Bayesian Ridge Regression

Given an input vector ⚠ $x_t$, the online Bayesian Ridge Regression predicts at each step ⚠ $T$ the normal distribution ⚠ $N({\gamma }_T,{\sigma }_T^2)$ with the mean and variance given by ⚠ $\$$\gamma_T = Y'_{T-1} X_{T-1} A_{T-1}^{-1} x_T , \quad \sigma_T^2 = \sigma^2 x_T' A_{T-1}^{-1} x_T + \sigma^2⚠ $\$$ for some ⚠ $a>0$ and the known noise variance ⚠ ${\sigma }^2$. Here ⚠ $X_t$ is the ⚠ $t\times n$ matrix of row vectors ⚠ $x_1ʹ,\dots ,x_tʹ$ and ⚠ $Y_t$ be the column vector of outcomes ⚠ $y_1,\dots ,y_t$. Here also ⚠ $A_t=X{ʹ}_t{X}_t+aI$.