Covariance

As explained in Bayesian Optimization in Action

Covariance is a measure of how two random variables vary together.

Covariance measures how two random variables vary together. In a multivariate Gaussian, the covariance matrix captures both each variable’s variance and the relationships between variables. section 2.2.1

Read the covariance matrix by position:

• Diagonal entries give the variances of the individual random variables. section 2.2.1
• Off-diagonal entries give the covariance between pairs of variables and relate to their correlation. section 2.2.1
• Positive covariance indicates that variables tend to move together; stronger positive correlation indicates a closer relationship. 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 figure to see how a mean vector represents individual variables while the covariance matrix represents their relationships. section 2.2.1

How Bayesian Optimization in Action covers it

How deep it goes

Defines covariance through matrix entries, distinguishing individual variances on the diagonal from pairwise relationships off the diagonal.

Examples

Uses a housing-price multivariate Gaussian and visualizations of samples, confidence intervals, and variable relationships.

What it assumes

Builds from normal distributions and random vectors before extending them to jointly model multiple variables.

What you can do after

Evaluate covariance matrices by identifying variable variances and pairwise relationships in multivariate Gaussian models.

How it connects

Connects covariance with multivariate Gaussian distributions, variance, correlation, mean vectors, and normal distributions.

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Also explained in

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Related concepts

• correlationcontrasts with · Correlation is a statistical relationship describing how two or more variables or quantities vary together, including the strength and direction of their association, without by itself implying causation.
• Gaussian mixture modelused by · A Gaussian mixture model is a probabilistic model representing a distribution as a weighted combination of multiple Gaussian component distributions.
• Pearson correlation coefficientused by · The Pearson correlation coefficient measures the strength and direction of the linear relationship between two numerical variables.
• radial basis function kernelmeasured by · A radial basis function kernel is a similarity function whose value depends on the distance between inputs and typically decreases as that distance increases.
• instrumental variableused by · An instrumental variable is a variable that affects a treatment or exposure while influencing an outcome only through that treatment or exposure, under causal identifying assumptions.
• varianceuses · Variance is the expected squared deviation of a random variable from its mean, measuring the spread of its 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.