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 Topic: Evolutionary Computation, Stuff that drives AEs nuts < Next Oldest | Next Newest >
Wesley R. Elsberry

Posts: 4991
Joined: May 2002

More "weasel" math:

Probability of a candidate changing a parent's correct base to an incorrect base = PCandidate_C2I =

(1 - (1 - (u * (K - 1) / K))C)

Probability that a population will have at least one candidate that preserves all the correct bases from the parent of the previous generation = PPopulation_C2C =

1 - (PCandidate_C2I )N

Checked via Monte Carlo methods, and using the N=50 and u=0.05 values that (IIRC) ROb was often using:

 Code Sample 1000 runs, 00 correct : p_c2c calc = 1.00000, MC = 1.00000; p_c2i calc = 0.00000, MC = 0.000001000 runs, N=50, u=0.05000, K=27, C=0, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.000001000 runs, 01 correct : p_c2c calc = 0.95185, MC = 0.93800; p_c2i calc = 0.04815, MC = 0.062001000 runs, N=50, u=0.05000, K=27, C=1, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.047261000 runs, 02 correct : p_c2c calc = 0.90602, MC = 0.90400; p_c2i calc = 0.09398, MC = 0.096001000 runs, N=50, u=0.05000, K=27, C=2, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.094761000 runs, 03 correct : p_c2c calc = 0.86240, MC = 0.85900; p_c2i calc = 0.13760, MC = 0.141001000 runs, N=50, u=0.05000, K=27, C=3, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.135921000 runs, 04 correct : p_c2c calc = 0.82088, MC = 0.82400; p_c2i calc = 0.17912, MC = 0.176001000 runs, N=50, u=0.05000, K=27, C=4, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.178101000 runs, 05 correct : p_c2c calc = 0.78135, MC = 0.80400; p_c2i calc = 0.21865, MC = 0.196001000 runs, N=50, u=0.05000, K=27, C=5, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.216181000 runs, 06 correct : p_c2c calc = 0.74373, MC = 0.75900; p_c2i calc = 0.25627, MC = 0.241001000 runs, N=50, u=0.05000, K=27, C=6, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.258481000 runs, 07 correct : p_c2c calc = 0.70792, MC = 0.74400; p_c2i calc = 0.29208, MC = 0.256001000 runs, N=50, u=0.05000, K=27, C=7, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.292261000 runs, 08 correct : p_c2c calc = 0.67384, MC = 0.67200; p_c2i calc = 0.32616, MC = 0.328001000 runs, N=50, u=0.05000, K=27, C=8, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.324601000 runs, 09 correct : p_c2c calc = 0.64139, MC = 0.62100; p_c2i calc = 0.35861, MC = 0.379001000 runs, N=50, u=0.05000, K=27, C=9, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.360861000 runs, 10 correct : p_c2c calc = 0.61051, MC = 0.61000; p_c2i calc = 0.38949, MC = 0.390001000 runs, N=50, u=0.05000, K=27, C=10, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.389661000 runs, 11 correct : p_c2c calc = 0.58112, MC = 0.59500; p_c2i calc = 0.41888, MC = 0.405001000 runs, N=50, u=0.05000, K=27, C=11, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.420601000 runs, 12 correct : p_c2c calc = 0.55314, MC = 0.54600; p_c2i calc = 0.44686, MC = 0.454001000 runs, N=50, u=0.05000, K=27, C=12, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.447941000 runs, 13 correct : p_c2c calc = 0.52650, MC = 0.52000; p_c2i calc = 0.47350, MC = 0.480001000 runs, N=50, u=0.05000, K=27, C=13, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.470501000 runs, 14 correct : p_c2c calc = 0.50115, MC = 0.50900; p_c2i calc = 0.49885, MC = 0.491001000 runs, N=50, u=0.05000, K=27, C=14, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.501301000 runs, 15 correct : p_c2c calc = 0.47702, MC = 0.45800; p_c2i calc = 0.52298, MC = 0.542001000 runs, N=50, u=0.05000, K=27, C=15, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.516581000 runs, 16 correct : p_c2c calc = 0.45406, MC = 0.48200; p_c2i calc = 0.54594, MC = 0.518001000 runs, N=50, u=0.05000, K=27, C=16, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.542701000 runs, 17 correct : p_c2c calc = 0.43219, MC = 0.41800; p_c2i calc = 0.56781, MC = 0.582001000 runs, N=50, u=0.05000, K=27, C=17, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.567081000 runs, 18 correct : p_c2c calc = 0.41139, MC = 0.41200; p_c2i calc = 0.58861, MC = 0.588001000 runs, N=50, u=0.05000, K=27, C=18, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.592181000 runs, 19 correct : p_c2c calc = 0.39158, MC = 0.35000; p_c2i calc = 0.60842, MC = 0.650001000 runs, N=50, u=0.05000, K=27, C=19, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.610701000 runs, 20 correct : p_c2c calc = 0.37272, MC = 0.37200; p_c2i calc = 0.62728, MC = 0.628001000 runs, N=50, u=0.05000, K=27, C=20, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.627621000 runs, 21 correct : p_c2c calc = 0.35478, MC = 0.33300; p_c2i calc = 0.64522, MC = 0.667001000 runs, N=50, u=0.05000, K=27, C=21, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.640481000 runs, 22 correct : p_c2c calc = 0.33770, MC = 0.32200; p_c2i calc = 0.66230, MC = 0.678001000 runs, N=50, u=0.05000, K=27, C=22, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.661461000 runs, 23 correct : p_c2c calc = 0.32144, MC = 0.31500; p_c2i calc = 0.67856, MC = 0.685001000 runs, N=50, u=0.05000, K=27, C=23, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.678541000 runs, 24 correct : p_c2c calc = 0.30596, MC = 0.28900; p_c2i calc = 0.69404, MC = 0.711001000 runs, N=50, u=0.05000, K=27, C=24, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.693801000 runs, 25 correct : p_c2c calc = 0.29123, MC = 0.28000; p_c2i calc = 0.70877, MC = 0.720001000 runs, N=50, u=0.05000, K=27, C=25, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.707921000 runs, 26 correct : p_c2c calc = 0.27721, MC = 0.27700; p_c2i calc = 0.72279, MC = 0.723001000 runs, N=50, u=0.05000, K=27, C=26, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.721541000 runs, 27 correct : p_c2c calc = 0.26386, MC = 0.23500; p_c2i calc = 0.73614, MC = 0.765001000 runs, N=50, u=0.05000, K=27, C=27, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.736441000 runs, 28 correct : p_c2c calc = 0.25116, MC = 0.24500; p_c2i calc = 0.74884, MC = 0.755001000 runs, N=50, u=0.05000, K=27, C=28, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.74932

The above completely explains why a list showing the best candidate from each generation is highly unlikely to show any change in a correct character when a candidate bearing it was selected as best in a previous generation. The proportion of candidates that had a change of a correct character to an incorrect one nonetheless rises to almost three-quarters of each generation when almost all characters are correct.

Now doing the Monte Carlo methods on the situation with N=12 and u=0.18, where I picked N and u in order to get a range of values for the population that went down to a relatively small probability.

 Code Sample 1000 runs, 00 correct : p_c2c calc = 1.00000, MC = 1.00000; p_c2i calc = 0.00000, MC = 0.000001000 runs, N=12, u=0.18000, K=27, C=0, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.000001000 runs, 01 correct : p_c2c calc = 0.82667, MC = 0.82700; p_c2i calc = 0.17333, MC = 0.173001000 runs, N=12, u=0.18000, K=27, C=1, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.170581000 runs, 02 correct : p_c2c calc = 0.68338, MC = 0.69300; p_c2i calc = 0.31662, MC = 0.307001000 runs, N=12, u=0.18000, K=27, C=2, p_pop_c2c calc = 1.00000, MC = 1.00000Proportion of candidates w/C2I bases = 0.322331000 runs, 03 correct : p_c2c calc = 0.56493, MC = 0.56600; p_c2i calc = 0.43507, MC = 0.434001000 runs, N=12, u=0.18000, K=27, C=3, p_pop_c2c calc = 0.99995, MC = 1.00000Proportion of candidates w/C2I bases = 0.441171000 runs, 04 correct : p_c2c calc = 0.46701, MC = 0.48600; p_c2i calc = 0.53299, MC = 0.514001000 runs, N=12, u=0.18000, K=27, C=4, p_pop_c2c calc = 0.99947, MC = 1.00000Proportion of candidates w/C2I bases = 0.535081000 runs, 05 correct : p_c2c calc = 0.38606, MC = 0.39800; p_c2i calc = 0.61394, MC = 0.602001000 runs, N=12, u=0.18000, K=27, C=5, p_pop_c2c calc = 0.99713, MC = 0.99700Proportion of candidates w/C2I bases = 0.611671000 runs, 06 correct : p_c2c calc = 0.31914, MC = 0.32100; p_c2i calc = 0.68086, MC = 0.679001000 runs, N=12, u=0.18000, K=27, C=6, p_pop_c2c calc = 0.99008, MC = 0.99200Proportion of candidates w/C2I bases = 0.679671000 runs, 07 correct : p_c2c calc = 0.26382, MC = 0.25100; p_c2i calc = 0.73618, MC = 0.749001000 runs, N=12, u=0.18000, K=27, C=7, p_pop_c2c calc = 0.97466, MC = 0.97200Proportion of candidates w/C2I bases = 0.734751000 runs, 08 correct : p_c2c calc = 0.21809, MC = 0.23600; p_c2i calc = 0.78191, MC = 0.764001000 runs, N=12, u=0.18000, K=27, C=8, p_pop_c2c calc = 0.94778, MC = 0.95500Proportion of candidates w/C2I bases = 0.783831000 runs, 09 correct : p_c2c calc = 0.18029, MC = 0.19100; p_c2i calc = 0.81971, MC = 0.809001000 runs, N=12, u=0.18000, K=27, C=9, p_pop_c2c calc = 0.90797, MC = 0.91400Proportion of candidates w/C2I bases = 0.814921000 runs, 10 correct : p_c2c calc = 0.14904, MC = 0.16000; p_c2i calc = 0.85096, MC = 0.840001000 runs, N=12, u=0.18000, K=27, C=10, p_pop_c2c calc = 0.85582, MC = 0.85600Proportion of candidates w/C2I bases = 0.856671000 runs, 11 correct : p_c2c calc = 0.12321, MC = 0.12900; p_c2i calc = 0.87679, MC = 0.871001000 runs, N=12, u=0.18000, K=27, C=11, p_pop_c2c calc = 0.79357, MC = 0.78600Proportion of candidates w/C2I bases = 0.880831000 runs, 12 correct : p_c2c calc = 0.10185, MC = 0.09800; p_c2i calc = 0.89815, MC = 0.902001000 runs, N=12, u=0.18000, K=27, C=12, p_pop_c2c calc = 0.72446, MC = 0.72100Proportion of candidates w/C2I bases = 0.899001000 runs, 13 correct : p_c2c calc = 0.08420, MC = 0.08100; p_c2i calc = 0.91580, MC = 0.919001000 runs, N=12, u=0.18000, K=27, C=13, p_pop_c2c calc = 0.65196, MC = 0.66100Proportion of candidates w/C2I bases = 0.914581000 runs, 14 correct : p_c2c calc = 0.06960, MC = 0.07600; p_c2i calc = 0.93040, MC = 0.924001000 runs, N=12, u=0.18000, K=27, C=14, p_pop_c2c calc = 0.57925, MC = 0.55400Proportion of candidates w/C2I bases = 0.931921000 runs, 15 correct : p_c2c calc = 0.05754, MC = 0.06100; p_c2i calc = 0.94246, MC = 0.939001000 runs, N=12, u=0.18000, K=27, C=15, p_pop_c2c calc = 0.50891, MC = 0.50000Proportion of candidates w/C2I bases = 0.943331000 runs, 16 correct : p_c2c calc = 0.04756, MC = 0.03900; p_c2i calc = 0.95244, MC = 0.961001000 runs, N=12, u=0.18000, K=27, C=16, p_pop_c2c calc = 0.44278, MC = 0.45400Proportion of candidates w/C2I bases = 0.950831000 runs, 17 correct : p_c2c calc = 0.03932, MC = 0.03800; p_c2i calc = 0.96068, MC = 0.962001000 runs, N=12, u=0.18000, K=27, C=17, p_pop_c2c calc = 0.38206, MC = 0.38200Proportion of candidates w/C2I bases = 0.961831000 runs, 18 correct : p_c2c calc = 0.03250, MC = 0.03600; p_c2i calc = 0.96750, MC = 0.964001000 runs, N=12, u=0.18000, K=27, C=18, p_pop_c2c calc = 0.32736, MC = 0.32200Proportion of candidates w/C2I bases = 0.968751000 runs, 19 correct : p_c2c calc = 0.02687, MC = 0.03200; p_c2i calc = 0.97313, MC = 0.968001000 runs, N=12, u=0.18000, K=27, C=19, p_pop_c2c calc = 0.27881, MC = 0.25300Proportion of candidates w/C2I bases = 0.976001000 runs, 20 correct : p_c2c calc = 0.02221, MC = 0.02000; p_c2i calc = 0.97779, MC = 0.980001000 runs, N=12, u=0.18000, K=27, C=20, p_pop_c2c calc = 0.23629, MC = 0.23400Proportion of candidates w/C2I bases = 0.978171000 runs, 21 correct : p_c2c calc = 0.01836, MC = 0.01800; p_c2i calc = 0.98164, MC = 0.982001000 runs, N=12, u=0.18000, K=27, C=21, p_pop_c2c calc = 0.19941, MC = 0.19100Proportion of candidates w/C2I bases = 0.983081000 runs, 22 correct : p_c2c calc = 0.01518, MC = 0.01600; p_c2i calc = 0.98482, MC = 0.984001000 runs, N=12, u=0.18000, K=27, C=22, p_pop_c2c calc = 0.16769, MC = 0.17300Proportion of candidates w/C2I bases = 0.984581000 runs, 23 correct : p_c2c calc = 0.01255, MC = 0.00600; p_c2i calc = 0.98745, MC = 0.994001000 runs, N=12, u=0.18000, K=27, C=23, p_pop_c2c calc = 0.14061, MC = 0.15400Proportion of candidates w/C2I bases = 0.986171000 runs, 24 correct : p_c2c calc = 0.01037, MC = 0.01400; p_c2i calc = 0.98963, MC = 0.986001000 runs, N=12, u=0.18000, K=27, C=24, p_pop_c2c calc = 0.11762, MC = 0.11200Proportion of candidates w/C2I bases = 0.990421000 runs, 25 correct : p_c2c calc = 0.00858, MC = 0.00600; p_c2i calc = 0.99142, MC = 0.994001000 runs, N=12, u=0.18000, K=27, C=25, p_pop_c2c calc = 0.09819, MC = 0.10400Proportion of candidates w/C2I bases = 0.990751000 runs, 26 correct : p_c2c calc = 0.00709, MC = 0.01200; p_c2i calc = 0.99291, MC = 0.988001000 runs, N=12, u=0.18000, K=27, C=26, p_pop_c2c calc = 0.08183, MC = 0.08000Proportion of candidates w/C2I bases = 0.992831000 runs, 27 correct : p_c2c calc = 0.00586, MC = 0.00700; p_c2i calc = 0.99414, MC = 0.993001000 runs, N=12, u=0.18000, K=27, C=27, p_pop_c2c calc = 0.06810, MC = 0.07400Proportion of candidates w/C2I bases = 0.993581000 runs, 28 correct : p_c2c calc = 0.00484, MC = 0.00600; p_c2i calc = 0.99516, MC = 0.994001000 runs, N=12, u=0.18000, K=27, C=28, p_pop_c2c calc = 0.05661, MC = 0.04400Proportion of candidates w/C2I bases = 0.99617

The above shows that in order to have low probabilities that the best candidate in a generation will retain all the characters that were correct in the parent, one must have small N and relatively high u values.

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"You can't teach an old dogma new tricks." - Dorothy Parker

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