Last-layer Bayes neural nets

Consider the original linear model. We have a (column) vector \(\mathbf{y}=[y_1,y_2,\dots,t_n]^T\) of \(n\) observations, an \(n\times p\) matrix \(\mathbf{X}\) of \(p\) covariates, where each column corresponds to a different covariate and each row to a different observation.

We assume the observations are assumed to related to the covariates by \[ \mathbf{y}=\mathbf{Xb}+\mathbf{e} \] where \(\mathbf{b}=[b_1,y_2,\dots,b_p]\) gives the parameters of the model which we don’t yet know, We call \(\mathbf{e}\) the “residual” vector. Legendre and Gauss pioneered the estimation of the parameters of a linear model by minimising the squared residuals, \(\mathbf{e}^T\mathbf{e}\), i.e. \[ \begin{aligned}\hat{\mathbf{b}} &=\operatorname{arg min}_\mathbf{b} (\mathbf{y}-\mathbf{Xb})^T (\mathbf{y}-\mathbf{Xb})\\ &=\operatorname{arg min}_\mathbf{b} \|\mathbf{y}-\mathbf{Xb}\|_2\\ &=\mathbf{X}^+\mathbf{y} \end{aligned} \] where we find the pseudo inverse \(\mathbf{X}^+\) using a numerical solver of some kind, using one of many carefully optimised methods that exists for least squares.

So far there is no statistical argument, merely function approximation.

However it turns out that if you assume that the \(\mathbf{e}_i\) are distributed randomly and independently i.i.d. errors in the observations (or at least indepenedent with constant variance), then there is also a statistical justification for this idea;

🏗 more exposition of these. Linkage to Maximum likelihood.

For now, handball to Lu (2022).