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