(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 74823, 2211]*) (*NotebookOutlinePosition[ 75470, 2233]*) (* CellTagsIndexPosition[ 75426, 2229]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ StyleBox[\(Handout\ 8. \ Example\ involving\ hierarchical\ \ model\ \ Binomial \((n, p)\), \ Beta \((k, k)\), and\ Hyperprior\ 1/\((k \((2 k - 1)\))\)\), "Title"], "\[IndentingNewLine]", StyleBox["\[IndentingNewLine]", "Title"], RowBox[{ RowBox[{ StyleBox["Plot", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["of", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["Beta", "Subtitle"], StyleBox[\((k, k)\), "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["for", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["some", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["integer", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox[\(k'\), "Subtitle"], RowBox[{ StyleBox["s", "Subtitle"], StyleBox[".", "Subtitle"], StyleBox[" ", "Subtitle"], StyleBox["\[IndentingNewLine]", "Subtitle"], StyleBox["\[IndentingNewLine]", "Subtitle"], \(bet[x_, \ m_, \ n_]\)}]}], ":=", " ", \(1/ Beta[m, n]\ x^\((m - 1)\)\ \((1 - x)\)^\((n - 1)\)\)}]}]}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(Plot[{bet[x, 1, 1], bet[x, 2, 2], bet[x, 3, 3], \[IndentingNewLine]bet[x, 5, 5], \ bet[x, \ 10, \ 10], bet[x, \ 20, 20]}, \[IndentingNewLine]{x, 0, 1}, \ PlotRange \[Rule] All]\)\(\ \)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 0.0147151 0.117373 [ [.21429 .00222 -9 -9 ] [.21429 .00222 9 0 ] [.40476 .00222 -9 -9 ] [.40476 .00222 9 0 ] [.59524 .00222 -9 -9 ] [.59524 .00222 9 0 ] [.78571 .00222 -9 -9 ] [.78571 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .13209 -6 -4.5 ] [.01131 .13209 0 4.5 ] [.01131 .24946 -6 -4.5 ] [.01131 .24946 0 4.5 ] [.01131 .36683 -6 -4.5 ] [.01131 .36683 0 4.5 ] [.01131 .48421 -6 -4.5 ] [.01131 .48421 0 4.5 ] [.01131 .60158 -6 -4.5 ] [.01131 .60158 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .01472 m .21429 .02097 L s [(0.2)] .21429 .00222 0 1 Mshowa .40476 .01472 m .40476 .02097 L s [(0.4)] .40476 .00222 0 1 Mshowa .59524 .01472 m .59524 .02097 L s [(0.6)] .59524 .00222 0 1 Mshowa .78571 .01472 m .78571 .02097 L s [(0.8)] .78571 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(1)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .2619 .01472 m .2619 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .45238 .01472 m .45238 .01847 L s .5 .01472 m .5 .01847 L s .54762 .01472 m .54762 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .7381 .01472 m .7381 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .13209 m .03006 .13209 L s [(1)] .01131 .13209 1 0 Mshowa .02381 .24946 m .03006 .24946 L s [(2)] .01131 .24946 1 0 Mshowa .02381 .36683 m .03006 .36683 L s [(3)] .01131 .36683 1 0 Mshowa .02381 .48421 m .03006 .48421 L s [(4)] .01131 .48421 1 0 Mshowa .02381 .60158 m .03006 .60158 L s [(5)] .01131 .60158 1 0 Mshowa .125 Mabswid .02381 .03819 m .02756 .03819 L s .02381 .06166 m .02756 .06166 L s .02381 .08514 m .02756 .08514 L s .02381 .10861 m .02756 .10861 L s .02381 .15556 m .02756 .15556 L s .02381 .17904 m .02756 .17904 L s .02381 .20251 m .02756 .20251 L s .02381 .22599 m .02756 .22599 L s .02381 .27294 m .02756 .27294 L s .02381 .29641 m .02756 .29641 L s .02381 .31988 m .02756 .31988 L s .02381 .34336 m .02756 .34336 L s .02381 .39031 m .02756 .39031 L s .02381 .41378 m .02756 .41378 L s .02381 .43726 m .02756 .43726 L s .02381 .46073 m .02756 .46073 L s .02381 .50768 m .02756 .50768 L s .02381 .53116 m .02756 .53116 L s .02381 .55463 m .02756 .55463 L s .02381 .57811 m .02756 .57811 L s .25 Mabswid .02381 0 m .02381 .61803 L s .5 Mabswid .02381 .13209 m .06244 .13209 L .10458 .13209 L .14415 .13209 L .18221 .13209 L .22272 .13209 L .26171 .13209 L .30316 .13209 L .34309 .13209 L .3815 .13209 L .42237 .13209 L .46172 .13209 L .49955 .13209 L .53984 .13209 L .57861 .13209 L .61984 .13209 L .65954 .13209 L .69774 .13209 L .73838 .13209 L .77751 .13209 L .81909 .13209 L .85916 .13209 L .89771 .13209 L .93871 .13209 L .97619 .13209 L s .02381 .01472 m .06244 .04212 L .10458 .06938 L .14415 .09246 L .18221 .11236 L .22272 .13108 L .26171 .14669 L .30316 .16069 L .34309 .17166 L .3815 .17987 L .40095 .18316 L .42237 .1861 L .44268 .18822 L .45178 .18897 L .46172 .18964 L .4671 .18993 L .4721 .19017 L .47727 .19037 L .48196 .19052 L .48658 .19063 L .4887 .19068 L .49093 .19071 L .49332 .19074 L .49438 .19075 L .49552 .19076 L .49675 .19077 L .49789 .19077 L .49859 .19077 L .49925 .19077 L .50049 .19077 L .50163 .19077 L .50286 .19077 L .50401 .19076 L .50508 .19075 L .50754 .19073 L .51014 .19069 L .51268 .19065 L .51504 .1906 L .5204 .19045 L .5293 .19011 L .53882 .1896 L .54906 .18891 L .56016 .18796 L .58032 .18577 L .60019 .18298 L .62123 .17936 L .65912 .17112 L .69946 .15988 L .73829 .14669 L .77956 .13009 L Mistroke .81932 .1116 L .85757 .09151 L .89827 .06762 L .93745 .0422 L .97619 .01472 L Mfstroke .02381 .01472 m .02499 .01472 L .02605 .01473 L .02729 .01476 L .02846 .0148 L .03053 .01489 L .03279 .01502 L .03527 .01521 L .0379 .01546 L .04262 .01603 L .05205 .01763 L .06244 .02005 L .0842 .02713 L .10458 .03593 L .14335 .05714 L .18458 .08404 L .22428 .11197 L .26248 .1389 L .30312 .16598 L .34225 .18914 L .38383 .20938 L .4047 .21751 L .4239 .22369 L .44265 .22845 L .45221 .23038 L .46245 .23206 L .46773 .23277 L .47272 .23335 L .47759 .23382 L .48209 .23417 L .48635 .23443 L .48852 .23453 L .49087 .23463 L .49321 .2347 L .49426 .23473 L .49541 .23475 L .49669 .23477 L .49738 .23478 L .49803 .23478 L .49919 .23479 L .50042 .23479 L .50166 .23478 L .50236 .23478 L .50301 .23477 L .50433 .23475 L .50575 .23473 L .50832 .23466 L .51067 .23457 L .51339 .23444 L .51636 .23427 L Mistroke .52177 .23387 L .53223 .23278 L .54192 .23139 L .56248 .22728 L .5811 .22221 L .62122 .20719 L .65983 .188 L .70088 .16343 L .74042 .13689 L .77844 .11002 L .81892 .08164 L .85788 .05639 L .87773 .04497 L .89929 .03411 L .91865 .02606 L .92925 .02245 L .93919 .01962 L .94767 .01769 L .95679 .01612 L .96173 .0155 L .96442 .01524 L .96692 .01504 L .96906 .01491 L .97135 .01481 L .97266 .01476 L .97389 .01474 L .97499 .01472 L .97619 .01472 L Mfstroke .02381 .01472 m .02499 .01472 L .02605 .01472 L .02729 .01472 L .02846 .01472 L .02954 .01472 L .03053 .01472 L .03163 .01472 L .03279 .01472 L .03395 .01472 L .0352 .01472 L .03638 .01472 L .03746 .01472 L .03873 .01472 L .04011 .01472 L .04262 .01473 L .04396 .01473 L .04523 .01473 L .04759 .01474 L .05014 .01475 L .0529 .01477 L .05532 .01479 L .05787 .01482 L .06244 .01488 L .06746 .01499 L .07283 .01514 L .07791 .01532 L .08253 .01554 L .08795 .01587 L .09385 .01631 L .10458 .0174 L .11398 .01871 L .12435 .02059 L .13519 .02313 L .14536 .02608 L .16554 .03375 L .18405 .04308 L .22367 .07062 L .26178 .10598 L .30234 .15029 L .34139 .19518 L .38289 .23977 L .40369 .25912 L .42287 .27442 L .44274 .28722 L .46133 .29602 L .47038 .29912 L .47503 .3004 L .4801 .30155 L .48297 .30209 L Mistroke .48563 .30251 L .48839 .30288 L .49088 .30314 L .49206 .30324 L .49334 .30334 L .49445 .30341 L .49566 .30347 L .49694 .30352 L .49765 .30353 L .4983 .30355 L .49948 .30356 L .50074 .30356 L .50196 .30354 L .50266 .30353 L .50331 .30351 L .50462 .30345 L .50604 .30338 L .5086 .30319 L .51094 .30295 L .51634 .3022 L .52202 .3011 L .53164 .2985 L .54215 .29462 L .5618 .28459 L .58276 .27021 L .62431 .23251 L .66435 .18863 L .70287 .14436 L .74384 .1003 L .78329 .06504 L .82123 .04019 L .84072 .0311 L .86162 .02399 L .87178 .02143 L .88284 .01923 L .89328 .01767 L .90295 .01659 L .90758 .01619 L .91256 .01583 L .92159 .01535 L .92682 .01515 L .93233 .01499 L .93704 .01489 L .93951 .01485 L .94219 .01482 L .94449 .01479 L .94661 .01478 L .9486 .01476 L .95079 .01475 L .95262 .01474 L Mistroke .95459 .01473 L .95667 .01473 L .95862 .01472 L .96087 .01472 L .96214 .01472 L .96333 .01472 L .96446 .01472 L .96549 .01472 L .96668 .01472 L .9678 .01472 L .96882 .01472 L .9699 .01472 L .97082 .01472 L .97183 .01472 L .97292 .01472 L .97409 .01472 L .97519 .01472 L .97619 .01472 L Mfstroke .02381 .01472 m .02499 .01472 L .02605 .01472 L .02729 .01472 L .02846 .01472 L .02954 .01472 L .03053 .01472 L .03163 .01472 L .03279 .01472 L .03395 .01472 L .0352 .01472 L .03638 .01472 L .03746 .01472 L .03873 .01472 L .03944 .01472 L .04011 .01472 L .04132 .01472 L .04262 .01472 L .04327 .01472 L .04396 .01472 L .04519 .01472 L .04588 .01472 L .04652 .01472 L .04722 .01472 L .04797 .01472 L .04862 .01472 L .04933 .01472 L .05061 .01472 L .05186 .01472 L .05302 .01472 L .05414 .01472 L .05519 .01472 L .05634 .01472 L .05759 .01472 L .05882 .01472 L .06012 .01472 L .06123 .01472 L .06244 .01472 L .06381 .01472 L .06529 .01472 L .06669 .01472 L .06797 .01472 L .07041 .01472 L .07164 .01472 L .073 .01472 L .07581 .01472 L .07738 .01472 L .07882 .01472 L .08162 .01472 L .08427 .01472 L Mistroke .08671 .01472 L .08931 .01472 L .09177 .01472 L .09399 .01472 L .09661 .01472 L .09945 .01472 L .10212 .01472 L .10458 .01473 L .10929 .01473 L .11194 .01474 L .11434 .01474 L .11902 .01476 L .12344 .01478 L .12859 .0148 L .13335 .01484 L .13858 .0149 L .14412 .01498 L .14902 .01507 L .15351 .01518 L .15845 .01534 L .16363 .01554 L .17237 .016 L .17732 .01635 L .18192 .01674 L .19247 .01793 L .20242 .01951 L .21135 .02142 L .22102 .02412 L .23051 .02753 L .24093 .03229 L .25194 .03866 L .26199 .04584 L .28148 .06396 L .30238 .09019 L .3422 .15989 L .38049 .24503 L .42124 .33696 L .44147 .37538 L .45138 .39111 L .46048 .40338 L .46914 .41296 L .47862 .42089 L .48356 .42391 L .48815 .42603 L .49069 .42691 L .49301 .42753 L .49429 .42779 L .49567 .42802 L .49699 .42818 L .4982 .42828 L Mistroke .49944 .42833 L .50063 .42832 L .5017 .42828 L .50286 .4282 L .50412 .42805 L .50546 .42784 L .50787 .42731 L .51032 .42658 L .51263 .42572 L .51694 .42365 L .5221 .42038 L .52685 .41665 L .53757 .40573 L .55641 .37894 L .57707 .34043 L .61752 .24967 L .65645 .16268 L .67627 .12451 L .69783 .08989 L .71854 .06394 L .73769 .04609 L .75637 .03371 L .767 .02856 L .77698 .02477 L .78682 .02188 L .79718 .01959 L .80602 .01814 L .81569 .01697 L .82098 .01649 L .82597 .01611 L .83531 .01559 L .84055 .01537 L .84537 .01521 L .85066 .01508 L .85627 .01497 L .8612 .0149 L .86571 .01485 L .87092 .01481 L .87586 .01478 L .88041 .01476 L .88462 .01475 L .88925 .01474 L .89183 .01473 L .89417 .01473 L .89907 .01472 L .90153 .01472 L .90423 .01472 L .90655 .01472 L .90902 .01472 L .91135 .01472 L Mistroke .91347 .01472 L .91614 .01472 L .91861 .01472 L .92132 .01472 L .92268 .01472 L .92418 .01472 L .9267 .01472 L .92812 .01472 L .92941 .01472 L .93061 .01472 L .93192 .01472 L .93431 .01472 L .93564 .01472 L .93686 .01472 L .9382 .01472 L .93962 .01472 L .94085 .01472 L .94201 .01472 L .94327 .01472 L .94462 .01472 L .94583 .01472 L .94714 .01472 L .94837 .01472 L .94952 .01472 L .95054 .01472 L .95168 .01472 L .95286 .01472 L .95398 .01472 L .9552 .01472 L .95589 .01472 L .95654 .01472 L .95785 .01472 L .95858 .01472 L .95924 .01472 L .96043 .01472 L .96172 .01472 L .96294 .01472 L .96407 .01472 L .96537 .01472 L .96658 .01472 L .96728 .01472 L .96792 .01472 L .96865 .01472 L .96935 .01472 L .97002 .01472 L .97074 .01472 L .97203 .01472 L .97282 .01472 L .97356 .01472 L .97423 .01472 L Mistroke .97495 .01472 L .97619 .01472 L Mfstroke .02381 .01472 m .02499 .01472 L .02605 .01472 L .02729 .01472 L .02846 .01472 L .02954 .01472 L .03053 .01472 L .03163 .01472 L .03279 .01472 L .03395 .01472 L .0352 .01472 L .03638 .01472 L .03746 .01472 L .03873 .01472 L .03944 .01472 L .04011 .01472 L .04132 .01472 L .04262 .01472 L .04327 .01472 L .04396 .01472 L .04519 .01472 L .04588 .01472 L .04652 .01472 L .04722 .01472 L .04797 .01472 L .04862 .01472 L .04933 .01472 L .05061 .01472 L .05186 .01472 L .05302 .01472 L .05414 .01472 L .05519 .01472 L .05634 .01472 L .05759 .01472 L .05882 .01472 L .06012 .01472 L .06123 .01472 L .06244 .01472 L .06315 .01472 L .06381 .01472 L .06453 .01472 L .06529 .01472 L .06657 .01472 L .06729 .01472 L .06797 .01472 L .06924 .01472 L .0704 .01472 L .07103 .01472 L .07173 .01472 L .073 .01472 L Mistroke .07376 .01472 L .07448 .01472 L .07583 .01472 L .07659 .01472 L .07728 .01472 L .07804 .01472 L .07885 .01472 L .07949 .01472 L .08019 .01472 L .08144 .01472 L .0822 .01472 L .08292 .01472 L .08427 .01472 L .08544 .01472 L .08673 .01472 L .08807 .01472 L .08933 .01472 L .09051 .01472 L .09158 .01472 L .09276 .01472 L .09401 .01472 L .09527 .01472 L .09663 .01472 L .0979 .01472 L .09906 .01472 L .10042 .01472 L .1019 .01472 L .1033 .01472 L .10458 .01472 L .10584 .01472 L .10703 .01472 L .10927 .01472 L .11052 .01472 L .11168 .01472 L .11303 .01472 L .1143 .01472 L .11639 .01472 L .1175 .01472 L .1187 .01472 L .11989 .01472 L .12114 .01472 L .12338 .01472 L .12467 .01472 L .12585 .01472 L .12711 .01472 L .12849 .01472 L .13093 .01472 L .13324 .01472 L .13454 .01472 L .13597 .01472 L Mistroke .13736 .01472 L .13885 .01472 L .14153 .01472 L .14404 .01472 L .14658 .01472 L .14928 .01472 L .15184 .01472 L .15415 .01472 L .15688 .01472 L .15983 .01472 L .16262 .01472 L .16518 .01472 L .1676 .01472 L .16985 .01472 L .17231 .01472 L .17496 .01472 L .17773 .01472 L .18036 .01472 L .1829 .01472 L .18528 .01473 L .18987 .01473 L .19231 .01474 L .19494 .01474 L .19755 .01475 L .2003 .01476 L .2052 .01478 L .20976 .0148 L .21401 .01483 L .21866 .01488 L .22363 .01496 L .22896 .01506 L .23388 .01519 L .23928 .01539 L .24499 .01567 L .24995 .016 L .25459 .01638 L .25968 .01692 L .26515 .01767 L .27041 .01858 L .27529 .01963 L .28109 .02118 L .28634 .02293 L .29608 .02725 L .30644 .03376 L .3165 .04248 L .32574 .05304 L .33589 .06798 L .34658 .08809 L .36627 .13831 L .38463 .2012 L Mistroke .42361 .37337 L .44347 .4642 L .45394 .50704 L .46504 .54588 L .47491 .57304 L .48053 .5849 L .48579 .59344 L .48826 .59656 L .49061 .59899 L .49272 .60071 L .49502 .6021 L .49633 .60266 L .49752 .60302 L .49817 .60316 L .49888 .60326 L .50017 .60332 L .50138 .60323 L .50266 .60297 L .50376 .60262 L .50496 .60211 L .50626 .60139 L .50746 .60058 L .51019 .59822 L .5126 .59554 L .51516 .59209 L .52446 .57451 L .52958 .56163 L .5344 .54763 L .54537 .50969 L .58404 .33734 L .60368 .24863 L .62517 .16573 L .63509 .13425 L .64569 .10568 L .66478 .06688 L .67502 .05207 L .68612 .03994 L .6966 .03161 L .70627 .02607 L .71094 .02399 L .71596 .02213 L .72502 .01955 L .73055 .01839 L .7358 .01752 L .74098 .01685 L .74566 .01636 L .75028 .01598 L .75527 .01566 L .76054 .0154 L .7655 .01521 L Mistroke .77016 .01508 L .77508 .01498 L .77928 .01491 L .78389 .01486 L .78641 .01483 L .78915 .01481 L .79156 .01479 L .79412 .01478 L .79896 .01476 L .80343 .01474 L .8059 .01474 L .80857 .01473 L .81338 .01473 L .81617 .01472 L .81911 .01472 L .82162 .01472 L .82436 .01472 L .8269 .01472 L .82965 .01472 L .83226 .01472 L .83465 .01472 L .837 .01472 L .83919 .01472 L .84148 .01472 L .844 .01472 L .84671 .01472 L .84928 .01472 L .85159 .01472 L .85411 .01472 L .85684 .01472 L .85822 .01472 L .85974 .01472 L .86243 .01472 L .86496 .01472 L .86738 .01472 L .86964 .01472 L .87212 .01472 L .8735 .01472 L .87477 .01472 L .87732 .01472 L .87875 .01472 L .88008 .01472 L .88252 .01472 L .88375 .01472 L .88511 .01472 L .88634 .01472 L .88751 .01472 L .88971 .01472 L .89093 .01472 L .89207 .01472 L Mistroke .8934 .01472 L .89466 .01472 L .89575 .01472 L .89693 .01472 L .89899 .01472 L .90015 .01472 L .90139 .01472 L .90244 .01472 L .9036 .01472 L .9049 .01472 L .90611 .01472 L .90744 .01472 L .90884 .01472 L .91006 .01472 L .9112 .01472 L .91246 .01472 L .91381 .01472 L .91494 .01472 L .91617 .01472 L .91746 .01472 L .91866 .01472 L .91979 .01472 L .92082 .01472 L .922 .01472 L .92312 .01472 L .92432 .01472 L .92501 .01472 L .92565 .01472 L .92695 .01472 L .92767 .01472 L .92834 .01472 L .92952 .01472 L .93081 .01472 L .93203 .01472 L .93315 .01472 L .93384 .01472 L .93458 .01472 L .93528 .01472 L .93592 .01472 L .93714 .01472 L .93842 .01472 L .93909 .01472 L .93981 .01472 L .9411 .01472 L .94189 .01472 L .94263 .01472 L .9433 .01472 L .94403 .01472 L .945 .01472 L .94588 .01472 L Mistroke .9469 .01472 L .94786 .01472 L .94875 .01472 L .94958 .01472 L .95049 .01472 L .95146 .01472 L .95252 .01472 L .95366 .01472 L .95475 .01472 L .95575 .01472 L .95665 .01472 L .95764 .01472 L .95867 .01472 L .95964 .01472 L .96066 .01472 L .96178 .01472 L .96284 .01472 L .96381 .01472 L .96469 .01472 L .96566 .01472 L .96667 .01472 L .96762 .01472 L .96867 .01472 L .96977 .01472 L .97071 .01472 L .97174 .01472 L .97286 .01472 L .97405 .01472 L .97517 .01472 L .97619 .01472 L Mfstroke 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg`3oool010000000 oooo0?ooo`0000090?ooo`030000003oool0oooo02<0oooo00@000000?ooo`3oool00000203oool5 000002<0oooo00@000000?ooo`3oool00000203oool010000000oooo0?ooo`00000T0?ooo`040000 003oool0oooo000000P0oooo00@000000?ooo`3oool00000;03oool00`000000oooo0?ooo`0;0?oo o`00>`3oool010000000oooo0?ooo`00000:0?ooo`030000003oool0oooo0280oooo00@000000?oo o`3oool00000203oool010000000oooo0?ooo`00000T0?ooo`040000003oool0oooo000000P0oooo 0`00000U0?ooo`040000003oool0oooo000000T0oooo0P00000]0?ooo`030000003oool0oooo00/0 oooo000k0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool000008`3oool01000 0000oooo0?ooo`0000090?ooo`030000003oool0000002@0oooo00@000000?ooo`3oool000002@3o ool00`000000oooo0?ooo`0T0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool0 0000;03oool00`000000oooo0?ooo`0;0?ooo`00?03oool2000000X0oooo0P00000U0?ooo`800000 2`3oool2000002D0oooo0P00000:0?ooo`<000009@3oool2000000X0oooo0P00000/0?ooo`800000 3@3oool00?l0oooo8@3oool000l0oooo00<000000?ooo`3oool0o`3oool?0?ooo`003`3oool00`00 0000oooo0?ooo`3o0?ooo`l0oooo000?0?ooo`030000003oool0oooo0?l0oooo3`3oool000l0oooo 00<000000?ooo`3oool0o`3oool?0?ooo`00203ooooo000001<000001P3oool000l0oooo0`000003 0?ooo`H0000000<0oooo0000003oool01P3oool6000000`0oooo00<000000?ooo`3oool020000002 0?ooo`030000003oool0oooo00T0oooo1@0000090?ooo`030000003oool0oooo00X0oooo00<00000 0?ooo`3oool02@3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0oooo00<00000 0?ooo`3oool02P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0oooo00<00000 0?ooo`3oool02P3oool00`000000oooo0?ooo`060?ooo`H000002`3oool00`000000oooo0?ooo`07 000000<0oooo00<000000?ooo`3oool02P3oool6000000L0oooo00<000000?ooo`3oool01@000003 0?ooo`<000003@3oool000l0oooo00@000000?ooo`3oool00000203oool3000000/0oooo1P00000B 0?ooo`<00000403oool4000001h0oooo00<000000?ooo`3oool0<@3oool00`000000oooo0?ooo`0M 0?ooo`<000004@3oool300000140oooo1P00000=0?ooo`<000001`3oool010000000oooo0?ooo`00 000=0?ooo`003`3oool01@000000oooo0?ooo`3oool0000000X0oooo0P00000?0?ooo`<000004@3o ool500000100oooo0P00001_0?ooo`030000003oool0oooo00l0oooo0`00000A0?ooo`<00000403o ool3000000T0oooo00<000000?ooo`3oool03`3oool000l0oooo00<000000?ooo`3oool00P3oool2 000000X0oooo0P00000@0?ooo`800000503oool200000100oooo0P00001[0?ooo`8000003`3oool3 00000180oooo0P00000@0?ooo`<000002P3oool200000180oooo000?0?ooo`030000003oool0oooo 00@0oooo00<000000?ooo`3oool02@3oool200000100oooo0P00000D0?ooo`800000403oool00`00 0000oooo0?ooo`1W0?ooo`030000003oool0oooo00d0oooo0P00000C0?ooo`800000403oool20000 00`0oooo00<000000?ooo`3oool04P3oool000l0oooo0P0000060?ooo`8000002`3oool200000100 oooo0P00000D0?ooo`8000003`3oool00`000000oooo0?ooo`1T0?ooo`8000003`3oool00`000000 oooo0?ooo`0A0?ooo`800000403oool2000000`0oooo0P00000E0?ooo`003`3oool00`000000oooo 0?ooo`070?ooo`030000003oool0oooo00X0oooo0P00000@0?ooo`800000503oool2000000d0oooo 00<000000?ooo`3oool0H`3oool00`000000oooo0?ooo`0=0?ooo`8000004`3oool00`000000oooo 0?ooo`0>0?ooo`8000003@3oool00`000000oooo0?ooo`0E0?ooo`003`3oool00`000000oooo0?oo o`080?ooo`800000303oool200000100oooo00<000000?ooo`3oool04`3oool00`000000oooo0?oo o`0;0?ooo`030000003oool0oooo0640oooo00<000000?ooo`3oool03@3oool00`000000oooo0?oo o`0A0?ooo`8000003`3oool2000000d0oooo0P00000H0?ooo`003`3oool00`000000oooo0?ooo`0: 0?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool03P3oool2000001@0oooo00<00000 0?ooo`3oool02`3oool200000600oooo00<000000?ooo`3oool03@3oool00`000000oooo0?ooo`0A 0?ooo`030000003oool0oooo00d0oooo0P00000>0?ooo`030000003oool0oooo01P0oooo000?0?oo o`030000003oool0oooo00/0oooo0P00000<0?ooo`800000403oool00`000000oooo0?ooo`0B0?oo o`030000003oool0oooo00`0oooo00<000000?ooo`3oool0G03oool00`000000oooo0?ooo`0=0?oo o`030000003oool0oooo0140oooo00<000000?ooo`3oool0303oool2000000h0oooo0P00000K0?oo o`003`3oool00`000000oooo0?ooo`0=0?ooo`030000003oool0oooo00/0oooo0P00000?0?ooo`80 00004`3oool00`000000oooo0?ooo`0<0?ooo`030000003oool0oooo05X0oooo00<000000?ooo`3o ool0303oool200000180oooo0P00000>0?ooo`030000003oool0oooo00`0oooo0P00000M0?ooo`00 3`3oool00`000000oooo0?ooo`0>0?ooo`8000003@3oool00`000000oooo0?ooo`0>0?ooo`800000 4P3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo05T0oooo00<000000?ooo`3oool0 303oool00`000000oooo0?ooo`0A0?ooo`030000003oool0oooo00`0oooo0P00000>0?ooo`030000 003oool0oooo01d0oooo000?0?ooo`8000004@3oool00`000000oooo0?ooo`0;0?ooo`800000403o ool00`000000oooo0?ooo`0@0?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool0F03o ool00`000000oooo0?ooo`0;0?ooo`030000003oool0oooo0100oooo0P00000>0?ooo`030000003o ool0oooo00`0oooo0P00000P0?ooo`003`3oool00`000000oooo0?ooo`0A0?ooo`8000003@3oool0 0`000000oooo0?ooo`0>0?ooo`8000004@3oool00`000000oooo0?ooo`0;0?ooo`030000003oool0 oooo05H0oooo00<000000?ooo`3oool02`3oool00`000000oooo0?ooo`0@0?ooo`030000003oool0 oooo00`0oooo0P00000>0?ooo`030000003oool0oooo0200oooo000?0?ooo`030000003oool0oooo 01<0oooo00<000000?ooo`3oool02`3oool200000100oooo00<000000?ooo`3oool03P3oool00`00 0000oooo0?ooo`0;0?ooo`030000003oool0oooo05H0oooo00<000000?ooo`3oool02P3oool00`00 0000oooo0?ooo`0@0?ooo`030000003oool0oooo00/0oooo0P00000>0?ooo`8000008`3oool000l0 oooo00<000000?ooo`3oool0503oool2000000d0oooo0P00000?0?ooo`030000003oool0oooo00h0 oooo00<000000?ooo`3oool02`3oool00`000000oooo0?ooo`1D0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool03`3oool2000000d0oooo00<000000?ooo`3oool03@3oool00`000000 oooo0?ooo`0S0?ooo`003`3oool00`000000oooo0?ooo`0F0?ooo`8000003@3oool00`000000oooo 0?ooo`0=0?ooo`030000003oool0oooo00h0oooo00<000000?ooo`3oool02P3oool00`000000oooo 0?ooo`1D0?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool03P3oool00`000000oooo 0?ooo`0;0?ooo`8000003P3oool2000002H0oooo000?0?ooo`030000003oool0oooo01P0oooo00<0 00000?ooo`3oool02`3oool2000000h0oooo00<000000?ooo`3oool03P3oool00`000000oooo0?oo o`0:0?ooo`030000003oool0oooo0580oooo00<000000?ooo`3oool02P3oool00`000000oooo0?oo o`0>0?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool0303oool2000002P0oooo000? 0?ooo`8000006P3oool2000000d0oooo00<000000?ooo`3oool0303oool00`000000oooo0?ooo`0= 0?ooo`030000003oool0oooo00X0oooo00<000000?ooo`3oool0DP3oool00`000000oooo0?ooo`09 0?ooo`030000003oool0oooo00h0oooo00<000000?ooo`3oool02P3oool2000000h0oooo00<00000 0?ooo`3oool0:03oool000l0oooo00<000000?ooo`3oool06`3oool2000000`0oooo0P00000=0?oo o`8000003P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo0500oooo00<000000?oo o`3oool02@3oool00`000000oooo0?ooo`0>0?ooo`030000003oool0oooo00X0oooo00<000000?oo o`3oool0303oool2000002/0oooo000?0?ooo`030000003oool0oooo01d0oooo00<000000?ooo`3o ool02`3oool00`000000oooo0?ooo`0<0?ooo`030000003oool0oooo00`0oooo00<000000?ooo`3o ool02@3oool00`000000oooo0?ooo`1@0?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3o ool03@3oool2000000`0oooo00<000000?ooo`3oool0303oool00`000000oooo0?ooo`0[0?ooo`00 3`3oool00`000000oooo0?ooo`0N0?ooo`800000303oool2000000d0oooo00<000000?ooo`3oool0 303oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo04h0oooo00<000000?ooo`3oool0 2@3oool00`000000oooo0?ooo`0<0?ooo`030000003oool0oooo00X0oooo0P00000=0?ooo`800000 ;P3oool000l0oooo00<000000?ooo`3oool0803oool2000000`0oooo0P00000<0?ooo`030000003o ool0oooo00/0oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`1>0?ooo`030000003o ool0oooo00P0oooo00<000000?ooo`3oool0303oool00`000000oooo0?ooo`0:0?ooo`030000003o ool0oooo00/0oooo0P00000`0?ooo`003`3oool00`000000oooo0?ooo`0R0?ooo`800000303oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool0203oool0 0`000000oooo0?ooo`1=0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool02`3oool0 0`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0oooo0P00000b0?ooo`003`3oool00`00 0000oooo0?ooo`0T0?ooo`8000002`3oool2000000/0oooo00<000000?ooo`3oool02P3oool00`00 0000oooo0?ooo`090?ooo`030000003oool0oooo04`0oooo00<000000?ooo`3oool0203oool00`00 0000oooo0?ooo`0;0?ooo`030000003oool0oooo00T0oooo0P00000;0?ooo`800000=03oool000l0 oooo0P00000W0?ooo`8000002`3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool0203oool00`000000oooo0?ooo`1<0?ooo`030000003oool0oooo00P0 oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00P0 oooo0`00000f0?ooo`003`3oool00`000000oooo0?ooo`0X0?ooo`8000002P3oool2000000X0oooo 00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo04/0oooo 00<000000?ooo`3oool0203oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00P0oooo 0P0000090?ooo`800000>@3oool000l0oooo00<000000?ooo`3oool0:P3oool2000000X0oooo00<0 00000?ooo`3oool0203oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00P0oooo00<0 00000?ooo`3oool0BP3oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo00T0oooo00<0 00000?ooo`3oool0203oool00`000000oooo0?ooo`070?ooo`800000>`3oool000l0oooo00<00000 0?ooo`3oool0;03oool2000000T0oooo00<000000?ooo`3oool0203oool00`000000oooo0?ooo`08 0?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool0B@3oool00`000000oooo0?ooo`08 0?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool0203oool00`000000oooo0?ooo`06 0?ooo`800000?@3oool000H0oooo1@0000040?ooo`030000003oool0oooo02h0oooo0P0000080?oo o`8000002@3oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo00P0oooo00<000000?oo o`3oool0B03oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo00T0oooo00<000000?oo o`3oool01P3oool2000000L0oooo0P00000o0?ooo`00203oool00`000000oooo0?ooo`040?ooo`03 0000003oool0oooo0300oooo0P0000080?ooo`030000003oool0oooo00H0oooo00<000000?ooo`3o ool0203oool00`000000oooo0?ooo`080?ooo`030000003oool0oooo04P0oooo00<000000?ooo`3o ool01`3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00H0oooo00<000000?ooo`3o ool0103oool300000440oooo00080?ooo`030000003oool0oooo00@0ooooo`000005000000d0oooo 00080?ooo`030000003oool0oooo00@0oooo00<000000?ooo`3oool0=@3oool2000000D0oooo0P00 00070?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0203oool00`000000oooo0?oo o`160?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0203oool00`000000oooo0?oo o`050?ooo`030000003oool0oooo00<0oooo0P0000160?ooo`00203oool00`000000oooo0?ooo`04 0?ooo`030000003oool0oooo03L0oooo0`0000040?ooo`030000003oool0oooo00D0oooo00<00000 0?ooo`3oool01`3oool00`000000oooo0?ooo`070?ooo`030000003oool0oooo04H0oooo00<00000 0?ooo`3oool01`3oool00`000000oooo0?ooo`070?ooo`030000003oool0oooo00@0oooo0P000003 0?ooo`<00000B03oool000L0oooo0P0000060?ooo`030000003oool0oooo03X0oooo0P0000030?oo o`8000001P3oool00`000000oooo0?ooo`060?ooo`030000003oool0oooo00L0oooo00<000000?oo o`3oool0A@3oool00`000000oooo0?ooo`070?ooo`030000003oool0oooo00L0oooo00<000000?oo o`3oool0103oool010000000oooo0?ooo`3oool2000004/0oooo000?0?ooo`030000003oool0oooo 03`0oooo0`0000020?ooo`030000003oool0oooo00@0oooo00<000000?ooo`3oool01P3oool00`00 0000oooo0?ooo`070?ooo`030000003oool0oooo04@0oooo00<000000?ooo`3oool01P3oool00`00 0000oooo0?ooo`070?ooo`030000003oool0oooo00@0oooo00<000000?ooo`0000000P00001=0?oo o`003`3oool00`000000oooo0?ooo`0o0?ooo`D000001@3oool00`000000oooo0?ooo`050?ooo`03 0000003oool0oooo00L0oooo00<000000?ooo`3oool0A03oool00`000000oooo0?ooo`060?ooo`03 0000003oool0oooo00H0oooo00<000000?ooo`3oool00`3oool400000500oooo000?0?ooo`800000 @`3oool4000000@0oooo00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`060?ooo`030000 003oool0oooo04@0oooo00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`060?ooo`030000 003oool0oooo00<0oooo0`00001B0?ooo`003`3oool00`000000oooo0?ooo`150?ooo`<000000`3o ool00`000000oooo0?ooo`040?ooo`030000003oool0oooo00H0oooo00<000000?ooo`3oool0@`3o ool00`000000oooo0?ooo`060?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool00P3o ool2000005D0oooo000?0?ooo`030000003oool0oooo04L0oooo1@0000070?ooo`030000003oool0 oooo00H0oooo00<000000?ooo`3oool0@P3oool00`000000oooo0?ooo`050?ooo`030000003oool0 oooo00D0oooo00<000000?ooo`3oool01000001F0?ooo`003`3oool00`000000oooo0?ooo`190?oo o`030000003oool0000000<00000103oool00`000000oooo0?ooo`060?ooo`030000003oool0oooo 0480oooo00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`040?ooo`@0000000<0oooo0000 003oool0E`3oool000l0oooo00<000000?ooo`3oool0BP3oool2000000030?ooo`000000oooo00@0 000000<0oooo0000003oool01P3oool00`000000oooo0?ooo`120?ooo`030000003oool0oooo00@0 oooo00@000000?ooo`3oool0oooo1@0000020?ooo`800000F@3oool000l0oooo00<000000?ooo`3o ool0C03oool00`000000oooo000000040?ooo`@000001P3oool00`000000oooo0?ooo`100?ooo`03 0000003oool0oooo00D0oooo100000030?ooo`040000003oool00000000005/0oooo000?0?ooo`03 0000003oool0oooo04d0oooo00<000000?ooo`0000001@3oool00`000000oooo0000000400000003 0?ooo`000000oooo0440oooo00<000000?ooo`3oool01@0000060?ooo`<00000G@3oool000l0oooo 0P00001?0?ooo`<00000103oool00`000000oooo0?ooo`040?ooo`D00000?@3oool5000000@0oooo 00<000000?ooo`3oool00`3oool2000005l0oooo000?0?ooo`030000003oool0oooo0500oooo00<0 00000?ooo`3oool00`3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo00D0000003oool01@000000oooo0?ooo`3oool00000 06h0oooo00080?ooo`030000003oool0oooo00@0oooo00<000000?ooo`3oool0G`3oool010000000 oooo0?ooo`00000g0?ooo`8000000`3oool00`000000oooo0?ooo`1/0?ooo`001P3oool010000000 oooo0?ooo`0000050?ooo`030000003oool0oooo05l0oooo00D000000?ooo`3oool0oooo0000000e 0?ooo`8000000`3oool00`000000oooo0?ooo`1]0?ooo`001`3oool2000000H0oooo00<000000?oo o`3oool0H03oool00`000000oooo0?ooo`02000003<0oooo00<000000?ooo`0000000`3oool00`00 0000oooo0?ooo`1]0?ooo`003`3oool00`000000oooo0?ooo`1P0?ooo`060000003oool0oooo0000 003oool00000<@3oool010000000oooo0?ooo`0000020?ooo`030000003oool0oooo06h0oooo000? 0?ooo`030000003oool0oooo0640oooo00<000000?ooo`0000000P3oool2000002h0oooo00D00000 0?ooo`3oool0oooo000000020?ooo`030000003oool0oooo06h0oooo000?0?ooo`800000HP3oool0 0`000000oooo000000040?ooo`030000003oool0oooo02X0oooo00D000000?ooo`3oool0oooo0000 00020?ooo`030000003oool0oooo06l0oooo000?0?ooo`030000003oool0oooo0680oooo00<00000 0?ooo`000000103oool00`000000oooo0?ooo`0W0?ooo`800000103oool010000000oooo0?ooo`00 001a0?ooo`003`3oool00`000000oooo0?ooo`1R0?ooo`030000003oool0000000D0oooo0P00000V 0?ooo`030000003oool0oooo00@0oooo00<000000?ooo`000000LP3oool000l0oooo00<000000?oo o`3oool0HP3oool00`000000oooo000000070?ooo`8000008P3oool2000000L0oooo00<000000?oo o`000000LP3oool000l0oooo00<000000?ooo`3oool0H`3oool2000000T0oooo0P00000N0?ooo`80 0000203oool00`000000oooo0000001c0?ooo`003`3oool00`000000oooo0?ooo`1S0?ooo`030000 003oool0000000X0oooo0P00000J0?ooo`8000002P3oool00`000000oooo0000001c0?ooo`003`3o ool00`000000oooo0?ooo`1T0?ooo`800000303oool2000001D0oooo0`00000<0?ooo`800000M03o ool000l0oooo0P00001U0?ooo`8000003P3oool00`000000oooo0?ooo`0?0?ooo`<000003`3oool2 000007@0oooo000?0?ooo`030000003oool0oooo06D0oooo00<000000?ooo`3oool03@3oool70000 00L0oooo0`00000A0?ooo`800000M@3oool000l0oooo00<000000?ooo`3oool0I@3oool2000001@0 oooo2000000D0?ooo`800000M@3oool000l0oooo00<000000?ooo`3oool0I@3oool200000300oooo 00<000000?ooo`3oool0M03oool000l0oooo00<000000?ooo`3oool0IP3oool00`000000oooo0?oo o`0^0?ooo`030000003oool0oooo07@0oooo000?0?ooo`030000003oool0oooo06H0oooo00<00000 0?ooo`3oool0;@3oool00`000000oooo0?ooo`1e0?ooo`003`3oool2000006L0oooo0P00000^0?oo o`030000003oool0oooo07D0oooo000?0?ooo`030000003oool0oooo06L0oooo00<000000?ooo`3o ool0:`3oool2000007L0oooo000?0?ooo`030000003oool0oooo06L0oooo0P00000/0?ooo`800000 M`3oool000l0oooo00<000000?ooo`3oool0I`3oool2000002/0oooo0P00001h0?ooo`003`3oool0 0`000000oooo0?ooo`1W0?ooo`030000003oool0000002X0oooo0P00001h0?ooo`003`3oool00`00 0000oooo0?ooo`1X0?ooo`800000:@3oool00`000000oooo0000001h0?ooo`003`3oool00`000000 oooo0?ooo`1X0?ooo`800000:@3oool00`000000oooo0000001h0?ooo`003`3oool2000006T0oooo 00<000000?ooo`0000009`3oool010000000oooo0?ooo`00001h0?ooo`003`3oool00`000000oooo 0?ooo`1X0?ooo`030000003oool0000002L0oooo00<000000?ooo`000000N@3oool000l0oooo00<0 00000?ooo`3oool0J@3oool00`000000oooo0000000U0?ooo`040000003oool0oooo000007T0oooo 000?0?ooo`030000003oool0oooo06T0oooo00<000000?ooo`0000009@3oool010000000oooo0?oo o`00001i0?ooo`001`3oool2000000H0oooo00<000000?ooo`3oool0J@3oool010000000oooo0?oo o`00000S0?ooo`050000003oool0oooo0?ooo`000000N@3oool000H0oooo00@000000?ooo`3oool0 00001@3oool00`000000oooo0?ooo`1Y0?ooo`040000003oool0oooo000002<0oooo00D000000?oo o`3oool0oooo0000001i0?ooo`00203oool00`000000oooo0?ooo`040?ooo`800000J`3oool01000 0000oooo0?ooo`00000Q0?ooo`050000003oool0oooo0?ooo`000000NP3oool000T0oooo00<00000 0?ooo`3oool00`3oool00`000000oooo0?ooo`1Z0?ooo`040000003oool0oooo00000240oooo00D0 00000?ooo`3oool0oooo0000001j0?ooo`001P3oool010000000oooo0?ooo`0000050?ooo`030000 003oool0oooo06X0oooo00D000000?ooo`3oool0oooo0000000O0?ooo`030000003oool0oooo0080 oooo00<000000?ooo`3oool0N03oool000L0oooo0P0000060?ooo`030000003oool0oooo06X0oooo 00<000000?ooo`3oool00P3oool00`000000oooo0?ooo`0L0?ooo`030000003oool0oooo0080oooo 00<000000?ooo`3oool0N03oool000l0oooo00<000000?ooo`3oool0JP3oool00`000000oooo0?oo o`020?ooo`030000003oool0oooo01/0oooo00<000000?ooo`3oool00P3oool00`000000oooo0?oo o`1i0?ooo`003`3oool00`000000oooo0?ooo`1[0?ooo`030000003oool0oooo0080oooo00<00000 0?ooo`3oool06P3oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo07T0oooo000?0?oo o`800000K03oool00`000000oooo0?ooo`020?ooo`030000003oool0oooo01T0oooo00<000000?oo o`3oool00`3oool00`000000oooo0?ooo`1i0?ooo`003`3oool00`000000oooo0?ooo`1[0?ooo`03 0000003oool0oooo00<0oooo00<000000?ooo`3oool05`3oool00`000000oooo0?ooo`040?ooo`03 0000003oool0oooo07T0oooo000?0?ooo`030000003oool0oooo06/0oooo00<000000?ooo`3oool0 103oool00`000000oooo0?ooo`0F0?ooo`030000003oool0oooo00@0oooo00<000000?ooo`3oool0 N@3oool000l0oooo00<000000?ooo`3oool0J`3oool00`000000oooo0?ooo`050?ooo`030000003o ool0oooo01@0oooo00<000000?ooo`3oool0103oool00`000000oooo0?ooo`1j0?ooo`003`3oool0 0`000000oooo0?ooo`1/0?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool04`3oool0 0`000000oooo0?ooo`040?ooo`030000003oool0oooo07X0oooo000?0?ooo`030000003oool0oooo 06`0oooo00<000000?ooo`3oool01P3oool00`000000oooo0?ooo`0A0?ooo`030000003oool0oooo 00D0oooo00<000000?ooo`3oool0NP3oool000l0oooo00<000000?ooo`3oool0K03oool00`000000 oooo0?ooo`060?ooo`030000003oool0oooo0100oooo00<000000?ooo`3oool01P3oool00`000000 oooo0?ooo`1j0?ooo`003`3oool2000006d0oooo00<000000?ooo`3oool01`3oool00`000000oooo 0?ooo`0>0?ooo`030000003oool0oooo00L0oooo00<000000?ooo`3oool0NP3oool000l0oooo00<0 00000?ooo`3oool0K03oool00`000000oooo0?ooo`080?ooo`8000003@3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo07/0oooo000?0?ooo`030000003oool0oooo06d0oooo00<00000 0?ooo`3oool02@3oool00`000000oooo0?ooo`070?ooo`<000002P3oool00`000000oooo0?ooo`1k 0?ooo`003`3oool00`000000oooo0?ooo`1]0?ooo`030000003oool0oooo00X0oooo0`0000040?oo o`8000003@3oool00`000000oooo0?ooo`1k0?ooo`003`3oool00`000000oooo0?ooo`1]0?ooo`03 0000003oool0oooo00d0oooo1000000?0?ooo`030000003oool0oooo07/0oooo000?0?ooo`030000 003oool0oooo06d0oooo00<000000?ooo`3oool07`3oool00`000000oooo0?ooo`1l0?ooo`003`3o ool2000006h0oooo00<000000?ooo`3oool07`3oool00`000000oooo0?ooo`1l0?ooo`003`3oool0 0`000000oooo0?ooo`1^0?ooo`030000003oool0oooo01h0oooo00<000000?ooo`3oool0O03oool0 00l0oooo00<000000?ooo`3oool0KP3oool00`000000oooo0?ooo`0N0?ooo`030000003oool0oooo 07`0oooo000?0?ooo`030000003oool0oooo06h0oooo00<000000?ooo`3oool07P3oool00`000000 oooo0?ooo`1l0?ooo`003`3oool00`000000oooo0?ooo`1^0?ooo`030000003oool0oooo01d0oooo 00<000000?ooo`3oool0O@3oool000l0oooo00<000000?ooo`3oool0KP3oool00`000000oooo0?oo o`0M0?ooo`030000003oool0oooo07d0oooo000?0?ooo`030000003oool0oooo06l0oooo00<00000 0?ooo`3oool0703oool00`000000oooo0?ooo`1m0?ooo`003`3oool200000700oooo00<000000?oo o`3oool0703oool00`000000oooo0?ooo`1m0?ooo`003`3oool00`000000oooo0?ooo`1_0?ooo`03 0000003oool0oooo01`0oooo00<000000?ooo`3oool0O@3oool000l0oooo00<000000?ooo`3oool0 K`3oool00`000000oooo0?ooo`0K0?ooo`030000003oool0oooo07h0oooo000?0?ooo`030000003o ool0oooo0700oooo00<000000?ooo`3oool06P3oool00`000000oooo0?ooo`1n0?ooo`00203oool3 000000@0oooo00<000000?ooo`3oool0L03oool00`000000oooo0?ooo`0J0?ooo`030000003oool0 oooo07h0oooo00090?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0L03oool00`00 0000oooo0?ooo`0J0?ooo`030000003oool0oooo07h0oooo00060?ooo`D00000103oool200000740 oooo00<000000?ooo`3oool06@3oool00`000000oooo0?ooo`1o0?ooo`001P3oool010000000oooo 0?ooo`0000050?ooo`030000003oool0oooo0740oooo00<000000?ooo`3oool0603oool00`000000 oooo0?ooo`1o0?ooo`001`3oool00`000000oooo000000050?ooo`030000003oool0oooo0740oooo 00<000000?ooo`3oool0603oool00`000000oooo0?ooo`1o0?ooo`00203oool2000000D0oooo00<0 00000?ooo`3oool0L@3oool00`000000oooo0?ooo`0H0?ooo`030000003oool0oooo07l0oooo000? 0?ooo`030000003oool0oooo0740oooo00<000000?ooo`3oool0603oool00`000000oooo0?ooo`1o 0?ooo`003`3oool00`000000oooo0?ooo`1b0?ooo`030000003oool0oooo01H0oooo00<000000?oo o`3oool0P03oool000l0oooo0P00001c0?ooo`030000003oool0oooo01H0oooo00<000000?ooo`3o ool0P03oool000l0oooo00<000000?ooo`3oool0LP3oool00`000000oooo0?ooo`0F0?ooo`030000 003oool0oooo0800oooo000?0?ooo`030000003oool0oooo07<0oooo00<000000?ooo`3oool05@3o ool00`000000oooo0?ooo`200?ooo`003`3oool00`000000oooo0?ooo`1c0?ooo`030000003oool0 oooo01@0oooo00<000000?ooo`3oool0P@3oool000l0oooo00<000000?ooo`3oool0L`3oool00`00 0000oooo0?ooo`0D0?ooo`030000003oool0oooo0840oooo000?0?ooo`030000003oool0oooo07<0 oooo00<000000?ooo`3oool0503oool00`000000oooo0?ooo`210?ooo`003`3oool00`000000oooo 0?ooo`1d0?ooo`030000003oool0oooo0180oooo00<000000?ooo`3oool0PP3oool000l0oooo0P00 001e0?ooo`030000003oool0oooo0180oooo00<000000?ooo`3oool0PP3oool000l0oooo00<00000 0?ooo`3oool0M03oool00`000000oooo0?ooo`0B0?ooo`030000003oool0oooo0880oooo000?0?oo o`030000003oool0oooo07@0oooo00<000000?ooo`3oool04P3oool00`000000oooo0?ooo`220?oo o`003`3oool00`000000oooo0?ooo`1e0?ooo`030000003oool0oooo0100oooo00<000000?ooo`3o ool0P`3oool000l0oooo00<000000?ooo`3oool0M@3oool00`000000oooo0?ooo`0@0?ooo`030000 003oool0oooo08<0oooo000?0?ooo`030000003oool0oooo07D0oooo00<000000?ooo`3oool0403o ool00`000000oooo0?ooo`230?ooo`003`3oool2000007L0oooo00<000000?ooo`3oool03P3oool0 0`000000oooo0?ooo`240?ooo`003`3oool00`000000oooo0?ooo`1f0?ooo`030000003oool0oooo 00h0oooo00<000000?ooo`3oool0Q03oool000l0oooo00<000000?ooo`3oool0M`3oool00`000000 oooo0?ooo`0=0?ooo`030000003oool0oooo08@0oooo000?0?ooo`030000003oool0oooo07L0oooo 00<000000?ooo`3oool0303oool00`000000oooo0?ooo`250?ooo`003`3oool00`000000oooo0?oo o`1h0?ooo`030000003oool0oooo00/0oooo00<000000?ooo`3oool0Q@3oool000l0oooo00<00000 0?ooo`3oool0N03oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo08H0oooo000?0?oo o`030000003oool0oooo07P0oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`260?oo o`003`3oool2000007X0oooo00<000000?ooo`3oool0203oool00`000000oooo0?ooo`270?ooo`00 3`3oool00`000000oooo0?ooo`1i0?ooo`030000003oool0oooo00P0oooo00<000000?ooo`3oool0 Q`3oool000l0oooo00<000000?ooo`3oool0NP3oool00`000000oooo0?ooo`060?ooo`030000003o ool0oooo08P0oooo000?0?ooo`030000003oool0oooo07X0oooo00<000000?ooo`3oool01P3oool0 0`000000oooo0?ooo`280?ooo`001`3oool2000000H0oooo00<000000?ooo`3oool0N`3oool00`00 0000oooo0?ooo`040?ooo`030000003oool0oooo08T0oooo00060?ooo`040000003oool0oooo0000 00D0oooo00<000000?ooo`3oool0N`3oool2000000@0oooo00<000000?ooo`3oool0RP3oool000T0 oooo00<000000?ooo`3oool00`3oool2000007d0oooo1@00002=0?ooo`001`3oool2000000H0oooo 00<000000?ooo`3oool0o`3oool?0?ooo`001`3oool00`000000oooo0?ooo`050?ooo`030000003o ool0oooo0?l0oooo3`3oool000L0oooo0`0000050?ooo`030000003oool0oooo0?l0oooo3`3oool0 00l0oooo00<000000?ooo`3oool0o`3oool?0?ooo`00\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0581768, -0.387681, \ 0.00385061, 0.0312444}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ StyleBox[\(Normalizing\ Constant\ for\ Hyperprior\), "Subsubtitle"], StyleBox[" ", "Subsubtitle"]}], "\[IndentingNewLine]", \(Sum[\ \ \ \ 1/\((\((2 k - 1)\)\ \ k)\), {k, 1, \ Infinity}]\), "\[IndentingNewLine]", }], "Input"], Cell[BoxData[ \(Log[4]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ StyleBox[\(Unconditional\ \(Prior : \ pi \((p)\)\)\), "Subsubtitle"], StyleBox[" ", "Subsubtitle"]}], "\[IndentingNewLine]", \(aa = Sum[\ \ Gamma[ 2\ k]/\((Gamma[k]^2)\)\ \ p^\((k - 1)\)\ \((1 - p)\)^\((k - 1)\)\ \ 1/\((2\ Log[2]\ \((2 k - 1)\)\ \ k)\), {k, 1, \ Infinity}]\)}], "Input"], Cell[BoxData[ \(\(-\(\(1 - \@\(1 - 4\ p + 4\ p\^2\)\)\/\(4\ \((\(-1\) + p)\)\ p\ Log[ 2]\)\)\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ StyleBox[\(Prior\ is\ bonafide\ distribution\), "Subsubtitle"], StyleBox[" ", "Subsubtitle"], "\[IndentingNewLine]", \(Integrate[\ \((1 - Abs[1 - 2\ p])\)/\((4\ Log[2]\ p\ \((1 - p)\))\), {p, 0, 1}]\), "\[IndentingNewLine]"}]}]], "Input"], Cell[BoxData[ \(1\)], "Output"] }, Open ]], Cell[BoxData[""], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[" ", "Subsubtitle"], RowBox[{ StyleBox[\(Here\ is\ the\ plot\ of\ prior\ from\ Handout\ 8\), "Subsubtitle"], "\[IndentingNewLine]", \(Plot[\((1 - Abs[1 - 2\ p])\)/\((4\ \ Log[2]\ p\ \((1 - p)\))\), {p, 0, 1}]\)}]}]], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.952381 -0.574998 0.817516 [ [.21429 .23002 -9 -9 ] [.21429 .23002 9 0 ] [.40476 .23002 -9 -9 ] [.40476 .23002 9 0 ] [.59524 .23002 -9 -9 ] [.59524 .23002 9 0 ] [.78571 .23002 -9 -9 ] [.78571 .23002 9 0 ] [.97619 .23002 -3 -9 ] [.97619 .23002 3 0 ] [.01131 .07901 -18 -4.5 ] [.01131 .07901 0 4.5 ] [.01131 .16077 -18 -4.5 ] [.01131 .16077 0 4.5 ] [.01131 .32427 -18 -4.5 ] [.01131 .32427 0 4.5 ] [.01131 .40602 -18 -4.5 ] [.01131 .40602 0 4.5 ] [.01131 .48777 -18 -4.5 ] [.01131 .48777 0 4.5 ] [.01131 .56952 -18 -4.5 ] [.01131 .56952 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .21429 .24252 m .21429 .24877 L s [(0.2)] .21429 .23002 0 1 Mshowa .40476 .24252 m .40476 .24877 L s [(0.4)] .40476 .23002 0 1 Mshowa .59524 .24252 m .59524 .24877 L s [(0.6)] .59524 .23002 0 1 Mshowa .78571 .24252 m .78571 .24877 L s [(0.8)] .78571 .23002 0 1 Mshowa .97619 .24252 m .97619 .24877 L s [(1)] .97619 .23002 0 1 Mshowa .125 Mabswid .07143 .24252 m .07143 .24627 L s .11905 .24252 m .11905 .24627 L s .16667 .24252 m .16667 .24627 L s .2619 .24252 m .2619 .24627 L s .30952 .24252 m .30952 .24627 L s .35714 .24252 m .35714 .24627 L s .45238 .24252 m .45238 .24627 L s .5 .24252 m .5 .24627 L s .54762 .24252 m .54762 .24627 L s .64286 .24252 m .64286 .24627 L s .69048 .24252 m .69048 .24627 L s .7381 .24252 m .7381 .24627 L s .83333 .24252 m .83333 .24627 L s .88095 .24252 m .88095 .24627 L s .92857 .24252 m .92857 .24627 L s .25 Mabswid 0 .24252 m 1 .24252 L s .02381 .07901 m .03006 .07901 L s [(0.8)] .01131 .07901 1 0 Mshowa .02381 .16077 m .03006 .16077 L s [(0.9)] .01131 .16077 1 0 Mshowa .02381 .32427 m .03006 .32427 L s [(1.1)] .01131 .32427 1 0 Mshowa .02381 .40602 m .03006 .40602 L s [(1.2)] .01131 .40602 1 0 Mshowa .02381 .48777 m .03006 .48777 L s [(1.3)] .01131 .48777 1 0 Mshowa .02381 .56952 m .03006 .56952 L s [(1.4)] .01131 .56952 1 0 Mshowa .125 Mabswid .02381 .01361 m .02756 .01361 L s .02381 .02996 m .02756 .02996 L s .02381 .04631 m .02756 .04631 L s .02381 .06266 m .02756 .06266 L s .02381 .09537 m .02756 .09537 L s .02381 .11172 m .02756 .11172 L s .02381 .12807 m .02756 .12807 L s .02381 .14442 m .02756 .14442 L s .02381 .17712 m .02756 .17712 L s .02381 .19347 m .02756 .19347 L s .02381 .20982 m .02756 .20982 L s .02381 .22617 m .02756 .22617 L s .02381 .25887 m .02756 .25887 L s .02381 .27522 m .02756 .27522 L s .02381 .29157 m .02756 .29157 L s .02381 .30792 m .02756 .30792 L s .02381 .34062 m .02756 .34062 L s .02381 .35697 m .02756 .35697 L s .02381 .37332 m .02756 .37332 L s .02381 .38967 m .02756 .38967 L s .02381 .42237 m .02756 .42237 L s .02381 .43872 m .02756 .43872 L s .02381 .45507 m .02756 .45507 L s .02381 .47142 m .02756 .47142 L s .02381 .50412 m .02756 .50412 L s .02381 .52047 m .02756 .52047 L s .02381 .53682 m .02756 .53682 L s .02381 .55317 m .02756 .55317 L s .02381 .58587 m .02756 .58587 L s .02381 .60223 m .02756 .60223 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .02381 .01472 m .06244 .03965 L .10458 .06936 L .14415 .10001 L .18221 .13236 L .22272 .17039 L .26171 .21107 L .30316 .25948 L .34309 .31211 L .3815 .36941 L .42237 .4391 L .46172 .51666 L .48147 .56026 L .49012 .58045 L .49468 .59141 L .49719 .59751 L .49842 .60053 L .49955 .60332 L .50085 .60232 L .50154 .60064 L .50226 .59886 L .50471 .59287 L .5095 .58135 L .51896 .55927 L .53984 .51337 L .57781 .43877 L .61824 .36983 L .65714 .31179 L .6985 .25742 L .73835 .21101 L .77668 .17099 L .81746 .13266 L .85673 .0993 L .89448 .07006 L .93468 .04159 L .97337 .01647 L .97619 .01472 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgP3oool000050?ooo`000000oooo0?ooo`000000203oool010000000oooo0?ooo`000004 0?ooo`030000003oool0oooo03<0oooo00<000000?ooo`3oool0U03oool00`000000oooo0?ooo`0k 0?ooo`000P3oool2000000X0oooo0`0000040?ooo`030000003oool0oooo03@0oooo00<000000?oo o`3oool0TP3oool00`000000oooo0?ooo`0l0?ooo`005@3oool2000003H0oooo00<000000?ooo`3o ool0T03oool00`000000oooo0?ooo`0m0?ooo`005@3oool00`000000oooo0?ooo`0f0?ooo`030000 003oool0oooo08h0oooo00<000000?ooo`3oool0?P3oool001D0oooo00<000000?ooo`3oool0=`3o ool00`000000oooo0?ooo`2<0?ooo`030000003oool0oooo03l0oooo000E0?ooo`030000003oool0 oooo03P0oooo00<000000?ooo`3oool0RP3oool00`000000oooo0?ooo`100?ooo`005@3oool00`00 0000oooo0?ooo`0i0?ooo`030000003oool0oooo08P0oooo00<000000?ooo`3oool0@@3oool001D0 oooo0P00000k0?ooo`030000003oool0oooo08H0oooo00<000000?ooo`3oool0@P3oool001D0oooo 00<000000?ooo`3oool0>`3oool00`000000oooo0?ooo`240?ooo`030000003oool0oooo04<0oooo 000E0?ooo`030000003oool0oooo02X0oooo0P0000040?ooo`8000000`3oool4000000<0oooo00<0 00000?ooo`3oool07`3oool2000000@0oooo0P0000050?ooo`<000008`3oool2000000@0oooo0P00 00030?ooo`<000008P3oool010000000oooo0?ooo`3oool2000000@0oooo0P0000040?ooo`800000 :`3oool5000000D0oooo000E0?ooo`030000003oool0oooo02T0oooo00@000000?ooo`3oool00000 203oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo01d0oooo00@000000?ooo`3oool0 00002`3oool00`000000oooo0?ooo`0Q0?ooo`040000003oool0oooo000000P0oooo00@000000?oo o`3oool00000803oool01@000000oooo0?ooo`3oool000000080oooo00<000000?ooo`3oool01P3o ool010000000oooo0?ooo`00000/0?ooo`030000003oool0oooo00D0oooo000E0?ooo`800000:P3o ool010000000oooo0?ooo`0000090?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0 703oool010000000oooo0?ooo`0000080?ooo`D000008P3oool010000000oooo0?ooo`0000080?oo o`040000003oool0oooo000001l0oooo00<000000?ooo`3oool00P3oool010000000oooo0?ooo`00 00080?ooo`040000003oool0oooo000002`0oooo00<000000?ooo`3oool01@3oool001D0oooo00<0 00000?ooo`3oool0:@3oool010000000oooo0?ooo`00000:0?ooo`030000003oool0oooo00D0oooo 00<000000?ooo`3oool06`3oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo0000 02<0oooo00@000000?ooo`3oool00000203oool3000001l0oooo00<000000?ooo`3oool00`3oool0 10000000oooo0?ooo`0000090?ooo`800000;@3oool00`000000oooo0?ooo`050?ooo`005@3oool0 0`000000oooo0?ooo`0Y0?ooo`040000003oool0oooo000000P0oooo00@000000?ooo`3oool00000 1`3oool00`000000oooo0?ooo`0J0?ooo`040000003oool0oooo000000T0oooo00<000000?ooo`00 00008`3oool010000000oooo0?ooo`0000090?ooo`030000003oool0oooo01d0oooo00<000000?oo o`3oool0103oool010000000oooo0?ooo`0000080?ooo`040000003oool0oooo000002`0oooo00<0 00000?ooo`3oool01@3oool001D0oooo00<000000?ooo`3oool0:P3oool2000000X0oooo0P000008 0?ooo`030000003oool0oooo01/0oooo0P00000;0?ooo`800000903oool2000000X0oooo0`00000L 0?ooo`030000003oool0oooo00H0oooo0P00000:0?ooo`800000;03oool2000000L0oooo000E0?oo o`800000@P3oool00`000000oooo0?ooo`1h0?ooo`030000003oool0oooo04T0oooo000E0?ooo`03 0000003oool0oooo0480oooo00<000000?ooo`3oool0MP3oool00`000000oooo0?ooo`1:0?ooo`00 5@3oool00`000000oooo0?ooo`130?ooo`030000003oool0oooo07@0oooo00<000000?ooo`3oool0 B`3oool001D0oooo00<000000?ooo`3oool0@`3oool00`000000oooo0?ooo`1c0?ooo`030000003o ool0oooo04`0oooo000E0?ooo`030000003oool0oooo04@0oooo00<000000?ooo`3oool0L@3oool0 0`000000oooo0?ooo`1=0?ooo`003P3ooooo000001<00000000E0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`050?ooo`030000003oool0oooo0080 oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool02@3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo0080 oooo00<000000?ooo`3oool01@3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool02P3oool00`000000oooo0?ooo`0:0?ooo`030000003oool0oooo00X0 oooo00<000000?ooo`3oool01@3oool001D0oooo00<000000?ooo`3oool0<@3oool00`000000oooo 0?ooo`0C0?ooo`030000003oool0oooo01/0oooo00<000000?ooo`3oool0<03oool00`000000oooo 0?ooo`0K0?ooo`030000003oool0oooo01<0oooo00<000000?ooo`3oool0<@3oool00`000000oooo 0?ooo`050?ooo`005@3oool00`000000oooo0?ooo`170?ooo`030000003oool0oooo06/0oooo00<0 00000?ooo`3oool0D03oool001D0oooo0P0000190?ooo`030000003oool0oooo06T0oooo00<00000 0?ooo`3oool0D@3oool001D0oooo00<000000?ooo`3oool0B@3oool00`000000oooo0?ooo`1W0?oo o`030000003oool0oooo0580oooo000E0?ooo`030000003oool0oooo04X0oooo00<000000?ooo`3o ool0IP3oool00`000000oooo0?ooo`1B0?ooo`005@3oool00`000000oooo0?ooo`1;0?ooo`030000 003oool0oooo06@0oooo00<000000?ooo`3oool0D`3oool001D0oooo00<000000?ooo`3oool0B`3o ool00`000000oooo0?ooo`1S0?ooo`030000003oool0oooo05@0oooo000E0?ooo`800000C@3oool0 0`000000oooo0?ooo`1Q0?ooo`030000003oool0oooo05D0oooo000E0?ooo`030000003oool0oooo 04d0oooo00<000000?ooo`3oool0H03oool00`000000oooo0?ooo`1E0?ooo`005@3oool00`000000 oooo0?ooo`1>0?ooo`030000003oool0oooo05h0oooo00<000000?ooo`3oool0EP3oool001D0oooo 00<000000?ooo`3oool0CP3oool00`000000oooo0?ooo`1M0?ooo`030000003oool0oooo05L0oooo 000E0?ooo`800000D03oool00`000000oooo0?ooo`1K0?ooo`030000003oool0oooo05P0oooo000E 0?ooo`030000003oool0oooo0500oooo00<000000?ooo`3oool0FP3oool00`000000oooo0?ooo`1H 0?ooo`005@3oool00`000000oooo0?ooo`1A0?ooo`030000003oool0oooo05P0oooo00<000000?oo o`3oool0F@3oool001D0oooo00<000000?ooo`3oool0DP3oool00`000000oooo0?ooo`1F0?ooo`03 0000003oool0oooo05X0oooo000E0?ooo`030000003oool0oooo0580oooo00<000000?ooo`3oool0 E@3oool00`000000oooo0?ooo`1K0?ooo`005@3oool2000005@0oooo00<000000?ooo`3oool0E03o ool00`000000oooo0?ooo`1K0?ooo`005@3oool00`000000oooo0?ooo`1D0?ooo`030000003oool0 oooo0580oooo00<000000?ooo`3oool0G03oool000030?ooo`000000000000<000000P3oool20000 00<0oooo1@0000030?ooo`030000003oool0oooo05D0oooo00<000000?ooo`3oool0D03oool00`00 0000oooo0?ooo`1M0?ooo`000`3oool00`000000oooo0?ooo`090?ooo`030000003oool0oooo00<0 oooo00<000000?ooo`3oool0E@3oool00`000000oooo0?ooo`1@0?ooo`030000003oool0oooo05d0 oooo00030?ooo`030000003oool0oooo00T0oooo00<000000?ooo`3oool00`3oool3000005H0oooo 00<000000?ooo`3oool0CP3oool00`000000oooo0?ooo`1N0?ooo`000`3oool00`000000oooo0?oo o`090?ooo`030000003oool0oooo00<0oooo00<000000?ooo`3oool0E`3oool00`000000oooo0?oo o`1<0?ooo`030000003oool0oooo05l0oooo00030?ooo`030000003oool0oooo00T0oooo00<00000 0?ooo`3oool00`3oool00`000000oooo0?ooo`1G0?ooo`030000003oool0oooo04`0oooo00<00000 0?ooo`3oool0G`3oool00080oooo0P00000:0?ooo`8000001@3oool00`000000oooo0?ooo`1H0?oo o`030000003oool0oooo04X0oooo00<000000?ooo`3oool0H03oool001D0oooo00<000000?ooo`3o ool0F@3oool00`000000oooo0?ooo`190?ooo`030000003oool0oooo0600oooo000E0?ooo`800000 FP3oool00`000000oooo0?ooo`180?ooo`030000003oool0oooo0640oooo000E0?ooo`030000003o ool0oooo05X0oooo00<000000?ooo`3oool0AP3oool00`000000oooo0?ooo`1R0?ooo`005@3oool0 0`000000oooo0?ooo`1K0?ooo`030000003oool0oooo04D0oooo00<000000?ooo`3oool0HP3oool0 01D0oooo00<000000?ooo`3oool0F`3oool00`000000oooo0?ooo`140?ooo`030000003oool0oooo 06<0oooo000E0?ooo`800000G@3oool00`000000oooo0?ooo`120?ooo`030000003oool0oooo06@0 oooo000E0?ooo`030000003oool0oooo05d0oooo00<000000?ooo`3oool0@@3oool00`000000oooo 0?ooo`1T0?ooo`005@3oool00`000000oooo0?ooo`1M0?ooo`030000003oool0oooo0400oooo00<0 00000?ooo`3oool0I@3oool001D0oooo00<000000?ooo`3oool0GP3oool00`000000oooo0?ooo`0o 0?ooo`030000003oool0oooo06D0oooo000E0?ooo`030000003oool0oooo05l0oooo00<000000?oo o`3oool0?@3oool00`000000oooo0?ooo`1V0?ooo`005@3oool200000600oooo00<000000?ooo`3o ool0?03oool00`000000oooo0?ooo`1W0?ooo`005@3oool00`000000oooo0?ooo`1P0?ooo`030000 003oool0oooo03/0oooo00<000000?ooo`3oool0I`3oool001D0oooo00<000000?ooo`3oool0H03o ool00`000000oooo0?ooo`0j0?ooo`030000003oool0oooo06P0oooo000E0?ooo`030000003oool0 oooo0640oooo00<000000?ooo`3oool0>@3oool00`000000oooo0?ooo`1X0?ooo`005@3oool20000 0680oooo00<000000?ooo`3oool0>03oool00`000000oooo0?ooo`1Y0?ooo`005@3oool00`000000 oooo0?ooo`1R0?ooo`030000003oool0oooo03H0oooo00<000000?ooo`3oool0JP3oool000030?oo o`000000000000<000000P3oool2000000<0oooo100000040?ooo`030000003oool0oooo06<0oooo 00<000000?ooo`3oool0=@3oool00`000000oooo0?ooo`1Z0?ooo`000`3oool00`000000oooo0?oo o`070?ooo`030000003oool0oooo00D0oooo00<000000?ooo`3oool0H`3oool00`000000oooo0?oo o`0d0?ooo`030000003oool0oooo06/0oooo00030?ooo`030000003oool0oooo00P0oooo00<00000 0?ooo`3oool0103oool3000006@0oooo00<000000?ooo`3oool0"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.0825959, 0.683594, \ 0.00385924, 0.00449589}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[" ", "Subtitle"], RowBox[{ StyleBox[\(Direct\ Bayes, \ Exact\ result\ for\ n = \(20\ and\ x = 12\)\), "Subtitle"], StyleBox[" ", "Subtitle"], "\[IndentingNewLine]", "\[IndentingNewLine]", "\[IndentingNewLine]", \(brule\ = \ Integrate[ p\ \ Gamma[ n + 1]/\((Gamma[x + 1]\ Gamma[ n - x + 1])\)\ \ p^\((x)\)\ \((1 - p)\)^\((n - x)\)\ \ \((1 - Abs[1 - 2\ p])\)/\((4\ Log[ 2]\ p\ \((1 - p)\))\)\ /. \ {n \[Rule] 20, x \[Rule] 12}, \ {p, 0, 1}]/\[IndentingNewLine]Integrate[\ Gamma[n + 1]/\((Gamma[x + 1]\ Gamma[ n - x + 1])\)\ \ p^\((x)\)\ \((1 - p)\)^\((n - x)\)\ \ \((1 - Abs[1 - 2\ p])\)/\((4\ Log[ 2]\ p\ \((1 - p)\))\) /. \ {n \[Rule] 20, x \[Rule] 12}, \ {p, 0, 1}]\), "\[IndentingNewLine]", "\[IndentingNewLine]", \(brule // N\)}]}]], "Input"], Cell[BoxData[ \(288241\/495798\)], "Output"], Cell[BoxData[ \(0.5813678151182538`\)], "Output"] }, Open ]], Cell[BoxData[ RowBox[{ StyleBox[\(Hierarchical\ Bayes\), "Subtitle"], StyleBox[" ", "Subtitle"], "\[IndentingNewLine]"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ StyleBox[\(First\ Beta\ Binomial\), "Subsubtitle"], StyleBox[" ", "Subsubtitle"], "\[IndentingNewLine]", \(Integrate[\ Gamma[n + 1]/\((Gamma[x + 1]\ Gamma[n - x + 1])\)\ p^ x\ \((1 - p)\)^\((n - x)\)\ * Gamma[2\ k]/\((Gamma[k]^2)\)\ p^\((k - 1)\)\ \((1 - p)\)^\((k - 1)\), {p, 0, 1}]\), "\[IndentingNewLine]", "\[IndentingNewLine]"}]}]], "Input"], Cell[BoxData[ \(\((Gamma[2\ k]\ Gamma[1 + n]\ If[ Re[k + x] > 0 && Re[x] < Re[ k + n], \(Gamma[k + n - x]\ Gamma[k + x]\)\/Gamma[2\ k + \ n], Integrate[\((1 - p)\)\^\(\(-1\) + k + n - x\)\ p\^\(\(-1\) + k + x\), {p, 0, 1}, Assumptions \[Rule] Re[k + n] \[LessEqual] Re[x] || Re[k + x] \[LessEqual] 0]])\)/\((Gamma[k]\^2\ Gamma[ 1 + n - x]\ Gamma[1 + x])\)\)], "Output"] }, Open ]], Cell[BoxData[ StyleBox[\(\(The\ result\ \(BetaBinomial : \ Gamma[2 k]/\((Gamma[k]^2)\)\ *\ \ Gamma[n + 1]/\((\ Gamma[x + 1]\ Gamma[n - x + 1])\)\ *\((Gamma[k + x]\ Gamma[ k + n - x])\)/ Gamma[2\ k + n]\)\)\(\ \)\(\[IndentingNewLine]\) \), "Subsubtitle"]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ StyleBox[\(Expectation\ in\ Hierarchical\ Model\), "Subsubtitle"], StyleBox[" ", "Subsubtitle"], "\[IndentingNewLine]", \(bruleh\ = \ Sum[\ \((k + x)\)/\((2\ k + n)\)\ \ Gamma[ 2 k]/\((Gamma[k]^2)\)\ *\ \ Gamma[n + 1]/\((\ Gamma[x + 1]\ Gamma[n - x + 1])\)\ *\((Gamma[ k + x]\ Gamma[k + n - x])\)/ Gamma[2\ k + n]\ \ \ \ \ 1/\((2\ Log[2]\ k\ \((2 k - 1)\))\), {k, 1, Infinity}]/ Sum[\ Gamma[2 k]/\((Gamma[k]^2)\)\ *\ \ Gamma[n + 1]/\((\ Gamma[x + 1]\ Gamma[n - x + 1])\)\ *\((Gamma[ k + x]\ Gamma[k + n - x])\)/ Gamma[2\ k + n]\ \ \ 1/\((2\ Log[2]\ k\ \((2 k - 1)\))\), {k, 1, Infinity}]\)}]}]], "Input"], Cell[BoxData[ RowBox[{\(General::"spell1"\), \(\(:\)\(\ \)\), "\<\"Possible spelling \ error: new symbol name \\\"\\!\\(bruleh\\)\\\" is similar to existing symbol \ \\\"\\!\\(brule\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"], Cell[BoxData[ \(\(x\ \((\(-1\) + HypergeometricPFQ[{\(-\(1\/2\)\), n - x, 1 + x}, {1\/2 \ + n\/2, 1 + n\/2}, 1])\)\)\/\(n\ \((\(-1\) + \ HypergeometricPFQ[{\(-\(1\/2\)\), n - x, x}, {1\/2 + n\/2, n\/2}, 1])\)\)\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{ StyleBox[\(The\ rule\ is\ the\ same\ as\ in\ Direct\ Bayes\), "Subsubtitle"], "\[IndentingNewLine]", \(bruleh /. {n \[Rule] 20, \ x \[Rule] 12} // N\)}]}]], "Input"], Cell[BoxData[ \(0.5813678151182538`\)], "Output"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 717}}, WindowSize->{754, 566}, WindowMargins->{{Automatic, 194}, {-41, Automatic}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[1754, 51, 1691, 52, 302, "Input"], Cell[CellGroupData[{ Cell[3470, 107, 248, 4, 70, "Input"], Cell[3721, 113, 42001, 1368, 186, 16034, 1043, "GraphicsData", "PostScript", \ "Graphics"], Cell[45725, 1483, 130, 3, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[45892, 1491, 308, 7, 70, "Input"], Cell[46203, 1500, 40, 1, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[46280, 1506, 394, 9, 70, "Input"], Cell[46677, 1517, 126, 2, 49, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[46840, 1524, 379, 9, 90, "Input"], Cell[47222, 1535, 35, 1, 29, "Output"] }, Open ]], Cell[47272, 1539, 26, 0, 30, "Input"], Cell[CellGroupData[{ Cell[47323, 1543, 339, 9, 50, "Input"], Cell[47665, 1554, 22123, 516, 186, 4098, 289, "GraphicsData", "PostScript", \ "Graphics"], Cell[69791, 2072, 130, 3, 29, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[69958, 2080, 1199, 26, 217, "Input"], Cell[71160, 2108, 48, 1, 42, "Output"], Cell[71211, 2111, 53, 1, 29, "Output"] }, Open ]], Cell[71279, 2115, 164, 5, 57, "Input"], Cell[CellGroupData[{ Cell[71468, 2124, 545, 12, 150, "Input"], Cell[72016, 2138, 483, 9, 98, "Output"] }, Open ]], Cell[72514, 2150, 343, 7, 129, "Input"], Cell[CellGroupData[{ Cell[72882, 2161, 969, 20, 150, "Input"], Cell[73854, 2183, 359, 5, 37, "Message"], Cell[74216, 2190, 230, 4, 59, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[74483, 2199, 268, 6, 70, "Input"], Cell[74754, 2207, 53, 1, 29, "Output"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)