Hidden Markov Model Practice Problems Here are 3 HMMs and several Viterbi matrices for potential observed sequences for each. See if you can construct each DP matrix before checking your answer. HMM #1 States: q0 q1 q2 StartProbs: q0=0.333 q1=0.333 q2=0.333 Transitions: q0->q0=0.100 q0->q1=0.900 q0->q2=0.000 q1->q0=0.750 q1->q1=0.000 q1->q2=0.250 q2->q0=1.000 q2->q1=0.000 q2->q2=0.000 Emissions: q0: H=0.500 T=0.500 q1: H=0.750 T=0.250 q2: H=0.250 T=0.750 Example Viterbi Matrices: H H T q0 0.166667 0.093750 0.042187 q1 0.250000 0.112500 0.021094 q2 0.083333 0.015625 0.021094 H H H q0 0.166667 0.093750 0.042187 q1 0.250000 0.112500 0.063281 q2 0.083333 0.015625 0.007031 H T H T H T q0 0.166667 0.093750 0.023438 0.023730 0.005933 0.006007 q1 0.250000 0.037500 0.063281 0.005273 0.016018 0.001335 q2 0.083333 0.046875 0.002344 0.011865 0.000330 0.003003 T T T T q0 0.166667 0.125000 0.014062 0.010547 q1 0.083333 0.037500 0.028125 0.003164 q2 0.250000 0.015625 0.007031 0.005273 =============================================================================== HMM #2: States: Biased Fair StartProbs: Biased=0.500 Fair=0.500 Transitions: Biased->Biased=0.900 Biased->Fair=0.100 Fair->Biased=0.100 Fair->Fair=0.900 Emissions: Biased: H=0.750 T=0.250 Fair: H=0.500 T=0.500 Example Viterbi Matrices: H H H H H Biased 0.375000 0.253125 0.170859 0.115330 0.077848 Fair 0.250000 0.112500 0.050625 0.022781 0.010252 T T T T T Biased 0.125000 0.028125 0.006328 0.001424 0.000570 Fair 0.250000 0.112500 0.050625 0.022781 0.010252 H H H H T T T T Biased 0.375000 0.253125 0.170859 0.115330 0.025949 0.005839 0.001314 0.000296 Fair 0.250000 0.112500 0.050625 0.022781 0.010252 0.004613 0.002076 0.000934 H T H T H T H T Biased 0.375000 0.084375 0.056953 0.012814 0.008650 0.001946 0.001314 0.000296 Fair 0.250000 0.112500 0.050625 0.022781 0.010252 0.004613 0.002076 0.000934 =============================================================================== HMM #3: States: Q0 Q1 Q2 Q3 StartProbs: Q0=1.000 Q1=0.000 Q2=0.000 Q3=0.000 Transitions: Q0->Q0=0.000 Q0->Q1=0.600 Q0->Q2=0.400 Q0->Q3=0.000 Q1->Q0=0.100 Q1->Q1=0.800 Q1->Q2=0.100 Q1->Q3=0.000 Q2->Q0=0.000 Q2->Q1=0.000 Q2->Q2=0.700 Q2->Q3=0.300 Q3->Q0=1.000 Q3->Q1=0.000 Q3->Q2=0.000 Q3->Q3=0.000 Emissions: Q0: a=0.250 c=0.250 g=0.250 t=0.250 Q1: a=0.300 c=0.200 g=0.200 t=0.300 Q2: a=0.100 c=0.400 g=0.400 t=0.100 Q3: a=1.000 c=0.000 g=0.000 t=0.000 Example Viterbi Matrices: a a a a Q0 0.250000 0.000000 0.001125 0.000750 Q1 0.000000 0.045000 0.010800 0.002592 Q2 0.000000 0.010000 0.000700 0.000108 Q3 0.000000 0.000000 0.003000 0.000210 c c c g Q0 0.250000 0.000000 0.000750 0.000120 Q1 0.000000 0.030000 0.004800 0.000768 Q2 0.000000 0.040000 0.011200 0.003136 Q3 0.000000 0.000000 0.000000 0.000000 t c c g Q0 0.250000 0.000000 0.000750 0.000120 Q1 0.000000 0.030000 0.004800 0.000768 Q2 0.000000 0.040000 0.011200 0.003136 Q3 0.000000 0.000000 0.000000 0.000000