December 30 2007 1:39:36.651 PM BAYES_DICE FORTRAN90 version Simple Bayesian Statistics demonstrations. BAYES_DICE Simple Bayesian Statistics demonstrations. Suppose we're watching a "system" and trying to analyze its behavior. Each time we observe the system, it rolls a die a certain number of times, and reports the results. We want to estimate THETA(1) through and THETA(6), the probabilities of each result. We treat the values of THETA as random variables themselves, controlled by a Dirichlet probability density function with parameters ALPHA(1) through ALPHA(6). We make an arbitrary guess for initial ALPHA values. We observe the system and adjust the ALPHA's using Bayes's Law. Run parameters: I, THETA_TRUE(I): 1 0.625000E-01 2 0.125000 3 0.250000 4 0.375000 5 0.625000E-01 6 0.125000 Number of observations = 4 Rolls per observation = 500 Initial parameters: I, ALPHA_INIT(I), THETA_INIT(I): 1 1.00000 0.166667 2 1.00000 0.166667 3 1.00000 0.166667 4 1.00000 0.166667 5 1.00000 0.166667 6 1.00000 0.166667 BAYES_DICE - Observation: 1 Observed data: 1 21 2 53 3 130 4 199 5 32 6 65 Post observation data: Total rolls: 500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 21 22.0000 0.434783E-01 2 53 54.0000 0.106719 3 130 131.000 0.258893 4 199 200.000 0.395257 5 32 33.0000 0.652174E-01 6 65 66.0000 0.130435 BAYES_DICE - Observation: 2 Observed data: 1 32 2 62 3 128 4 181 5 29 6 68 Post observation data: Total rolls: 1000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 53 54.0000 0.536779E-01 2 115 116.000 0.115308 3 258 259.000 0.257455 4 380 381.000 0.378728 5 61 62.0000 0.616302E-01 6 133 134.000 0.133201 BAYES_DICE - Observation: 3 Observed data: 1 33 2 65 3 112 4 192 5 29 6 69 Post observation data: Total rolls: 1500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 86 87.0000 0.577689E-01 2 180 181.000 0.120186 3 370 371.000 0.246348 4 572 573.000 0.380478 5 90 91.0000 0.604250E-01 6 202 203.000 0.134794 BAYES_DICE - Observation: 4 Observed data: 1 33 2 60 3 131 4 191 5 21 6 64 Post observation data: Total rolls: 2000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 119 120.000 0.598205E-01 2 240 241.000 0.120140 3 501 502.000 0.250249 4 763 764.000 0.380857 5 111 112.000 0.558325E-01 6 266 267.000 0.133101 BAYES_DICE Simple Bayesian Statistics demonstrations. Suppose we're watching a "system" and trying to analyze its behavior. Each time we observe the system, it rolls a die a certain number of times, and reports the results. We want to estimate THETA(1) through and THETA(6), the probabilities of each result. We treat the values of THETA as random variables themselves, controlled by a Dirichlet probability density function with parameters ALPHA(1) through ALPHA(6). We make an arbitrary guess for initial ALPHA values. We observe the system and adjust the ALPHA's using Bayes's Law. Run parameters: I, THETA_TRUE(I): 1 0.625000E-01 2 0.125000 3 0.250000 4 0.375000 5 0.625000E-01 6 0.125000 Number of observations = 4 Rolls per observation = 5000 Initial parameters: I, ALPHA_INIT(I), THETA_INIT(I): 1 1.00000 0.166667 2 1.00000 0.166667 3 1.00000 0.166667 4 1.00000 0.166667 5 1.00000 0.166667 6 1.00000 0.166667 BAYES_DICE - Observation: 1 Observed data: 1 314 2 632 3 1214 4 1874 5 327 6 639 Post observation data: Total rolls: 5000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 314 315.000 0.629245E-01 2 632 633.000 0.126448 3 1214 1215.00 0.242709 4 1874 1875.00 0.374551 5 327 328.000 0.655214E-01 6 639 640.000 0.127847 BAYES_DICE - Observation: 2 Observed data: 1 331 2 624 3 1244 4 1868 5 304 6 629 Post observation data: Total rolls: 10000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 645 646.000 0.645613E-01 2 1256 1257.00 0.125625 3 2458 2459.00 0.245753 4 3742 3743.00 0.374076 5 631 632.000 0.631621E-01 6 1268 1269.00 0.126824 BAYES_DICE - Observation: 3 Observed data: 1 298 2 605 3 1256 4 1866 5 334 6 641 Post observation data: Total rolls: 15000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 943 944.000 0.629082E-01 2 1861 1862.00 0.124084 3 3714 3715.00 0.247568 4 5608 5609.00 0.373784 5 965 966.000 0.643743E-01 6 1909 1910.00 0.127282 BAYES_DICE - Observation: 4 Observed data: 1 320 2 647 3 1255 4 1831 5 312 6 635 Post observation data: Total rolls: 20000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1263 1264.00 0.631810E-01 2 2508 2509.00 0.125412 3 4969 4970.00 0.248425 4 7439 7440.00 0.371888 5 1277 1278.00 0.638808E-01 6 2544 2545.00 0.127212 BAYES_DICE Simple Bayesian Statistics demonstrations. Suppose we're watching a "system" and trying to analyze its behavior. Each time we observe the system, it rolls a die a certain number of times, and reports the results. We want to estimate THETA(1) through and THETA(6), the probabilities of each result. We treat the values of THETA as random variables themselves, controlled by a Dirichlet probability density function with parameters ALPHA(1) through ALPHA(6). We make an arbitrary guess for initial ALPHA values. We observe the system and adjust the ALPHA's using Bayes's Law. Run parameters: I, THETA_TRUE(I): 1 0.625000E-01 2 0.125000 3 0.250000 4 0.375000 5 0.625000E-01 6 0.125000 Number of observations = 40 Rolls per observation = 500 Initial parameters: I, ALPHA_INIT(I), THETA_INIT(I): 1 1.00000 0.166667 2 1.00000 0.166667 3 1.00000 0.166667 4 1.00000 0.166667 5 1.00000 0.166667 6 1.00000 0.166667 BAYES_DICE - Observation: 1 Observed data: 1 30 2 63 3 117 4 186 5 32 6 72 Post observation data: Total rolls: 500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 30 31.0000 0.612648E-01 2 63 64.0000 0.126482 3 117 118.000 0.233202 4 186 187.000 0.369565 5 32 33.0000 0.652174E-01 6 72 73.0000 0.144269 BAYES_DICE - Observation: 2 Observed data: 1 29 2 51 3 116 4 210 5 29 6 65 Post observation data: Total rolls: 1000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 59 60.0000 0.596421E-01 2 114 115.000 0.114314 3 233 234.000 0.232604 4 396 397.000 0.394632 5 61 62.0000 0.616302E-01 6 137 138.000 0.137177 BAYES_DICE - Observation: 3 Observed data: 1 32 2 75 3 113 4 184 5 25 6 71 Post observation data: Total rolls: 1500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 91 92.0000 0.610890E-01 2 189 190.000 0.126162 3 346 347.000 0.230412 4 580 581.000 0.385790 5 86 87.0000 0.577689E-01 6 208 209.000 0.138778 BAYES_DICE - Observation: 4 Observed data: 1 23 2 63 3 136 4 183 5 29 6 66 Post observation data: Total rolls: 2000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 114 115.000 0.573280E-01 2 252 253.000 0.126122 3 482 483.000 0.240778 4 763 764.000 0.380857 5 115 116.000 0.578265E-01 6 274 275.000 0.137089 BAYES_DICE - Observation: 5 Observed data: 1 20 2 74 3 99 4 224 5 25 6 58 Post observation data: Total rolls: 2500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 134 135.000 0.538707E-01 2 326 327.000 0.130487 3 581 582.000 0.232243 4 987 988.000 0.394254 5 140 141.000 0.562650E-01 6 332 333.000 0.132881 BAYES_DICE - Observation: 6 Observed data: 1 28 2 73 3 111 4 178 5 47 6 63 Post observation data: Total rolls: 3000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 162 163.000 0.542249E-01 2 399 400.000 0.133067 3 692 693.000 0.230539 4 1165 1166.00 0.387891 5 187 188.000 0.625416E-01 6 395 396.000 0.131737 BAYES_DICE - Observation: 7 Observed data: 1 29 2 63 3 136 4 179 5 26 6 67 Post observation data: Total rolls: 3500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 191 192.000 0.547633E-01 2 462 463.000 0.132059 3 828 829.000 0.236452 4 1344 1345.00 0.383628 5 213 214.000 0.610382E-01 6 462 463.000 0.132059 BAYES_DICE - Observation: 8 Observed data: 1 22 2 61 3 124 4 211 5 22 6 60 Post observation data: Total rolls: 4000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 213 214.000 0.534199E-01 2 523 524.000 0.130804 3 952 953.000 0.237893 4 1555 1556.00 0.388417 5 235 236.000 0.589116E-01 6 522 523.000 0.130554 BAYES_DICE - Observation: 9 Observed data: 1 27 2 75 3 117 4 183 5 34 6 64 Post observation data: Total rolls: 4500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 240 241.000 0.534842E-01 2 598 599.000 0.132934 3 1069 1070.00 0.237461 4 1738 1739.00 0.385930 5 269 270.000 0.599201E-01 6 586 587.000 0.130271 BAYES_DICE - Observation: 10 Observed data: 1 32 2 68 3 120 4 189 5 37 6 54 Post observation data: Total rolls: 5000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 272 273.000 0.545346E-01 2 666 667.000 0.133240 3 1189 1190.00 0.237715 4 1927 1928.00 0.385138 5 306 307.000 0.613264E-01 6 640 641.000 0.128046 BAYES_DICE - Observation: 11 Observed data: 1 34 2 46 3 121 4 190 5 32 6 77 Post observation data: Total rolls: 5500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 306 307.000 0.557574E-01 2 712 713.000 0.129495 3 1310 1311.00 0.238104 4 2117 2118.00 0.384671 5 338 339.000 0.615692E-01 6 717 718.000 0.130403 BAYES_DICE - Observation: 12 Observed data: 1 34 2 61 3 144 4 176 5 37 6 48 Post observation data: Total rolls: 6000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 340 341.000 0.567766E-01 2 773 774.000 0.128871 3 1454 1455.00 0.242258 4 2293 2294.00 0.381951 5 375 376.000 0.626041E-01 6 765 766.000 0.127539 BAYES_DICE - Observation: 13 Observed data: 1 28 2 66 3 135 4 181 5 21 6 69 Post observation data: Total rolls: 6500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 368 369.000 0.567169E-01 2 839 840.000 0.129112 3 1589 1590.00 0.244390 4 2474 2475.00 0.380418 5 396 397.000 0.610206E-01 6 834 835.000 0.128343 BAYES_DICE - Observation: 14 Observed data: 1 28 2 63 3 142 4 181 5 28 6 58 Post observation data: Total rolls: 7000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 396 397.000 0.566657E-01 2 902 903.000 0.128890 3 1731 1732.00 0.247217 4 2655 2656.00 0.379104 5 424 425.000 0.606623E-01 6 892 893.000 0.127462 BAYES_DICE - Observation: 15 Observed data: 1 32 2 51 3 148 4 176 5 35 6 58 Post observation data: Total rolls: 7500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 428 429.000 0.571543E-01 2 953 954.000 0.127098 3 1879 1880.00 0.250466 4 2831 2832.00 0.377298 5 459 460.000 0.612843E-01 6 950 951.000 0.126699 BAYES_DICE - Observation: 16 Observed data: 1 36 2 84 3 120 4 181 5 21 6 58 Post observation data: Total rolls: 8000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 464 465.000 0.580814E-01 2 1037 1038.00 0.129653 3 1999 2000.00 0.249813 4 3012 3013.00 0.376343 5 480 481.000 0.600799E-01 6 1008 1009.00 0.126030 BAYES_DICE - Observation: 17 Observed data: 1 32 2 71 3 129 4 172 5 36 6 60 Post observation data: Total rolls: 8500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 496 497.000 0.584293E-01 2 1108 1109.00 0.130379 3 2128 2129.00 0.250294 4 3184 3185.00 0.374442 5 516 517.000 0.607806E-01 6 1068 1069.00 0.125676 BAYES_DICE - Observation: 18 Observed data: 1 32 2 76 3 117 4 171 5 38 6 66 Post observation data: Total rolls: 9000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 528 529.000 0.587386E-01 2 1184 1185.00 0.131579 3 2245 2246.00 0.249389 4 3355 3356.00 0.372640 5 554 555.000 0.616256E-01 6 1134 1135.00 0.126027 BAYES_DICE - Observation: 19 Observed data: 1 34 2 69 3 130 4 173 5 31 6 63 Post observation data: Total rolls: 9500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 562 563.000 0.592258E-01 2 1253 1254.00 0.131917 3 2375 2376.00 0.249947 4 3528 3529.00 0.371239 5 585 586.000 0.616453E-01 6 1197 1198.00 0.126026 BAYES_DICE - Observation: 20 Observed data: 1 37 2 64 3 125 4 180 5 37 6 57 Post observation data: Total rolls: 10000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 599 600.000 0.599640E-01 2 1317 1318.00 0.131721 3 2500 2501.00 0.249950 4 3708 3709.00 0.370678 5 622 623.000 0.622626E-01 6 1254 1255.00 0.125425 BAYES_DICE - Observation: 21 Observed data: 1 38 2 61 3 133 4 182 5 35 6 51 Post observation data: Total rolls: 10500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 637 638.000 0.607272E-01 2 1378 1379.00 0.131258 3 2633 2634.00 0.250714 4 3890 3891.00 0.370360 5 657 658.000 0.626309E-01 6 1305 1306.00 0.124310 BAYES_DICE - Observation: 22 Observed data: 1 27 2 57 3 142 4 174 5 37 6 63 Post observation data: Total rolls: 11000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 664 665.000 0.604216E-01 2 1435 1436.00 0.130474 3 2775 2776.00 0.252226 4 4064 4065.00 0.369344 5 694 695.000 0.631474E-01 6 1368 1369.00 0.124387 BAYES_DICE - Observation: 23 Observed data: 1 24 2 49 3 125 4 204 5 26 6 72 Post observation data: Total rolls: 11500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 688 689.000 0.598818E-01 2 1484 1485.00 0.129063 3 2900 2901.00 0.252129 4 4268 4269.00 0.371024 5 720 721.000 0.626630E-01 6 1440 1441.00 0.125239 BAYES_DICE - Observation: 24 Observed data: 1 21 2 73 3 131 4 182 5 33 6 60 Post observation data: Total rolls: 12000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 709 710.000 0.591371E-01 2 1557 1558.00 0.129768 3 3031 3032.00 0.252540 4 4450 4451.00 0.370731 5 753 754.000 0.628019E-01 6 1500 1501.00 0.125021 BAYES_DICE - Observation: 25 Observed data: 1 28 2 65 3 124 4 195 5 33 6 55 Post observation data: Total rolls: 12500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 737 738.000 0.590117E-01 2 1622 1623.00 0.129778 3 3155 3156.00 0.252359 4 4645 4646.00 0.371502 5 786 787.000 0.629298E-01 6 1555 1556.00 0.124420 BAYES_DICE - Observation: 26 Observed data: 1 33 2 61 3 121 4 185 5 41 6 59 Post observation data: Total rolls: 13000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 770 771.000 0.592803E-01 2 1683 1684.00 0.129479 3 3276 3277.00 0.251961 4 4830 4831.00 0.371444 5 827 828.000 0.636629E-01 6 1614 1615.00 0.124173 BAYES_DICE - Observation: 27 Observed data: 1 33 2 63 3 132 4 188 5 33 6 51 Post observation data: Total rolls: 13500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 803 804.000 0.595291E-01 2 1746 1747.00 0.129350 3 3408 3409.00 0.252406 4 5018 5019.00 0.371613 5 860 861.000 0.637494E-01 6 1665 1666.00 0.123353 BAYES_DICE - Observation: 28 Observed data: 1 24 2 60 3 116 4 193 5 42 6 65 Post observation data: Total rolls: 14000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 827 828.000 0.591175E-01 2 1806 1807.00 0.129016 3 3524 3525.00 0.251678 4 5211 5212.00 0.372126 5 902 903.000 0.644724E-01 6 1730 1731.00 0.123590 BAYES_DICE - Observation: 29 Observed data: 1 35 2 76 3 121 4 182 5 26 6 60 Post observation data: Total rolls: 14500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 862 863.000 0.594926E-01 2 1882 1883.00 0.129808 3 3645 3646.00 0.251344 4 5393 5394.00 0.371846 5 928 929.000 0.640425E-01 6 1790 1791.00 0.123466 BAYES_DICE - Observation: 30 Observed data: 1 30 2 84 3 123 4 186 5 26 6 51 Post observation data: Total rolls: 15000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 892 893.000 0.595095E-01 2 1966 1967.00 0.131081 3 3768 3769.00 0.251166 4 5579 5580.00 0.371851 5 954 955.000 0.636412E-01 6 1841 1842.00 0.122751 BAYES_DICE - Observation: 31 Observed data: 1 27 2 53 3 141 4 191 5 33 6 55 Post observation data: Total rolls: 15500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 919 920.000 0.593319E-01 2 2019 2020.00 0.130272 3 3909 3910.00 0.252160 4 5770 5771.00 0.372179 5 987 988.000 0.637173E-01 6 1896 1897.00 0.122340 BAYES_DICE - Observation: 32 Observed data: 1 30 2 61 3 137 4 183 5 27 6 62 Post observation data: Total rolls: 16000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 949 950.000 0.593527E-01 2 2080 2081.00 0.130014 3 4046 4047.00 0.252843 4 5953 5954.00 0.371986 5 1014 1015.00 0.634137E-01 6 1958 1959.00 0.122392 BAYES_DICE - Observation: 33 Observed data: 1 33 2 60 3 132 4 182 5 34 6 59 Post observation data: Total rolls: 16500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 982 983.000 0.595541E-01 2 2140 2141.00 0.129710 3 4178 4179.00 0.253181 4 6135 6136.00 0.371744 5 1048 1049.00 0.635526E-01 6 2017 2018.00 0.122259 BAYES_DICE - Observation: 34 Observed data: 1 31 2 67 3 113 4 206 5 27 6 56 Post observation data: Total rolls: 17000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1013 1014.00 0.596260E-01 2 2207 2208.00 0.129837 3 4291 4292.00 0.252382 4 6341 6342.00 0.372927 5 1075 1076.00 0.632718E-01 6 2073 2074.00 0.121957 BAYES_DICE - Observation: 35 Observed data: 1 30 2 66 3 119 4 196 5 26 6 63 Post observation data: Total rolls: 17500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1043 1044.00 0.596367E-01 2 2273 2274.00 0.129898 3 4410 4411.00 0.251971 4 6537 6538.00 0.373472 5 1101 1102.00 0.629498E-01 6 2136 2137.00 0.122072 BAYES_DICE - Observation: 36 Observed data: 1 29 2 48 3 123 4 197 5 33 6 70 Post observation data: Total rolls: 18000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1072 1073.00 0.595912E-01 2 2321 2322.00 0.128957 3 4533 4534.00 0.251805 4 6734 6735.00 0.374042 5 1134 1135.00 0.630345E-01 6 2206 2207.00 0.122570 BAYES_DICE - Observation: 37 Observed data: 1 35 2 59 3 125 4 174 5 33 6 74 Post observation data: Total rolls: 18500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1107 1108.00 0.598725E-01 2 2380 2381.00 0.128661 3 4658 4659.00 0.251756 4 6908 6909.00 0.373338 5 1167 1168.00 0.631147E-01 6 2280 2281.00 0.123257 BAYES_DICE - Observation: 38 Observed data: 1 34 2 72 3 106 4 200 5 27 6 61 Post observation data: Total rolls: 19000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1141 1142.00 0.600863E-01 2 2452 2453.00 0.129065 3 4764 4765.00 0.250710 4 7108 7109.00 0.374040 5 1194 1195.00 0.628749E-01 6 2341 2342.00 0.123224 BAYES_DICE - Observation: 39 Observed data: 1 25 2 73 3 131 4 188 5 34 6 49 Post observation data: Total rolls: 19500 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1166 1167.00 0.598277E-01 2 2525 2526.00 0.129499 3 4895 4896.00 0.251000 4 7296 7297.00 0.374090 5 1228 1229.00 0.630063E-01 6 2390 2391.00 0.122578 BAYES_DICE - Observation: 40 Observed data: 1 34 2 69 3 129 4 169 5 34 6 65 Post observation data: Total rolls: 20000 Estimated parameters: I, COUNT(I), ALPHA_POST(I), THETA_POST(I): 1 1200 1201.00 0.600320E-01 2 2594 2595.00 0.129711 3 5024 5025.00 0.251175 4 7465 7466.00 0.373188 5 1262 1263.00 0.631311E-01 6 2455 2456.00 0.122763 BAYES_DICE: Normal end of execution. December 30 2007 1:39:36.730 PM