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Hidden Markov Models

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Explore the applications and computational problems of Hidden Markov Models in identifying coding regions in DNA sequences, separating "bad" pixels in images, and more. Learn about the Baum-Welch algorithm and how to determine transition and emission probabilities.

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Hidden Markov Models

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  1. Hidden Markov Models Usman Roshan CS 675 Machine Learning

  2. HMM applications • Determine coding and non-coding regions in DNA sequences • Separating “bad” pixels from “good” ones in image files • Classification of DNA sequences

  3. Hidden Markov Models • Alphabet of symbols: • Set of states that emit symbols from the alphabet: • Set of probabilities • State transition: • Emission probabilities:

  4. Loaded die problem

  5. Loaded die automata aFL F L aFF aLL eF(i) eL(i) aLF

  6. Loaded die problem • Consider the following rolls: Observed : 21665261 Underlying die : FFLLFLLF • What is the probability that the underlying path generated the observed sequence?

  7. HMM computational problems

  8. Probabilities unknown but hidden sequence known • Akl: number of transitions from state k to l • Ek(b): number of times state k emits symbol b

  9. Probabilities known but hidden sequence unknown • Problem: Given an HMM and a sequence of rolls, find the most probably underlying generating path. • Let be the sequence of rolls. • Let VF(i) denote the probability of the most probable path of that ends in state F. (Define VL(i) similarly.)

  10. Probabilities known but hidden sequence unknown • Initialize: • Recurrence: for i=0..n-1

  11. Probabilities and hidden sequence unknown • Use Expected-Maximization algorithm (also known as EM algorithm) • Very popular and many applications • For HMMs also called Baum-Welch algorithm • Outline: • Start with random assignment to transition and emission probabilities • Find expected transition and emission probabilities • Estimate actual transition and emission probabilities from expected values in previous step • Go to step 2 if probabilities not converged

  12. HMM forward probabilities • Consider the total probability of all hidden sequences under a given HMM. • Let fL(i) be the sum of the probabilities of all hidden sequences upto i that end in the state L. • Then fL(i) is given by • We calculate fF(i) in the same way. • We call these forward probabilities: • f(i) = fL(i)+fF(i)+fB(i)

  13. HMM backward probabilities • Similarly we can calculate backward probabilties bL(i). • Let bL(i) be the sum of the probabilities of all hidden sequences from i to the end that start in state L. • Then bL(i) is given by • We calculate bF(i) in the same way. • We call these forward probabilities: • b(i) = bL(i)+bF(i)+bB(i)

  14. Baum Welch • How do we calculate expected transition and emission probabilities? • Consider the fair-loaded die problem. What is the expected transition of fair (F) to loaded (L)? • To answer we have to count the number of times F transitions to L in all possible hidden sequences and multiply each by the probability of the hidden sequence

  15. Baum Welch • For example suppose input is 12 • What are the different hidden sequences? • What is the probability of each? • What is the total probability? • What is the probability of all hidden sequences where the first state is F? • Can we determine these answers automatically with forward and backward probabilities?

  16. Baum Welch General formula for expected number of transitions from state k to l. General formula for expected number of emissions of b from state k. Equations from Durbin et. al., 1998

  17. Baum Welch • Initialize random values for all parameters • Calculate forward and backward probabilities • Calculate new model parameters • Did the new probabilities (parameters) change by more than 0.001? If yes goto step 2. Otherwise stop.

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