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Mean vector

As explained in Bayesian Optimization in Action

A mean vector is a vector containing the expected value of each component of a multivariate probability distribution.

A mean vector collects the expected value of every component in a multivariate Gaussian distribution. For X = [X₁, X₂, …, Xₙ] following N(μ, Σ), μ is the mean vector, and μᵢ is the expected value of Xᵢ. section 2.2.1

Think of each component as a quantity being modeled together: the mean vector places the center of each quantity, while the covariance matrix describes individual variance and relationships between quantities. section 2.2.1

Housing prices along a street

A high-dimensional Gaussian can represent prices for many houses, with one variable per house and one corresponding entry in the mean vector. A mean-normalized model uses the zero vector. After observing the price of the 10th house, nearby variables receive updated mean values closer to that observation because nearby houses are more correlated. section 2.2.3

How Bayesian Optimization in Action covers it

How deep it goes
Defines component-wise expectations and contrasts them with covariance, then illustrates posterior mean updates without deriving them.
Examples
Uses housing prices, a 20-dimensional street model, zero-vector normalization, and conditioning on the 10th variable.
What it assumes
Assumes familiarity with vectors and random variables while introducing multivariate Gaussian notation and covariance roles.
What you can do after
Evaluate the center of each variable in a multivariate Gaussian and interpret how observations update related predictions.
How it connects
Connects multivariate Gaussian distributions, covariance matrices, Bayesian updates, and Gaussian processes.
Cover of Bayesian Optimization in Action
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Related concepts

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.