1. Hidden Markov Models Usman Roshan BNFO 601

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

3. Loaded die problem

4. Loaded die automata F L a FL a LF e F(i) e L(i) a FF a LL

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

6. Optimization Problem: Given an HMM and a sequence of rolls, find the most probably underlying generating path. Let be the sequence of rolls. Let V F (i) denote the probability of the most probable path of that ends in state F. (Define V L (i) similarly.)

7. Optimization Initialize : Recursions: for i=0..n-1

8. Parameter learning (when generating sequence is known) A kl : number of transitions from state k to l E k (b): number of times state k emits symbol b

9. Sequence alignment

10. Alignment recursions Initialization Recursions

11. Alignment recursions Termination