Analytical Bayesian Batch Estimation for Linear Regression

Summary

This article derives a batch Bayesian method for estimating the intercept and slope of a univariate linear regression. It assumes normally distributed observation noise with known variance and assigns the regression parameters a normal prior with a specified mean and covariance. Given the data, the likelihood and prior combine to produce a normal posterior, whose mean and covariance are calculated using matrix operations over the full dataset.

The article illustrates the method with simulated linear data, then describes implementing the posterior calculation using standard numerical linear algebra tools. Because the normal prior is conjugate to the Gaussian likelihood, the posterior has an analytical form and does not require methods such as Markov chain Monte Carlo for this model. The example demonstrates parameter recovery under its chosen simulation and prior; it is not evidence about real market data or trading performance. The method also relies on its stated linearity and noise assumptions, and its batch calculation uses all observations together rather than updating parameters recursively as each new observation arrives.

Key ideas

• The regression parameters are assigned a multivariate normal prior, and observation noise is assumed Gaussian with known variance.
• The normal prior and Gaussian likelihood produce a normal posterior distribution.
• Posterior mean and covariance can be calculated analytically from the prior and the full dataset.
• The article demonstrates the calculation on simulated linear data with known parameters.
• The batch method depends on its modelling assumptions and does not itself establish trading value.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.