How is this different from the numerical-analysis discipline in mathematics?

It uses the same mathematical tools but focuses on the problems and conditioning that arise specifically in statistics, such as least squares with collinear predictors or evaluating likelihoods, and on the accuracy implications for statistical estimates.

Why does numerical stability matter for a correct statistical model?

Even a correct model must be solved on a computer in finite precision. An unstable algorithm can amplify rounding error, so a well-posed estimate can still come out inaccurate unless a numerically sound method is used.

Numerical Methods in Statistics

Numerical methods in statistics are the matrix computations, quadrature, and accuracy considerations that make statistical estimation reliable and efficient when carried out in finite-precision arithmetic.

Definition

Numerical methods in statistics is the study of finite-precision algorithms, especially matrix computations and quadrature, used to evaluate and solve the linear-algebraic and integral problems that arise in fitting and analyzing statistical models.

Scope

This area covers the numerical linear algebra behind regression and multivariate analysis, the matrix decompositions used to solve and stabilize statistical computations, and numerical integration for evaluating likelihoods and posterior quantities. The treatment emphasizes the statistical motivation and the conditioning issues specific to statistical models, complementing the general numerical-analysis discipline.

Sub-topics

Core questions

How are regression and multivariate computations organized to be accurate and stable?
Which matrix decompositions solve the normal equations and related problems reliably?
How are likelihoods and posterior integrals evaluated numerically?
How does conditioning of a statistical problem affect the accuracy of its solution?

Key theories

Stable solution of statistical linear systems: Statistical computations such as least squares are best solved through orthogonal decompositions rather than the normal equations, which improves numerical stability when predictors are nearly collinear.
Numerical evaluation of statistical integrals: Likelihoods, marginal densities and posterior expectations often require quadrature; choosing rules suited to the integrand's shape and tails controls the accuracy of the resulting statistical quantities.

Clinical relevance

The reliability of regression coefficients, covariance estimates, and likelihood evaluations rests on the underlying numerical methods; poor conditioning or unstable algorithms can produce inaccurate estimates or convergence failures even when the statistical model is correct.

History

As statistical computing matured, the field absorbed numerical-analysis results on stable matrix factorizations and quadrature and adapted them to the structure of statistical problems, producing dedicated texts on numerical methods for statisticians from the 1990s onward.

Key figures

Kenneth Lange
John Monahan
Gene Golub
Charles Van Loan

Seminal works

lange2010
monahan2011

Frequently asked questions

How is this different from the numerical-analysis discipline in mathematics?: It uses the same mathematical tools but focuses on the problems and conditioning that arise specifically in statistics, such as least squares with collinear predictors or evaluating likelihoods, and on the accuracy implications for statistical estimates.
Why does numerical stability matter for a correct statistical model?: Even a correct model must be solved on a computer in finite precision. An unstable algorithm can amplify rounding error, so a well-posed estimate can still come out inaccurate unless a numerically sound method is used.