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Bayesian Estimation in Graphical Models

Bayesian estimation is a probabilistic approach for estimating model parameters in graphical models by incorporating prior knowledge through Bayes' theorem. Unlike frequentist methods, which rely solely on observed data, Bayesian estimation updates prior beliefs with evidence to obtain a posterior distribution over parameters.

Bayesian Parameter Estimation

In probabilistic graphical models, Bayesian estimation is used to determine the values of unknown parameters ($\theta$) given observed data ($X$):

$$P(\theta | X) = \frac{P(X | \theta) P(\theta)}{P(X)}$$

Where:

• $P(\theta)$ is the prior distribution (initial belief about parameters).

• $P(X | \theta)$ is the likelihood (how well parameters explain the data).

• $P(X)$ is the marginal likelihood (normalization constant).

• $P(\theta | X)$ is the posterior distribution (updated belief).

Inference Using Bayesian Estimation

1. Prior Selection: Choose a prior distribution (e.g., Gaussian, Dirichlet).

2. Likelihood Computation: Define how data is generated given parameters.

3. Posterior Computation: Use Bayes' theorem to update beliefs.

4. Prediction & Decision Making: Use the posterior for inference in graphical models.

Applications in Graphical Models

• Hidden Markov Models (HMMs): Learning transition probabilities.

• Bayesian Networks: Updating belief states dynamically.

• Markov Random Fields (MRFs): Estimating dependencies between variables.

Bayesian estimation provides a flexible and robust framework for parameter learning and inference, especially when dealing with uncertainty or limited data.