August 23 2008 5:27:12.624 PM BAYES_WEIGHT Simple Bayesian Statistics demonstrations. Suppose an event is a roll of a die. We have two dice to choose from, each with a different characteristic probability density function (PDF). We know the form of both PDF's. For each event, the probability we will choose die #1 or die #2 is W1 or W2. But W1 and W2 are unknown to us, and we must deduce them from the results of the rolls. Run parameters: Number of components is 2 Exact weights: 1 0.750000 2 0.250000 I, THETA_TRUE(I,1) THETA_TRUE(I,2): 1 0.625000E-01 0.562500 2 0.125000 0.625000E-01 3 0.250000 0.625000E-01 4 0.375000 0.00000 5 0.625000E-01 0.187500 6 0.125000 0.125000 Number of observations = 1000 Rolls per observation = 10 Initial parameters: I, ALPHA(I), WEIGHT_EST(I): 1 1.00000 0.500000 2 1.00000 0.500000 BAYES_BETA - Observation: 1 Current and summary events 1 2 2 2 2 2 3 2 2 4 3 3 5 1 1 6 0 0 Total 10 10 (Unknown) component calls and probability: 1 7 0.700000 2 3 0.300000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 1.80000 0.600000 2 1.20000 0.400000 BAYES_BETA - Observation: 2 Current and summary events 1 1 3 2 0 2 3 2 4 4 5 8 5 0 1 6 2 2 Total 10 20 (Unknown) component calls and probability: 1 14 0.700000 2 6 0.300000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 2.80000 0.700000 2 1.20000 0.300000 BAYES_BETA - Observation: 3 Current and summary events 1 1 4 2 0 2 3 1 5 4 5 13 5 0 1 6 3 5 Total 10 30 (Unknown) component calls and probability: 1 21 0.700000 2 9 0.300000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 3.70323 0.740645 2 1.29677 0.259355 BAYES_BETA - Observation: 4 Current and summary events 1 3 7 2 1 3 3 0 5 4 3 16 5 2 3 6 1 6 Total 10 40 (Unknown) component calls and probability: 1 26 0.650000 2 14 0.350000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 3.94410 0.657350 2 2.05590 0.342650 BAYES_BETA - Observation: 5 Current and summary events 1 1 8 2 3 6 3 2 7 4 4 20 5 0 3 6 0 6 Total 10 50 (Unknown) component calls and probability: 1 35 0.700000 2 15 0.300000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 4.73735 0.676765 2 2.26265 0.323235 BAYES_BETA - Observation: 6 Current and summary events 1 1 9 2 2 8 3 3 10 4 1 21 5 3 6 6 0 6 Total 10 60 (Unknown) component calls and probability: 1 43 0.716667 2 17 0.283333 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 5.14839 0.643549 2 2.85161 0.356451 BAYES_BETA - Observation: 7 Current and summary events 1 0 9 2 0 8 3 1 11 4 4 25 5 4 10 6 1 7 Total 10 70 (Unknown) component calls and probability: 1 51 0.728571 2 19 0.271429 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 5.52410 0.613789 2 3.47590 0.386211 BAYES_BETA - Observation: 8 Current and summary events 1 0 9 2 0 8 3 4 15 4 3 28 5 0 10 6 3 10 Total 10 80 (Unknown) component calls and probability: 1 60 0.750000 2 20 0.250000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 6.38818 0.638817 2 3.61183 0.361183 BAYES_BETA - Observation: 9 Current and summary events 1 1 10 2 2 10 3 3 18 4 1 29 5 1 11 6 2 12 Total 10 90 (Unknown) component calls and probability: 1 67 0.744444 2 23 0.255556 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 7.26433 0.660394 2 3.73567 0.339606 BAYES_BETA - Observation: 10 Current and summary events 1 2 12 2 2 12 3 1 19 4 3 32 5 1 12 6 1 13 Total 10 100 (Unknown) component calls and probability: 1 75 0.750000 2 25 0.250000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 7.44201 0.620167 2 4.55799 0.379833 BAYES_BETA - Observation: 100 Current and summary events 1 3 172 2 1 105 3 2 199 4 3 286 5 1 93 6 0 145 Total 10 1000 (Unknown) component calls and probability: 1 765 0.765000 2 235 0.235000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 75.6291 0.741461 2 26.3709 0.258539 BAYES_BETA - Observation: 200 Current and summary events 1 4 351 2 1 188 3 1 401 4 2 602 5 1 193 6 1 265 Total 10 2000 (Unknown) component calls and probability: 1 1522 0.761000 2 478 0.239000 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 153.008 0.757466 2 48.9918 0.242534 BAYES_BETA - Observation: 300 Current and summary events 1 4 562 2 3 292 3 0 600 4 2 867 5 1 284 6 0 395 Total 10 3000 (Unknown) component calls and probability: 1 2258 0.752667 2 742 0.247333 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 224.447 0.743202 2 77.5529 0.256798 BAYES_BETA - Observation: 400 Current and summary events 1 0 735 2 1 387 3 4 805 4 3 1164 5 2 369 6 0 540 Total 10 4000 (Unknown) component calls and probability: 1 3033 0.758250 2 967 0.241750 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 298.097 0.741536 2 103.902 0.258464 BAYES_BETA - Observation: 500 Current and summary events 1 2 920 2 0 509 3 2 1013 4 4 1415 5 0 472 6 2 671 Total 10 5000 (Unknown) component calls and probability: 1 3786 0.757200 2 1214 0.242800 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 370.170 0.737390 2 131.830 0.262610 BAYES_BETA - Observation: 600 Current and summary events 1 1 1104 2 1 629 3 1 1205 4 2 1714 5 1 576 6 4 772 Total 10 6000 (Unknown) component calls and probability: 1 4557 0.759500 2 1443 0.240500 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 445.648 0.740280 2 156.352 0.259720 BAYES_BETA - Observation: 700 Current and summary events 1 2 1298 2 0 743 3 2 1422 4 2 1979 5 0 669 6 4 889 Total 10 7000 (Unknown) component calls and probability: 1 5300 0.757143 2 1700 0.242857 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 521.217 0.742475 2 180.782 0.257525 BAYES_BETA - Observation: 800 Current and summary events 1 3 1485 2 0 855 3 1 1619 4 3 2232 5 1 779 6 2 1030 Total 10 8000 (Unknown) component calls and probability: 1 6038 0.754750 2 1962 0.245250 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 597.185 0.744619 2 204.815 0.255381 BAYES_BETA - Observation: 900 Current and summary events 1 2 1686 2 0 957 3 2 1832 4 4 2517 5 2 863 6 0 1145 Total 10 9000 (Unknown) component calls and probability: 1 6798 0.755333 2 2202 0.244667 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 670.162 0.742974 2 231.838 0.257026 BAYES_BETA - Observation: 1000 Current and summary events 1 1 1880 2 1 1083 3 3 2034 4 4 2778 5 1 962 6 0 1263 Total 10 10000 (Unknown) component calls and probability: 1 7544 0.754400 2 2456 0.245600 Estimated parameters: I, ALPHA(I), WEIGHT_EST(I): 1 745.042 0.743555 2 256.957 0.256445 BAYES_WEIGHT: Normal end of execution. August 23 2008 5:27:12.657 PM