Introduction to Bayesian reasoning
- Introduction to Bayesian Reasoning
- Historical Perspective of Bayesian Reasoning
- Understanding Priors
- Implementing Priors
- Advanced Bayesian Inference
- Bayesian Networks
- Bayesian Data Analysis
- Introduction to Bayesian Software
- Handling Complex Bayesian Models
- Bayesian Perspective on Learning
- Case Study: Bayesian Methods in Finance
- Case Study: Bayesian Methods in Healthcare
- Wrap Up & Real World Bayesian Applications
Bayesian Networks
Basic Structure of Bayesian Networks
Directed graph with no directed cycles.
Bayesian Networks, also known as Belief Networks, are a type of probabilistic graphical model that uses a directed acyclic graph (DAG) to represent a set of variables and their conditional dependencies. They are a powerful tool for encoding probabilistic relationships among variables of interest.
Definition of Bayesian Networks
A Bayesian Network is a graphical representation of the probabilistic relationships among a set of variables. Each node in the network represents a variable, and the edges between the nodes represent the probabilistic dependencies between the variables. The absence of an edge indicates that the corresponding variables are conditionally independent.
Understanding Nodes and Edges
In a Bayesian Network, nodes represent random variables that can be either observed quantities, latent variables, unknown parameters or hypotheses. Edges, on the other hand, represent direct dependencies between the variables. If there is an edge from node A to node B, it means that the variable represented by node B is directly dependent on the variable represented by node A.
Conditional Independence and D-separation
One of the key concepts in Bayesian Networks is conditional independence. Two nodes are conditionally independent given a third node if the state of the third node provides all the information necessary to predict the relationship between the first two nodes.
D-separation is a criterion in a Bayesian Network that helps to determine whether a set of nodes is independent of another set of nodes, given a third set of nodes. If the nodes are d-separated, they are conditionally independent.
Understanding Directed Acyclic Graphs (DAGs)
A Directed Acyclic Graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges, with each edge directed from one vertex to another, such that following those directions will never form a closed loop.
In the context of Bayesian Networks, a DAG is used to represent the conditional dependencies between variables. Each edge in the DAG corresponds to a direct conditional dependency, and the direction of the edge indicates the direction of the dependency.
Joint Probability Distribution
The joint probability distribution of a set of variables is a probability distribution that gives the probability that each of a set of variables falls in any particular range. In a Bayesian Network, the joint probability distribution of all the variables can be computed as the product of the local conditional probability distributions specified by each node.
In conclusion, understanding the basic structure of Bayesian Networks is crucial for their application in decision making. The concepts of nodes, edges, conditional independence, D-separation, DAGs, and joint probability distribution form the foundation of Bayesian Networks.