Hidden Markov Model (HMM)
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The document provides an overview of Markov Models and Hidden Markov Models (HMMs), detailing their definitions, applications, and problems associated with them. Markov Models are stochastic processes where future states depend only on the current state, while HMMs involve hidden states and observable symbols, with specific algorithms like Baum-Welch and Viterbi used for evaluation, decoding, and learning. Applications of HMMs span various fields, including speech recognition, financial modeling, and time series analysis.
Hidden Markov Model (HMM)
- 1. 01 Markov Model HIDDEN MARKOV MODEL 02 HMMs 03 Applications 04 Problems of HMMs
- 2. Markov Model 01 What is Markov Model? Example of Markov Model
- 3. Markov Model 01 • Stochastic Method • Randomly Changing Systems • Next State Is Only Dependent On The Current State
- 4. Markov Models 01 • Assume there are three types of weather: • Weather prediction is about the what would be the weather tomorrow: • Based on the observations on the past • Weather at day n is • 𝑞𝑛 depends on the weather of the past days (𝑞𝑛−1, 𝑞𝑛−2,….) Sunny Rainy Foggy
- 5. Markov Model 01 • We want to find that: P (𝑞𝑛|𝑞𝑛−1, 𝑞𝑛−2, …. , 𝑞1) Means given the past weathers what is the probability of any possible weather of today.
- 6. Markov Model 01 Today’s weather Tomorrows Weather 0.8 0.05 0.15 0.2 0.6 0.2 0.2 0.3 0.5
- 7. Examples: • If the weather yesterday was rainy and today is foggy, what is the probability that tomorrow it will be sunny? P (𝑞3 = | 𝑞2 = , 𝑞1 = )= P (𝑞3 = | 𝑞2 = ) = 0.2 Markov assumption Markov Model 01
- 8. Hidden Markov Model 02 History What is HMMs? Variants of HMMs Example of HMMs
- 9. Hidden Markov Model 02 • Introduced in the 1960s • Baum and Petrie
- 10. Hidden Markov Model 02 Has a set of states each of which as limited number of transitions and emissions Each transition between states has an assigned probability Each model start from start state and ends in end state
- 12. Variants of HMMs 02 profile-HMMs pair-HMMs context-sensitive HMMs
- 13. Hidden Markov Model 02 • Suppose that you are locked in a room for several days, • You try to predict the weather outside • The only piece of evidence you have is whether the person who comes into the room bringing your daily meal is carrying an umbrella or not.
- 14. Hidden Markov Model 02 • Assume probabilities as seen in the table: Weather Probability of Umbrella Sunny 0.1 Rainy 0.8 Foggy 0.3 Probability P(𝑥𝑖|𝑞𝑖) of carrying an umbrella (𝑥𝑖 = true) based on the weather 𝑞𝑖 on some day i
- 15. Hidden Markov Model 02 • Finding the probability of a certain weather 𝑞𝑛 ∈ { sunny, rainy, foggy } • Is based on the observations 𝒙𝒊:
- 16. Hidden Markov Model 02 • Using Bayes rule: P(𝑞𝑖|𝑥𝑖) = P(𝑥𝑖|𝑞𝑖)P(𝑞𝑖) P(𝑥𝑖) • For n days: P(𝑞1, . . . ,𝑞𝑛|𝑥1, . . . , 𝑥𝑛) = P(𝑥1, . . . ,𝑥𝑛|𝑞1, . . . , 𝑞𝑛)P(𝑞1, . . . ,𝑞𝑛) P(𝑥1, . . . , 𝑥𝑛)
- 17. Hidden Markov Model 02 - Examples: • Suppose the day you were locked in it was sunny. The next day, the caretaker carried an umbrella into the room. • You would like to know, what the weather was like on this second day.
- 18. Hidden Markov Model 02 An HMM is characterized by: • N, the number if hidden states • M, the number of distinct observation symbols per state • {𝑎𝑖𝑗}, the state transition probability distribution • {𝑏𝑗𝑘}, the observation symbol probability distribution • {π𝑖 = P(𝑤(1) = 𝑤𝑖)}, the initial state distribution • Θ = ({𝑎𝑖𝑗}, {𝑏𝑗𝑘}, {π𝑖}), the complete parameter set of the model.
- 19. Problems of HMMs i Evaluating Problem Problem s ii Decoding Problem iii Leaning Problem 03
- 20. Problems 03 • Evaluation problem: Given the model, compute the probability that a particular output sequence was produced by that model (solved by the forward algorithm). • Decoding problem: Given the model, find the most likely sequence of hidden states which could have generated a given output sequence (solved by the Viterbi algorithm), • Learning problem: Given a set of output sequences, find the most likely set of state transition and output probabilities (solved by the Baum- Welch algorithm.)
- 21. Evolution Problem 03 Given model λ = (A, B, π), what is the probability of occurrence of a particular observation sequence O ={O1, O2,... Or}. i.e determine the likelihood P(O/λ) Our goal is to compute the like likelihood of on observation sequence O = O1, O2, O3.... Given a particular HMM model λ = A, B, π.
- 22. Decoding Problem 03 Decoding problem of Hidden Markov Model, One of the three fundamental problems to be solved under HMM is Decoding problem, Decoding problem is the way to figure out the best hidden state sequence using HMM Given an HMM λ = (A, B, π) and an observation sequence O = o1, o2, …, oT, how do we choose the corresponding optimal hidden state sequence (most likely sequence) Q = q1, q2, …, qT that can best explain the observations.
- 23. Decoding Problem 03 Goal: Find single best state sequence. q* = argmaxq P(q | O, λ) = arg maxq P(q, O | λ) Define i.e. the best score (highest probability) along a single path, at time t, which accounts for the first t observations and ends in state Si.
- 24. Learning Problem 03 Given a sequence of observation O = o1, o2, …, oT, estimate the transition and emission probabilities that are most likely to give O. that is, using the observation sequence and HMM general structure, determine the HMM model λ = (A, B, π) that best fit training data. Question answered by Learning problem: Given a model structure and a set of sequences, find the model that best fits the data. Baum-Welch Algorithm: The Baum-Welch algorithm is a specific form of the EM algorithm tailored for HMMs. It is used for unsupervised learning, where you have access to a sequence of observations but not to the corresponding hidden states. It iteratively refines the model's parameters (A, B, and π) until convergence.
- 26. Learning Problem 03 Baum-Welch Algorithm Time complexity: O(N2 T) · (# iterations) Guaranteed to increase likelihood P(O | λ) via EM but not guaranteed to find globally optimal λ * Practical Issues • Use multiple training sequences (sum over them) • Apply smoothing to avoid zero counts and improve generalization (add pseudocounts)
- 27. Applications 04 Computational finance speed analysis Speech recognition Speech synthesis Part-of-speech tagging Document separation in scanning solutions Machine translation Handwriting recognition Time series analysis Activity recognition Sequence classification Transportation forecasting
- 28. References ● https://www.cs.hmc.edu/~yjw/teaching/cs158/lectures/17_19_HMMs.pdf ● https://www.exploredatabase.com/2020/04/decoding-problem-of-hidden-markov- model.html ● https://www.javatpoint.com/hidden-markov-model-in-machine-learning ● https://youtu.be/KcXOT-PJy1U?si=5EYnzh-WBfUMftC2 ● https://youtu.be/F5Wrn_UX4L8?si=OCw-K5NIDELyAW0z ● https://youtu.be/Io_VNym0vkI?si=D-II7GsLRb7RUaEo ● https://www.slideshare.net/shivangisaxena566/hidden-markov-model-ppt ● https://www.javatpoint.com/hidden-markov-model-in-machine-learning
- 29. Any Questions?
- 30. Thank You!