Summary, Main Concepts, and Takeaways

Summary of concepts

First, let's remember some common concepts that will be highlighted throughout this lesson.

Descriptive graph: A descriptive graph is a visual representation of a set of data or relationships between variables. It can be used to identify trends, patterns, and relationships in the data.

Bayesian network: A Bayesian network is a graphical model that represents the relationships between random variables and their probabilities. It uses conditional probability distributions to model the dependencies between variables.

Random variables: A random variable is a variable whose value is subject to randomness or uncertainty. In a Bayesian network, each node represents a random variable.

Conditional probability distribution: A conditional probability distribution is a probability distribution that describes the likelihood of an event, given the occurrence of a related event. It is used to represent the dependencies between variables in a Bayesian network.

Bayesian networks: BNs use conditional probabilities to describe events. Any belief about the uncertainty of an event or hypothesis $H$ is assumed provisional. This is called prior probability or $P(H)$. This prior probability is updated by the new experience $E$ providing a revised belief about the uncertainty of $H$. The new probability is called posterior probability given the evidence or $P(H|E)$.

Bayes’ theorem describes the relationship between posterior probability and the prior probability. $P(E|H)$ is the probability of the evidence being true given that the hypothesis is true, and $P(E)$ represents how probable is the new evidence under all possible hypotheses.

Now that we have defined these concepts, let's move on to the process of transforming descriptive graphs into Bayesian networks.

A common cause graph is a type of causal graph in which a single variable, the common cause, influences two or more other variables. In other words, the common cause is a shared or confounding variable that affects multiple variables in the graph. This shared variable can create an association between the influenced variables, even if there is no direct causal relationship between them.

A causal-chain graph is a graphical representation of the causal relationships between variables in a system. It's a type of directed acyclic graph (DAG), where nodes represent variables, and directed edges (arrows) represent causal relationships between them. The graph is called a causal-chain graph because it captures the chains of cause-and-effect relationships among the variables.

A common-effect graph is a graphical representation used in causal modeling, specifically in the context of Bayesian networks. In a common-effect graph, two or more independent variables (or causes) are linked to a single dependent variable (or effect), indicating that the effect is a result of the combination of the causes. The term common effect comes from the fact that multiple variables share a common effect in this structure.

The size of the conditional probability distribution (CPD) table for a child node depends on the number of parent nodes the number of their states, and the number of states of the child node: