Contents
How to use linear basis models in Bayesian regression?
This article is an introduction to Bayesian regression with linear basis function models. After a short overview of the relevant mathematical results and their intuition, Bayesian linear regression is implemented from scratch with NumPy followed by an example how scikit-learn can be used to obtain equivalent results.
How is the posterior predictive distribution used in Bayesian regression?
The posterior predictive distribution includes uncertainty about parameters w into predictions by weighting the conditional distribution p(t | x, w, β) with the posterior probability of weights p(w | t, α, β) over the entire weight parameter space.
How is the MSE calculated in Bayesian regression?
The MSE, ˆσ2, may be calculated through squaring the residuals of the output of bodyfat.lm. If this model is correct, the residuals and fitted values should be uncorrelated, and the expected value of the residuals is zero. We apply the scatterplot of residuals versus fitted values, which provides an additional visual check of the model adequacy.
Are there degrees of freedom in Bayesian regression?
They both have degrees of freedom n − 2. Let us now turn to the Bayesian version and show that under the reference prior, we will obtain the posterior distributions of α and β analogous with the frequentist OLS results.
How to calculate design matrix with linear basis function?
The design matrix Φ can be computed from observations X and a parametric basis function with function expand. This function also prepends a column vector 1 according to ϕ 0(x) = 1. For straight line fitting, we use a linear regression model of the form y(x, w) = w 0 + w 1x and do Bayesian inference for model parameters w.
How to fit a Gaussian basis function to a noisy dataset?
The following example demonstrates how to fit a Gaussian basis function model to a noisy sinusoidal dataset. It uses 9 Gaussian basis functions with mean values equally distributed over [0, 1] each having a standard deviation of 0.1.