 mt (2 components, mean: 0.013979) s 1 sg 0 1159 2259 0 0 -1159 898 4613 4 MP PP -2259 0 0 1159 2259 0 0 -1159 898 4613 5 MP stroke DO 4 w SO 6 w 0 sg 898 4613 mt 3157 4613 L 898 3454 mt 3157 3454 L 3157 4613 mt 3157 3454 L 898 4613 mt 898 3454 L 898 4613 mt 3157 4613 L 898 4613 mt 898 3454 L 898 4613 mt 898 4590 L 898 3454 mt 898 3477 L 865 4759 mt (0) s 1463 4613 mt 1463 4590 L 1463 3454 mt 1463 3477 L 1347 4759 mt (0.02) s 2028 4613 mt 2028 4590 L 2028 3454 mt 2028 3477 L 1912 4759 mt (0.04) s 2592 4613 mt 2592 4590 L 2592 3454 mt 2592 3477 L 2476 4759 mt (0.06) s 3157 4613 mt 3157 4590 L 3157 3454 mt 3157 3477 L 3041 4759 mt (0.08) s 898 4613 mt 921 4613 L 3157 4613 mt 3134 4613 L 797 4657 mt (0) s 898 4227 mt 921 4227 L 3157 4227 mt 3134 4227 L 597 4271 mt (1000) s 898 3840 mt 921 3840 L 3157 3840 mt 3134 3840 L 597 3884 mt (2000) s 898 3454 mt 921 3454 L 3157 3454 mt 3134 3454 L 597 3498 mt (3000) s 898 3454 mt 3157 3454 L 898 4613 mt 3157 4613 L 898 4613 mt 898 3454 L 3157 4613 mt 3157 3454 L gs 898 3454 2260 1160 MR c np c8 0 115 141 0 0 -115 969 4613 4 MP PP 0 sg -141 0 0 115 141 0 0 -115 969 4613 5 MP stroke c8 0 529 141 0 0 -529 1110 4613 4 MP PP 0 sg -141 0 0 529 141 0 0 -529 1110 4613 5 MP stroke c8 0 644 141 0 0 -644 1251 4613 4 MP PP 0 sg -141 0 0 644 141 0 0 -644 1251 4613 5 MP stroke c8 0 392 141 0 0 -392 1392 4613 4 MP PP 0 sg -141 0 0 392 141 0 0 -392 1392 4613 5 MP stroke c8 0 170 142 0 0 -170 1533 4613 4 MP PP 0 sg -142 0 0 170 142 0 0 -170 1533 4613 5 MP stroke c8 0 50 141 0 0 -50 1675 4613 4 MP PP 0 sg -141 0 0 50 141 0 0 -50 1675 4613 5 MP stroke c8 0 20 141 0 0 -20 1816 4613 4 MP PP 0 sg -141 0 0 20 141 0 0 -20 1816 4613 5 MP stroke c8 0 7 141 0 0 -7 1957 4613 4 MP PP 0 sg -141 0 0 7 141 0 0 -7 1957 4613 5 MP stroke c8 0 2 141 0 0 -2 2098 4613 4 MP PP 0 sg -141 0 0 2 141 0 0 -2 2098 4613 5 MP stroke c8 0 2 141 0 0 -2 2239 4613 4 MP PP 0 sg -141 0 0 2 141 0 0 -2 2239 4613 5 MP stroke c8 0 0 142 0 0 0 2380 4613 4 MP PP 0 sg -142 0 0 0 142 0 0 0 2380 4613 5 MP stroke c8 0 0 141 0 0 0 2522 4613 4 MP PP 0 sg -141 0 0 0 141 0 0 0 2522 4613 5 MP stroke gr 1423 3358 mt (1 component, mean: 0.015731) s 1 sg 0 1159 2260 0 0 -1159 3994 4613 4 MP PP -2260 0 0 1159 2260 0 0 -1159 3994 4613 5 MP stroke DO 4 w SO 6 w 0 sg 3994 4613 mt 6254 4613 L 3994 3454 mt 6254 3454 L 6254 4613 mt 6254 3454 L 3994 4613 mt 3994 3454 L 3994 4613 mt 6254 4613 L 3994 4613 mt 3994 3454 L 3994 4613 mt 3994 4590 L 3994 3454 mt 3994 3477 L 3961 4759 mt (1) s 4559 4613 mt 4559 4590 L 4559 3454 mt 4559 3477 L 4526 4759 mt (2) s 5124 4613 mt 5124 4590 L 5124 3454 mt 5124 3477 L 5091 4759 mt (3) s 5689 4613 mt 5689 4590 L 5689 3454 mt 5689 3477 L 5656 4759 mt (4) s 6254 4613 mt 6254 4590 L 6254 3454 mt 6254 3477 L 6221 4759 mt (5) s 3994 4613 mt 4017 4613 L 6254 4613 mt 6231 4613 L 3726 4657 mt (0.01) s 3994 4227 mt 4017 4227 L 6254 4227 mt 6231 4227 L 3659 4271 mt (0.012) s 3994 3840 mt 4017 3840 L 6254 3840 mt 6231 3840 L 3659 3884 mt (0.014) s 3994 3454 mt 4017 3454 L 6254 3454 mt 6231 3454 L 3659 3498 mt (0.016) s 3994 3454 mt 6254 3454 L 3994 4613 mt 6254 4613 L 3994 4613 mt 3994 3454 L 6254 4613 mt 6254 3454 L gs 3994 3454 2261 1160 MR c np gr 36 36 3994 3506 FO 36 36 4559 3844 FO 36 36 5124 4057 FO 36 36 5689 4222 FO 36 36 6254 4344 FO gs 3994 3454 2261 1160 MR c np gr 4836 4901 mt (no. of components) s 3605 4324 mt -90 rotate (mean error) s 90 rotate end eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF %%EndDocument @endspecial 450 2989 a(Figure)e(1:)48 b Fb(Plots)17 b(of)i(histograms)g(of)f(the)h(normalized)g(error)f(between)h(the)g (true)f(log)h(lik)o(elihood)g(and)g(the)g(lo)n(wer)450 3072 y(bound,)24 b(for)d(v)n(arious)i(numbers)g(of)f(mixture)f (components.)34 b(Also)22 b(sho)n(wn)g(is)g(the)f(mean)i(v)n(alues)f (tak)o(en)h(from)f(the)450 3155 y(histograms,)d(plotted)h(against)f (the)g(number)g(of)g(components.)450 3397 y Fp(Ne)o(xt)27 b(we)h(consider)e(the)i(impact)f(of)g(using)g(mixture)f(distrib)n 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n(ution)h(o)o(v)o(er)450 4037 y Fl(\(0)p Fm(;)14 b Fl(1\))p Fp(.)24 b(Again)17 b(the)h(netw)o(orks)f(are)h(suf)n(\002ciently)e (small)j(that)f(the)f(e)o(xact)h(log)f(lik)o(elihood)f(for)i(the)f (trained)450 4128 y(models)f(can)h(be)g(e)n(v)n(aluated)f(directly)-5 b(.)22 b(A)c(Hinton)e(diagram)g(of)h(the)g(hidden-to-output)12 b(weights)17 b(for)g(the)450 4219 y(eight)j(units)h(in)g(a)h(netw)o (ork)d(trained)h(with)h(5)g(mixture)f(components)e(is)k(sho)n(wn)e(in)h (Figure)f(2.)27 b(Figure)20 b(3)450 4311 y(sho)n(ws)h(a)f(plot)h(of)f (the)g(true)g(log)h(lik)o(elihood)d(v)o(ersus)i(the)h(number)e Fm(M)29 b Fp(of)20 b(components)f(in)h(the)h(mixture)450 4402 y(for)27 b(a)g(set)h(of)f(e)o(xperiments)e(in)j(which,)g(for)e (each)h(v)n(alue)g(of)g Fm(M)9 b Fp(,)29 b(the)e(model)f(w)o(as)i (trained)e(10)h(times)450 4493 y(starting)21 b(from)g(dif)n(ferent)f (random)g(parameter)g(initializations.)29 b(These)22 b(results)g(indicate)f(that,)h(as)h(the)450 4585 y(number)k(of)i (mixture)f(components)e(is)k(increased,)g(the)f(learning)f(algorithm)f (is)j(able)f(to)g(\002nd)f(a)i(set)450 4676 y(of)25 b(netw)o(ork)f (parameters)f(ha)n(ving)h(a)i(lar)o(ger)e(lik)o(elihood,)g(and)h(hence) f(that)h(the)g(impro)o(v)o(ed)d(\003e)o(xibility)450 4767 y(of)f(the)g(approximating)d(distrib)n(ution)i(is)i(indeed)e (translated)h(into)g(an)g(impro)o(v)o(ed)d(training)i(algorithm.)450 4859 y(W)-7 b(e)18 b(are)e(currently)f(applying)f(the)j(mixture)e (formalism)h(to)g(the)h(lar)o(ge-scale)e(problem)g(of)h(hand-written) 450 4950 y(digit)k(classi\002cation.)p eop %%Page: 7 7 7 6 bop 1123 367 a @beginspecial 92 @llx 358 @lly 543 @urx 458 @ury 1984 @rwi @setspecial %%BeginDocument: hinton_diag1.eps %!PS-Adobe-3.0 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /user/cs_neural/lawrennd/hinton_diag1.eps %%CreationDate: 05/20/97 09:50:22 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: (atend) %%BoundingBox: (atend) %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 92 358 543 458 MathWorks begin bpage %%EndPageSetup %%BeginObject: graph1 1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 898 1845 5404 1196 rc 25 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 1 j 1 sg 0 0 6912 5185 rf 6 w gs 898 1962 1079 1079 rc 0 sg -136 0 0 -136 136 0 0 136 965 2028 5 MP PP 0 0 -136 0 0 -136 136 0 0 136 965 2028 6 MP stroke 136 0 0 -136 965 2164 3 MP stroke -136 0 0 136 1101 2028 3 MP stroke -190 0 0 -190 190 0 0 190 1207 2001 5 MP PP 0 0 -190 0 0 -190 190 0 0 190 1207 2001 6 MP stroke 190 0 0 -190 1207 2191 3 MP stroke -190 0 0 190 1397 2001 3 MP stroke 57 0 0 -57 1543 2125 3 MP stroke -57 0 0 57 1600 2068 3 MP stroke -142 0 0 -142 142 0 0 142 1770 2025 5 MP PP 0 0 -142 0 0 -142 142 0 0 142 1770 2025 6 MP stroke 142 0 0 -142 1770 2167 3 MP stroke -142 0 0 142 1912 2025 3 MP stroke -208 0 0 -208 208 0 0 208 929 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gs 2324 1962 1079 1079 rc -165 0 0 -165 165 0 0 165 2376 2014 5 MP PP 0 0 -165 0 0 -165 165 0 0 165 2376 2014 6 MP stroke 165 0 0 -165 2376 2179 3 MP stroke -165 0 0 165 2541 2014 3 MP stroke -100 0 0 -100 100 0 0 100 2678 2046 5 MP PP 0 0 -100 0 0 -100 100 0 0 100 2678 2046 6 MP stroke 100 0 0 -100 2678 2146 3 MP stroke -100 0 0 100 2778 2046 3 MP stroke -237 0 0 -237 237 0 0 237 2879 1978 5 MP PP 0 0 -237 0 0 -237 237 0 0 237 2879 1978 6 MP stroke 237 0 0 -237 2879 2215 3 MP stroke -237 0 0 237 3116 1978 3 MP stroke -105 0 0 -105 105 0 0 105 3215 2044 5 MP PP 0 0 -105 0 0 -105 105 0 0 105 3215 2044 6 MP stroke 105 0 0 -105 3215 2149 3 MP stroke -105 0 0 105 3320 2044 3 MP stroke -140 0 0 -140 140 0 0 140 2389 2296 5 MP PP 0 0 -140 0 0 -140 140 0 0 140 2389 2296 6 MP stroke 140 0 0 -140 2389 2436 3 MP stroke -140 0 0 140 2529 2296 3 MP stroke -101 0 0 -101 101 0 0 101 2678 2315 5 MP PP 0 0 -101 0 0 -101 101 0 0 101 2678 2315 6 MP stroke 101 0 0 -101 2678 2416 3 MP stroke -101 0 0 101 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2569 3 MP stroke -192 0 0 -192 192 0 0 192 2363 2809 5 MP PP 0 0 -192 0 0 -192 192 0 0 192 2363 2809 6 MP stroke 192 0 0 -192 2363 3001 3 MP stroke -192 0 0 192 2555 2809 3 MP stroke -160 0 0 -160 160 0 0 160 2648 2825 5 MP PP 0 0 -160 0 0 -160 160 0 0 160 2648 2825 6 MP stroke 160 0 0 -160 2648 2985 3 MP stroke -160 0 0 160 2808 2825 3 MP stroke -255 0 0 -255 255 0 0 255 2870 2777 5 MP PP 0 0 -255 0 0 -255 255 0 0 255 2870 2777 6 MP stroke 255 0 0 -255 2870 3032 3 MP stroke -255 0 0 255 3125 2777 3 MP stroke -164 0 0 -164 164 0 0 164 3185 2823 5 MP PP 0 0 -164 0 0 -164 164 0 0 164 3185 2823 6 MP stroke 164 0 0 -164 3185 2987 3 MP stroke -164 0 0 164 3349 2823 3 MP stroke 0 -1078 -1078 0 0 1078 1078 0 2324 1962 5 MP stroke gr gs 3750 1962 1079 1079 rc 89 0 0 -89 3840 2141 3 MP stroke -89 0 0 89 3929 2052 3 MP stroke -156 0 0 -156 156 0 0 156 4076 2018 5 MP PP 0 0 -156 0 0 -156 156 0 0 156 4076 2018 6 MP stroke 156 0 0 -156 4076 2174 3 MP stroke -156 0 0 156 4232 2018 3 MP stroke -84 0 0 -84 84 0 0 84 4382 2054 5 MP PP 0 0 -84 0 0 -84 84 0 0 84 4382 2054 6 MP stroke 84 0 0 -84 4382 2138 3 MP stroke -84 0 0 84 4466 2054 3 MP stroke -227 0 0 -227 227 0 0 227 4580 1983 5 MP PP 0 0 -227 0 0 -227 227 0 0 227 4580 1983 6 MP stroke 227 0 0 -227 4580 2210 3 MP stroke -227 0 0 227 4807 1983 3 MP stroke 143 0 0 -143 3813 2437 3 MP stroke -143 0 0 143 3956 2294 3 MP stroke -60 0 0 -60 60 0 0 60 4124 2336 5 MP PP 0 0 -60 0 0 -60 60 0 0 60 4124 2336 6 MP stroke 60 0 0 -60 4124 2396 3 MP stroke -60 0 0 60 4184 2336 3 MP stroke 98 0 0 -98 4375 2415 3 MP stroke -98 0 0 98 4473 2317 3 MP stroke -165 0 0 -165 165 0 0 165 4611 2283 5 MP PP 0 0 -165 0 0 -165 165 0 0 165 4611 2283 6 MP stroke 165 0 0 -165 4611 2448 3 MP stroke -165 0 0 165 4776 2283 3 MP stroke -141 0 0 -141 141 0 0 141 3814 2565 5 MP PP 0 0 -141 0 0 -141 141 0 0 141 3814 2565 6 MP stroke 141 0 0 -141 3814 2706 3 MP stroke -141 0 0 141 3955 2565 3 MP stroke -211 0 0 -211 211 0 0 211 4049 2530 5 MP PP 0 0 -211 0 0 -211 211 0 0 211 4049 2530 6 MP stroke 211 0 0 -211 4049 2741 3 MP stroke -211 0 0 211 4260 2530 3 MP stroke -187 0 0 -187 187 0 0 187 4330 2542 5 MP PP 0 0 -187 0 0 -187 187 0 0 187 4330 2542 6 MP stroke 187 0 0 -187 4330 2729 3 MP stroke -187 0 0 187 4517 2542 3 MP stroke -255 0 0 -255 255 0 0 255 4566 2508 5 MP PP 0 0 -255 0 0 -255 255 0 0 255 4566 2508 6 MP stroke 255 0 0 -255 4566 2763 3 MP stroke -255 0 0 255 4821 2508 3 MP stroke 96 0 0 -96 3837 2953 3 MP stroke -96 0 0 96 3933 2857 3 MP stroke -133 0 0 -133 133 0 0 133 4088 2838 5 MP PP 0 0 -133 0 0 -133 133 0 0 133 4088 2838 6 MP stroke 133 0 0 -133 4088 2971 3 MP stroke -133 0 0 133 4221 2838 3 MP stroke -69 0 0 -69 69 0 0 69 4389 2870 5 MP PP 0 0 -69 0 0 -69 69 0 0 69 4389 2870 6 MP stroke 69 0 0 -69 4389 2939 3 MP stroke -69 0 0 69 4458 2870 3 MP stroke -219 0 0 -219 219 0 0 219 4584 2795 5 MP PP 0 0 -219 0 0 -219 219 0 0 219 4584 2795 6 MP stroke 219 0 0 -219 4584 3014 3 MP stroke -219 0 0 219 4803 2795 3 MP stroke 0 -1078 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stroke -109 0 0 109 5904 2311 3 MP stroke -159 0 0 -159 159 0 0 159 6040 2286 5 MP PP 0 0 -159 0 0 -159 159 0 0 159 6040 2286 6 MP stroke 159 0 0 -159 6040 2445 3 MP stroke -159 0 0 159 6199 2286 3 MP stroke -173 0 0 -173 173 0 0 173 5224 2549 5 MP PP 0 0 -173 0 0 -173 173 0 0 173 5224 2549 6 MP stroke 173 0 0 -173 5224 2722 3 MP stroke -173 0 0 173 5397 2549 3 MP stroke 94 0 0 -94 5533 2682 3 MP stroke -94 0 0 94 5627 2588 3 MP stroke 110 0 0 -110 5795 2690 3 MP stroke -110 0 0 110 5905 2580 3 MP stroke -165 0 0 -165 165 0 0 165 6037 2553 5 MP PP 0 0 -165 0 0 -165 165 0 0 165 6037 2553 6 MP stroke 165 0 0 -165 6037 2718 3 MP stroke -165 0 0 165 6202 2553 3 MP stroke -226 0 0 -226 226 0 0 226 5198 2792 5 MP PP 0 0 -226 0 0 -226 226 0 0 226 5198 2792 6 MP stroke 226 0 0 -226 5198 3018 3 MP stroke -226 0 0 226 5424 2792 3 MP stroke -199 0 0 -199 199 0 0 199 6020 2805 5 MP PP 0 0 -199 0 0 -199 199 0 0 199 6020 2805 6 MP stroke 199 0 0 -199 6020 3004 3 MP stroke -199 0 0 199 6219 2805 3 MP stroke 0 -1078 -1078 0 0 1078 1078 0 5176 1962 5 MP stroke gr end eplot %%EndObject graph 1 epage end showpage %%Trailer %%BoundingBox: 92 358 543 458 %%Pages: 001 %%EOF %%EndDocument @endspecial 445 x @beginspecial 92 @llx 358 @lly 543 @urx 458 @ury 1984 @rwi @setspecial %%BeginDocument: hinton_diag2.eps %!PS-Adobe-3.0 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /user/cs_neural/lawrennd/hinton_diag2.eps %%CreationDate: 05/20/97 09:53:58 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%LanguageLevel: 2 %%Pages: (atend) %%BoundingBox: (atend) %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 150 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 92 358 543 458 MathWorks begin bpage %%EndPageSetup %%BeginObject: graph1 1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 898 1845 5404 1196 rc 25 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 1 j 1 sg 0 0 6912 5185 rf 6 w gs 898 1962 1079 1079 rc 0 sg -142 0 0 -142 142 0 0 142 962 2025 5 MP PP 0 0 -142 0 0 -142 142 0 0 142 962 2025 6 MP stroke 142 0 0 -142 962 2167 3 MP stroke -142 0 0 142 1104 2025 3 MP stroke -186 0 0 -186 186 0 0 186 1209 2003 5 MP PP 0 0 -186 0 0 -186 186 0 0 186 1209 2003 6 MP stroke 186 0 0 -186 1209 2189 3 MP stroke -186 0 0 186 1395 2003 3 MP stroke -255 0 0 -255 255 0 0 255 1444 1969 5 MP PP 0 0 -255 0 0 -255 255 0 0 255 1444 1969 6 MP stroke 255 0 0 -255 1444 2224 3 MP stroke -255 0 0 255 1699 1969 3 MP stroke -164 0 0 -164 164 0 0 164 1759 2014 5 MP PP 0 0 -164 0 0 -164 164 0 0 164 1759 2014 6 MP stroke 164 0 0 -164 1759 2178 3 MP stroke -164 0 0 164 1923 2014 3 MP stroke 157 0 0 -157 954 2444 3 MP stroke -157 0 0 157 1111 2287 3 MP stroke 103 0 0 -103 1251 2417 3 MP stroke -103 0 0 103 1354 2314 3 MP stroke -154 0 0 -154 154 0 0 154 1495 2289 5 MP PP 0 0 -154 0 0 -154 154 0 0 154 1495 2289 6 MP stroke 154 0 0 -154 1495 2443 3 MP stroke -154 0 0 154 1649 2289 3 MP stroke 134 0 0 -134 1774 2433 3 MP stroke -134 0 0 134 1908 2299 3 MP stroke 166 0 0 -166 950 2718 3 MP stroke -166 0 0 166 1116 2552 3 MP stroke 102 0 0 -102 1251 2686 3 MP stroke -102 0 0 102 1353 2584 3 MP stroke -171 0 0 -171 171 0 0 171 1486 2550 5 MP PP 0 0 -171 0 0 -171 171 0 0 171 1486 2550 6 MP stroke 171 0 0 -171 1486 2721 3 MP stroke -171 0 0 171 1657 2550 3 MP stroke 132 0 0 -132 1775 2701 3 MP stroke -132 0 0 132 1907 2569 3 MP stroke 94 0 0 -94 986 2952 3 MP stroke -94 0 0 94 1080 2858 3 MP stroke -81 0 0 -81 81 0 0 81 1262 2864 5 MP PP 0 0 -81 0 0 -81 81 0 0 81 1262 2864 6 MP stroke 81 0 0 -81 1262 2945 3 MP stroke -81 0 0 81 1343 2864 3 MP stroke -186 0 0 -186 186 0 0 186 1479 2812 5 MP PP 0 0 -186 0 0 -186 186 0 0 186 1479 2812 6 MP stroke 186 0 0 -186 1479 2998 3 MP stroke -186 0 0 186 1665 2812 3 MP stroke 25 0 0 -25 1829 2917 3 MP stroke -25 0 0 25 1854 2892 3 MP stroke 0 -1078 -1078 0 0 1078 1078 0 898 1962 5 MP stroke gr 0 sg gs 2324 1962 1079 1079 rc -217 0 0 -217 217 0 0 217 2350 1988 5 MP PP 0 0 -217 0 0 -217 217 0 0 217 2350 1988 6 MP stroke 217 0 0 -217 2350 2205 3 MP stroke -217 0 0 217 2567 1988 3 MP stroke 41 0 0 -41 2708 2117 3 MP stroke -41 0 0 41 2749 2076 3 MP stroke -110 0 0 -110 110 0 0 110 2943 2041 5 MP PP 0 0 -110 0 0 -110 110 0 0 110 2943 2041 6 MP stroke 110 0 0 -110 2943 2151 3 MP stroke -110 0 0 110 3053 2041 3 MP stroke 148 0 0 -148 3193 2170 3 MP stroke -148 0 0 148 3341 2022 3 MP stroke -199 0 0 -199 199 0 0 199 2359 2266 5 MP PP 0 0 -199 0 0 -199 199 0 0 199 2359 2266 6 MP stroke 199 0 0 -199 2359 2465 3 MP stroke -199 0 0 199 2558 2266 3 MP stroke 94 0 0 -94 2681 2413 3 MP stroke -94 0 0 94 2775 2319 3 MP stroke -89 0 0 -89 89 0 0 89 2953 2321 5 MP PP 0 0 -89 0 0 -89 89 0 0 89 2953 2321 6 MP stroke 89 0 0 -89 2953 2410 3 MP stroke -89 0 0 89 3042 2321 3 MP stroke 145 0 0 -145 3195 2438 3 MP stroke -145 0 0 145 3340 2293 3 MP stroke -255 0 0 -255 255 0 0 255 2331 2508 5 MP PP 0 0 -255 0 0 -255 255 0 0 255 2331 2508 6 MP stroke 255 0 0 -255 2331 2763 3 MP stroke 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2324 1962 5 MP stroke gr gs 3750 1962 1079 1079 rc -183 0 0 -183 183 0 0 183 3793 2005 5 MP PP 0 0 -183 0 0 -183 183 0 0 183 3793 2005 6 MP stroke 183 0 0 -183 3793 2188 3 MP stroke -183 0 0 183 3976 2005 3 MP stroke -255 0 0 -255 255 0 0 255 4027 1969 5 MP PP 0 0 -255 0 0 -255 255 0 0 255 4027 1969 6 MP stroke 255 0 0 -255 4027 2224 3 MP stroke -255 0 0 255 4282 1969 3 MP stroke -200 0 0 -200 200 0 0 200 4324 1996 5 MP PP 0 0 -200 0 0 -200 200 0 0 200 4324 1996 6 MP stroke 200 0 0 -200 4324 2196 3 MP stroke -200 0 0 200 4524 1996 3 MP stroke -212 0 0 -212 212 0 0 212 4587 1990 5 MP PP 0 0 -212 0 0 -212 212 0 0 212 4587 1990 6 MP stroke 212 0 0 -212 4587 2202 3 MP stroke -212 0 0 212 4799 1990 3 MP stroke -216 0 0 -216 216 0 0 216 3777 2258 5 MP PP 0 0 -216 0 0 -216 216 0 0 216 3777 2258 6 MP stroke 216 0 0 -216 3777 2474 3 MP stroke -216 0 0 216 3993 2258 3 MP stroke -251 0 0 -251 251 0 0 251 4029 2240 5 MP PP 0 0 -251 0 0 -251 251 0 0 251 4029 2240 6 MP stroke 251 0 0 -251 4029 2491 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0 0 146 4766 2562 3 MP stroke -53 0 0 -53 53 0 0 53 3858 2878 5 MP PP 0 0 -53 0 0 -53 53 0 0 53 3858 2878 6 MP stroke 53 0 0 -53 3858 2931 3 MP stroke -53 0 0 53 3911 2878 3 MP stroke -118 0 0 -118 118 0 0 118 4095 2846 5 MP PP 0 0 -118 0 0 -118 118 0 0 118 4095 2846 6 MP stroke 118 0 0 -118 4095 2964 3 MP stroke -118 0 0 118 4213 2846 3 MP stroke 39 0 0 -39 4404 2924 3 MP stroke -39 0 0 39 4443 2885 3 MP stroke -82 0 0 -82 82 0 0 82 4652 2864 5 MP PP 0 0 -82 0 0 -82 82 0 0 82 4652 2864 6 MP stroke 82 0 0 -82 4652 2946 3 MP stroke -82 0 0 82 4734 2864 3 MP stroke 0 -1078 -1078 0 0 1078 1078 0 3750 1962 5 MP stroke gr gs 5176 1962 1079 1079 rc 137 0 0 -137 5242 2165 3 MP stroke -137 0 0 137 5379 2028 3 MP stroke -239 0 0 -239 239 0 0 239 5461 1977 5 MP PP 0 0 -239 0 0 -239 239 0 0 239 5461 1977 6 MP stroke 239 0 0 -239 5461 2216 3 MP stroke -239 0 0 239 5700 1977 3 MP stroke -91 0 0 -91 91 0 0 91 5804 2051 5 MP PP 0 0 -91 0 0 -91 91 0 0 91 5804 2051 6 MP stroke 91 0 0 -91 5804 2142 3 MP 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