From Btry4790
Problem 1:
log(p(x_L))=-43.89539
j i m_ji(x_p=A) m_ji(x_p=C) m_ji(x_p=G) m_ji(x_p=T)
1 37 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
2 37 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
3 38 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
4 39 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
5 42 5.200000e-02 1.100000e-01 4.200000e-02 8.420000e-01
6 41 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
7 40 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
8 40 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
9 46 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
10 45 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
11 44 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
12 44 5.200000e-02 1.100000e-01 4.200000e-02 8.420000e-01
13 48 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
14 48 5.200000e-02 1.100000e-01 4.200000e-02 8.420000e-01
15 50 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
16 51 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
17 51 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
18 52 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
19 53 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
20 54 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
21 58 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
22 56 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
23 56 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
24 57 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
25 60 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
26 59 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
27 59 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
28 62 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
29 62 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
30 65 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
31 67 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
32 66 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
33 66 8.500000e-02 2.800000e-02 8.080000e-01 2.900000e-02
34 69 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
35 70 3.200000e-02 8.160000e-01 2.800000e-02 7.600000e-02
36 71 8.310000e-01 4.600000e-02 1.220000e-01 5.300000e-02
37 38 5.753351e-01 3.421820e-02 9.645194e-02 3.955736e-02
38 39 4.735343e-02 5.339694e-03 6.901123e-02 5.890675e-03
39 43 3.344017e-02 2.280648e-03 1.162363e-02 2.611294e-03
40 41 2.252533e-02 5.440429e-01 1.964496e-02 5.554546e-02
41 42 1.507131e-02 3.627672e-01 1.313998e-02 3.734798e-02
42 43 3.610356e-03 3.607265e-02 2.979628e-03 2.956865e-02
43 47 1.099189e-04 8.214826e-05 4.825996e-05 7.866844e-05
44 45 3.882718e-02 1.116904e-02 1.142803e-02 4.039848e-02
45 46 2.705883e-02 2.178010e-03 5.167220e-03 3.592368e-03
46 47 1.875246e-02 1.154699e-03 3.263444e-03 1.377961e-03
47 49 1.734959e-06 1.885545e-07 3.859363e-07 2.122972e-07
48 49 7.682648e-03 8.039275e-02 6.354160e-03 6.082532e-02
49 50 1.244146e-08 1.447150e-08 4.574390e-09 1.280238e-08
50 55 8.695606e-09 1.109053e-09 1.759402e-09 1.186055e-09
51 52 2.252533e-02 5.440429e-01 1.964496e-02 5.554546e-02
52 53 1.507131e-02 3.627672e-01 1.313998e-02 3.734798e-02
53 54 1.005222e-02 2.418954e-01 8.763836e-03 2.492357e-02
54 55 1.566262e-03 5.843910e-03 6.045838e-03 1.373977e-03
55 64 1.251418e-11 6.392262e-12 1.050625e-11 2.895018e-12
56 57 6.156624e-02 1.934480e-02 5.284528e-01 2.008369e-02
57 58 4.069049e-02 1.270250e-02 3.456859e-01 1.319163e-02
58 61 2.288595e-03 8.899231e-03 8.311988e-03 1.981625e-03
59 60 6.719778e-02 7.229416e-03 8.836750e-02 7.994401e-03
60 61 4.735321e-02 3.188537e-03 1.555067e-02 3.654268e-03
61 63 1.023288e-04 3.255530e-05 1.187597e-04 1.774598e-05
62 63 5.753351e-01 3.421820e-02 9.645194e-02 3.955736e-02
63 64 4.996955e-05 4.015133e-06 1.649854e-05 4.128205e-06
64 65 5.358238e-16 5.587645e-17 2.175676e-16 5.018270e-17
65 68 3.724958e-16 2.361556e-17 7.595351e-17 2.680384e-17
66 67 6.156624e-02 1.934480e-02 5.284528e-01 2.008369e-02
67 68 4.069049e-02 1.270250e-02 3.456859e-01 1.319163e-02
68 69 1.485525e-17 1.716069e-18 2.308730e-17 1.885267e-18
69 70 1.050559e-17 7.221284e-19 3.788316e-18 8.260838e-19
70 71 3.105018e-19 5.061739e-19 1.458567e-19 1.185399e-19
Problem 2:
i p(x_i = A|x_E) p(x_i = C|x_E) p(x_i = G|x_E) p(x_i = T|x_E)
37 0.9745562358 0.0002899438 2.455571e-02 0.0005981057
38 0.8137531074 0.0014198315 1.822214e-01 0.0026056873
39 0.9367693955 0.0047623856 5.067617e-02 0.0077920523
40 0.0003918148 0.9941464492 1.180012e-04 0.0053437348
41 0.0067736264 0.9447484503 1.256357e-03 0.0472215664
42 0.1548247996 0.3633255517 1.661298e-02 0.4652366654
43 0.8509836693 0.0463009323 3.675454e-02 0.0659608583
44 0.9153322719 0.0046518350 1.321655e-02 0.0667993416
45 0.9861688598 0.0010850070 5.837835e-03 0.0069082984
46 0.9863343872 0.0021471616 6.528804e-03 0.0049896468
47 0.9208634945 0.0251827960 1.958290e-02 0.0343708077
48 0.1544336404 0.3853689683 1.585550e-02 0.4443418882
49 0.8516249502 0.0518085178 3.154461e-02 0.0650219204
50 0.9408954451 0.0126903984 3.616086e-02 0.0102532958
51 0.0001291227 0.9983537741 6.535212e-05 0.0014517511
52 0.0009471381 0.9953293857 6.553647e-04 0.0030681115
53 0.0185310960 0.9508655950 1.630195e-02 0.0143013566
54 0.3889973884 0.1419631787 4.444187e-01 0.0246207114
55 0.8526457944 0.0183676060 1.215361e-01 0.0074504972
56 0.0062294500 0.0002300865 9.933558e-01 0.0001846464
57 0.0464428521 0.0050323722 9.467007e-01 0.0018240500
58 0.3983148620 0.1372529234 4.428052e-01 0.0216270065
59 0.7937129455 0.0007694868 2.041739e-01 0.0013436467
60 0.8961348608 0.0011347613 1.016369e-01 0.0010934612
61 0.8359555978 0.0097438828 1.503050e-01 0.0039955426
62 0.9903931068 0.0002120293 9.014648e-03 0.0003802160
63 0.9422504575 0.0012455932 5.546870e-02 0.0010352455
64 0.9306215966 0.0018904017 6.608438e-02 0.0014036215
65 0.9361535070 0.0005819502 6.246647e-02 0.0007980773
66 0.0098431914 0.0000889344 9.899198e-01 0.0001480403
67 0.0850937706 0.0004306645 9.137819e-01 0.0006937048
68 0.7918770541 0.0022362036 2.037722e-01 0.0021144989
69 0.9204383582 0.0109757614 6.424713e-02 0.0043387529
70 0.8095040326 0.1188715535 5.127654e-02 0.0203478712
71 0.8810721349 0.0552504366 4.222450e-02 0.0214529311
See attached tree for maximum marginals
Problem 3:
log(MAP) = -48.82849
The maxima of the marginal distributions differ from the maximum joint configuration of the unobserved variables
at nodes 54 and 58. Unless the joint distribution is the product of the marginals, the maxima of the marginals will
not necessarily by the maximum of the joint configuration (as we saw in Bishop problem 8.27).
Problem 4:
-33091.32
Problem 5:
t log_likelihood
0.00000000 -Inf
0.05263158 -36609.87
0.10526316 -34026.88
0.15789474 -33724.63**
0.21052632 -34288.76
0.26315789 -35268.91
0.31578947 -36458.32
0.36842105 -37741.43
0.42105263 -39045.01
0.47368421 -40318.88
0.52631579 -41527.54
0.57894737 -42646.41
0.63157895 -43660.26
0.68421053 -44562.09
0.73684211 -45352.01
0.78947368 -46035.55
0.84210526 -46621.76
0.89473684 -47121.35
0.94736842 -47545.40
1.00000000 -47904.41
approximate mle of t:
0.15789
Problem 5 (extra credit)
t_mle = 0.144
