Learning, Uncertainty, and Information: Learning Parameters
This document delves into the intricacies of Hidden Markov Models (HMMs), emphasizing the learning of parameters from data. It covers essential topics, including the noisy channel model, the forward and backward algorithms, the Viterbi algorithm for decoding, and the Baum-Welch algorithm for parameter estimation. By exploring transition and emission probabilities, the document seeks to address common issues in HMMs, such as data sparseness and the need for labeled datasets. This guide aims to equip learners with foundational knowledge required to effectively utilize HMMs in various applications.
Learning, Uncertainty, and Information: Learning Parameters
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Presentation Transcript
Learning, Uncertainty, and Information:Learning Parameters Big Ideas November 10, 2004
Roadmap • Noisy-channel model: Redux • Hidden Markov Models • The Model • Decoding the best sequence • Training the model (EM) • N-gram models: Modeling sequences • Shannon, Information Theory, and Perplexity • Conclusion
Bayes and the Noisy Channel • Generative and sequence
Hidden Markov Models (HMMs) • An HMM is: • 1) A set of states: • 2) A set of transition probabilities: • Where aij is the probability of transition qi -> qj • 3)Observation probabilities: • The probability of observing ot in state i • 4) An initial probability dist over states: • The probability of starting in state i • 5) A set of accepting states
Three Problems for HMMs • Find the probability of an observation sequence given a model • Forward algorithm • Find the most likely path through a model given an observed sequence • Viterbi algorithm (decoding) • Find the most likely model (parameters) given an observed sequence • Baum-Welch (EM) algorithm
Learning HMMs • Issue: Where do the probabilities come from? • Supervised/manual construction • Solution: Learn from data • Trains transition (aij), emission (bj), and initial (πi) probabilities • Typically assume state structure is given • Unsupervised
Manual Construction • Manually labeled data • Observation sequences, aligned to • Ground truth state sequences • Compute (relative) frequencies of state transitions • Compute frequencies of observations/state • Compute frequencies of initial states • Bootstrapping: iterate tag, correct, reestimate, tag. • Problem: • Labeled data is expensive, hard/impossible to obtain, may be inadequate to fully estimate • Sparseness problems
Unsupervised Learning • Re-estimation from unlabeled data • Baum-Welch aka forward-backward algorithm • Assume “representative” collection of data • E.g. recorded speech, gene sequences, etc • Assign initial probabilities • Or estimate from very small labeled sample • Compute state sequences given the data • I.e. use forward algorithm • Update transition, emission, initial probabilities
Updating Probabilities • Intuition: • Observations identify state sequences • Adjust probability of transitions/emissions • Make closer to those consistent with observed • Increase P(Observations|Model) • Functionally • For each state i, what proportion of transitions from state i go to state j • For each state i, what proportion of observations match O? • How often is state i the initial state?
Estimating Transitions • Consider updating transition aij • Compute probability of all paths using aij • Compute probability of all paths through i (w/ and w/o i->j) i j
Forward Probability Where α is the forward probability, t is the time in utterance, i,j are states in the HMM, aij is the transition probability, bj(ot) is the probability of observing ot in state bj N is the max state, T is the last time
Backward Probability Where β is the backward probability, t is the time in sequence, i,j are states in the HMM, aij is the transition probability, bj(ot) is the probability of observing ot in state bj N is the final state, and T is the last time
Re-estimating • Estimate transitions from i->j • Estimate observations in j • Estimate initial i
