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Class 5: HMMs and Profile HMMs

This review explores Hidden Markov Models (HMMs) and Profile HMMs, focusing on their key components: hidden states and observations. HMMs are probabilistic models that represent sequences through stochastic automata, with practical applications in sequence alignment. We cover the Viterbi algorithm for finding the most probable sequence of hidden states and discuss the potential to utilize Pair-HMMs, which model the joint distribution of aligned sequences, including handling gaps. The overview also highlights the robustness of HMMs compared to traditional alignment methods.

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Class 5: HMMs and Profile HMMs

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  1. Class 5:HMMs and Profile HMMs .

  2. Review of HMM • Hidden Markov Models • Probabilistic models of sequences • Consist of two parts: • Hidden statesThese act like a stochastic automata • ObservationsThese are determined (stochastically) by the hidden state

  3. 0.95 0.9 1: 1/10 2: 1/10 3: 1/10 4: 1/10 5: 1/10 6: 1/2 1: 1/6 2: 1/6 3: 1/6 4: 1/6 5: 1/6 6: 1/6 0.05 1.0 0.1 Begin Loaded Fair Example Possible Sequence:

  4. Hidden Markov Models Two components: • A Markov chain of hidden statesH1,…,Hn with L values • P(Hi+1=k |Hi=l ) = Akl • ObservationsX1,…,Xn • Assumption: Xidepends only on hidden state Hi • P(Xi=a |Hi=k ) = Bka

  5. HMM Three aspects: • Representation • Computation • Viterbi algorithm • Forward-Backward algorithm • Learning

  6. 1.0 Begin AA 0.21 AC 0.01 AG 0.05 AT 0.04 CA 0.02 …. 1.0 Match Example: pair-HMM • We want to model the joint distribution of two aligned sequences • We start with ungapped alignment

  7. Begin Pair-HMM • This model is equivalentto ungapped models wetreated two classes ago • Can we add gaps? 1.0 AA 0.21 AC 0.01 AG 0.05 AT 0.04 CA 0.02 …. 1.0 Match

  8. Begin Adding GAP States  A- 0.2 C- 0.4 G- 0.3 T- 0.1 1- 1-2  Gap Y AA 0.21 AC 0.01 AG 0.05 AT 0.04 CA 0.02 ….   1- -A 0.2 -C 0.4 -G 0.3 -T 0.1  Match Gap X

  9. Gapped Alignment What happens if we do not observe skips? • Suppose input is AAT and ATATT Each sequence of hidden states determines an alignment!!

  10. Viterbi in Pair-HMM • Finding most probable sequence of hidden states is exactly global sequence alignment

  11. Scoring Alignments with HMMs • Viterbi finds most probable alignment • The probability of this alignment can be small… • Using HMM algorithm we can compute the probability of generating the two sequences • This sums over all possible alignments of the two strings • Such methods are more sensitive than standard alignment procedures • We can easily extend the pair-HMM for dealing with local alignment

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