Marginal probability

As explained in Causal AI

A marginal probability is the probability of a variable or event obtained without conditioning on other variables by summing or integrating them out.

A marginal probability describes one variable without conditioning on another. Obtain it from a joint distribution by summing over the other variable, or integrating it for continuous variables. section 2.1.3 section 2.1.4

For a joint distribution P(X, Y):

• Compute P(X) by summing P(X, Y) across all values of Y. section 2.1.3
• Compute P(Y) by summing across all values of X. section 2.1.3
• Use marginal distributions to connect joint distributions with conditional probability through the law of total probability. section 2.1.4

Marginalizing a tabular distribution

A joint table assigns probabilities to six combinations of X and Y. Summing over Y produces P(X) = {0.45, 0.30, 0.25}; summing over X produces P(Y) = {0.60, 0.40}. In pgmpy, DiscreteFactor.marginalize(variables=['Y'], inplace=False) returns the new marginal table without modifying the joint distribution. section 2.1.3

Figure 2.5 Marginalizing over Y yields the marginal distribution of X.

Shows how summing over Y yields the marginal distribution of X. section 2.1.3

How Causal AI covers it

How deep it goes

Explains discrete summation, continuous integration, notation, and pgmpy operations, but stops short of broader multivariate inference.

Examples

Uses a six-cell X-Y probability table and pgmpy DiscreteFactor marginalize outputs.

What it assumes

Assumes joint and conditional probability notation, while motivating marginal distributions through explicit tables and sums.

What you can do after

Apply it via pgmpy to derive marginal tables from discrete joint distributions without altering the original factor.

How it connects

Connects marginal probability with joint probability, conditional probability, the law of total probability, the chain rule, and Bayes rule.

liveBookmanning.com

Read Causal AI on liveBook Get the book at manning.com

Also explained in

• 

  liveBookmanning.com
  Outlier Detection in PythonA marginal distribution describes the probability behavior of one variable after accounting for or aggregating over the other variables.
• 

  liveBookmanning.com
  Math and Architectures of Deep LearningThe probability distribution for one variable obtained by summing out the other variables from a joint distribution.

Compare how 3 books explain marginal probability →

Related concepts

• marginal likelihoodequivalent to · Marginal likelihood, or Bayesian evidence, is the probability or density of observed data under a model after marginalizing over unknown parameters or latent variables according to their prior distributions.
• message passingproduced by · A communication or computation technique in which entities exchange messages to transmit or propagate information, coordinate work, or update state.
• joint probability distributionused by · A joint probability distribution specifies probabilities or densities for combinations of values taken simultaneously by multiple random variables.
• marginalizationproduced by · Marginalization is the operation of summing or integrating over variables or alternatives that are not retained to obtain a distribution or quantity for the remaining variables.

This explainer was generated by Manning's knowledge graph from the text of Causal AI. The section links point to the parts of the book it draws on.