Random vector

As explained in Bayesian Optimization in Action

A random vector is an ordered collection of random variables represented as a vector.

A random vector is an ordered collection of random variables grouped into one vector. In multivariate Gaussian modeling, this grouping captures both each variable’s behavior and relationships between variables. section 2.2.1

For a Gaussian random vector X = [X₁, X₂, …, Xₙ] distributed as N(μ, Σ):

• The mean vector μ contains the expected value of each corresponding random variable. section 2.2.1
• The covariance matrix Σ contains individual variances on its diagonal and covariances between variables elsewhere. section 2.2.1
• Each component follows a normal distribution, while covariance describes how components vary together. section 2.2.1

Figure 2.4 An MVN distribution groups multiple normally distributed random variables together. While the mean vector of the MVN concatenates the mean values, the covariance matrix models the correlations between the individual variables.

Use the diagram to see how multiple normally distributed variables combine into a mean vector and covariance matrix. section 2.2.1

How Bayesian Optimization in Action covers it

How deep it goes

Defines Gaussian random vectors and interprets mean-vector and covariance-matrix entries, without developing broader probability theory.

Examples

Uses quantities such as height, IQ, income, and birth weight, plus a housing-price visualization with sampled vectors.

What you can do after

Evaluate how a multivariate Gaussian represents individual variables and their correlations through vector and matrix parameters.

How it connects

Connects random vectors to multivariate Gaussian distributions, normal distributions, mean vectors, covariance matrices, and correlation.

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Related concepts

• causal adjustmentuses · Causal adjustment estimates a causal effect by conditioning on variables that block noncausal pathways and averaging over their target-population distribution.
• vectorkind of · A vector is an ordered collection of components representing an element of a vector space.
• multinomial distributionproduced by · A multinomial distribution is a probability distribution for counts of outcomes in multiple mutually exclusive categories across a fixed number of independent trials.
• covariance matrixproduces · A covariance matrix is a square matrix whose diagonal entries are variances and whose off-diagonal entries are pairwise covariances among variables.
• random variablekind of · A random variable is a function that assigns numerical values to outcomes of a random process and is characterized by a probability distribution.

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