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 Bayesian Learning: Inference & 

EM Algorithm Explained

Bayesian learning updates beliefs using Bayes' theorem, refining models with new data. Inference estimates hidden variables, while the Expectation-Maximization (EM) algorithm iteratively finds maximum likelihood estimates for models with latent variables. EM alternates between expectation (E-step) and maximization (M-step) to improve parameter estimates until convergence, making it powerful for probabilistic learning.

Bayesian Learning & Inference

Bayesian learning is a statistical approach that updates probabilities as more evidence (data) becomes available. It relies on Bayes' Theorem:

P(θ∣D)=P(D∣θ)P(θ)P(D)P(\theta | D) = \frac{P(D | \theta) P(\theta)}{P(D)}

where:

  • P(θ∣D)P(\theta | D) is the posterior probability of parameters θ\theta given data DD.
  • P(D∣θ)P(D | \theta) is the likelihood (probability of data given parameters).
  • P(θ)P(\theta) is the prior belief about parameters.
  • P(D)P(D) is the evidence (normalizing constant).

Bayesian inference refines parameter estimates using this principle, making it effective for probabilistic modeling.

Expectation-Maximization (EM) Algorithm

The EM algorithm is used when data has latent (hidden) variables, making direct estimation difficult. It iterates between two steps:

  1. Expectation (E-step): Computes expected values of hidden variables based on current parameter estimates.
  2. Maximization (M-step): Updates parameters to maximize the expected likelihood.

This process continues until convergence, refining model parameters iteratively. EM is widely used in clustering (e.g., Gaussian Mixture Models), missing data problems, and probabilistic learning.

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