Part 10: Continuing our Discussion on the Learning Problem and the Baum-Welch Algorithm
3 — Maximization Step (M): It involves updating the parameters of the Hidden Markov Model (HMM) to maximize the likelihood of the observed data. This is done based on the statistics computed in the Expectation step. Here’s a bit more detail:
The formulas for updating these parameters are as follows:
let’s break down the notation used in these formulas:
Note — 1(xt = vk): This is an indicator function that equals 1 if the observation at time t equals k, and 0 otherwise. The function is defined as follows:
Here, xt represents the observation at time t, and vk represents a particular observation from the set of all possible observations. It essentially checks whether the observation at time t is the one we’re interested in (i.e., equals vk). If it is, the function returns 1; if it’s not, the function returns 0. This way, it contributes to the count of times the particular observation vk is emitted when the system is in a certain state. This ensures that only the visits to state i that produce observation k are counted in the numerator.
Let’s continue with the example to update the parameters — initial state probabilities (πi), transition probabilities (aij), and emission probabilities bj(o)) — using the formulas provided. Given our observed sequence: [‘Rainy’, ‘Sunny’, ‘Rainy’], and the previously computed values for γ and ξ:
Similarly we can calculate –
Now, lets have look at updated model parameters reflect the revised probabilities after considering the observations.
4- Iteration: After calculating the forward and backward probabilities, the next step is to reestimate the model parameters (transition probabilities, emission probabilities, and initial state probabilities) using the Baum-Welch update equations. This is where the model learns from the observed data to adjust its parameters and better fit the data. The E-step and M-step are repeated until the model parameters converge, meaning they stop changing significantly beyond a certain threshold. This indicates that the algorithm has found a set of parameters that provides a good fit for the observed data.
Note — The Baum-Welch algorithm guarantees that the likelihood of the observed data will not decrease with each iteration. This means that the algorithm always improves the likelihood of the observed data in each step. However, it does not guarantee that the global maximum likelihood will be found; the algorithm may converge to a local maximum instead. A local maximum is a point in the parameter space where the likelihood of the observed data is higher than at neighboring points, but lower than at other points in the parameter space. Therefore, even though the Baum-Welch algorithm improves the likelihood of the observed data in each step, it may not always find the best possible set of parameters.
For more examples and a deeper understanding of Hidden Markov Models (HMMs), I highly recommend checking out this comprehensive article.
Here’s a simple implementation of Hidden Markov Model (HMM) in Python:
Here’s a simple implementation of Part of Speech (POS) tagging with Hidden Markov Model. For a detailed understanding of the topic “Part of Speech (POS) tagging with Hidden Markov Model,” please refer to this article — POS Tagging with Hidden Markov Model.
Here are some Real-World Applications of Hidden Markov Models:
Ending note: As we draw the curtains on this comprehensive guide, I hope you’ve gained valuable insights and a deeper understanding of HMM. The potential applications of HMMs are vast and varied, from natural language processing to bioinformatics, and the possibilities are truly endless.
In the spirit of continuous learning and discovery, I encourage you to experiment with HMMs and explore other machine learning algorithms as well. Don’t hesitate to apply these models to your own projects and problems. The more you practice, the better you’ll become. If you found this content informative and engaging, consider subscribing to the blog for more such insightful posts.
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