1	HMM Parameters and Probabilities
2	Important Issues re Interpretation of Sequences in the Context of the Model Forward decoding: What is the most probable path underlying the observed sequence? The Viterbi algorithm What is the full probability of all paths (and hence the probability of any path) underlying the observed sequence? The Forward Algorithm Posterior Decoding Given an observed sequence, what is the a posteriori probability that the observed sequence came from a particular state? The Backward Algorithm
3	Important Issues re Training When the training data are labeled Maximum Likelihood Estimation When the training paths are unknown EM techniques: Baum-Welch algorithm
4	Most Probable Path
5	Viterbi Algorithm
6	Model with 3 Coding Regions (States)
7	OBSERVED SEQUENCE
8	Viterbi Algorithm
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15	Closely probable paths, and the probability of any path The Forward Algorithm Define a function relating all joint probabilities up to the observation of interest,x_i
16	The Forward Algorithm Consider the Forward Algorithm variable (function) f , vis à vis the Viterbi variable v_k(i) v is an ad hoc argmax times emission f is an ad hoc joint probability times emission times transition
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19	Implementations Both the Forward Algorithm and the Viterbi Algorithm can be implemented with DYNAMIC PROGRAMMING Otherwise they are hard problems (exponential) Use logs to prevent underflow
20	Posterior Decoding Did observation x_i come from state k given the observed sequence x. Restated, what is P(p_i=k\|x) Need the Backward Algorithm
21	Backward Algorithm Recurse backward from the end of the sequence
22	The Posterior Probability
23	Posterior Probability
24	Training Data with Unknown Paths using EM: Baum-Welch Expectation Step Estimate transition and emission probabilities using frequency counts (the states will look alike) Maximization step Run the model with these estimates, using Posterior decoding Re-estimate the parameters using maximum likelihood estimator Iterate