Machine Learning

12.9 Appendix to Chapter 12

In this appendix, a number of results concerning the Gaussian pdf are derived. The reader is advised to work on these derivations to get familiar with the tools that are heavily used in Bayesian inference. Because the derived formulas are applicable to different parts of the book and to different variables, we denote the involved random vectors as z and t and one can substitute notation accordingly, depending on the notational needs for each case.

12.9.1 PDFs with Exponent of Quadratic Form

Let

$\begin{array}{l}\hfill p(\mathit{z})=expF(\mathit{z}),\end{array}$

  (12.110)

where

$\begin{array}{l}\hfill F(\mathit{z})=-\frac{1}{2}{\mathit{z}}^{\text{T}}Q\mathit{z}+{\mathit{z}}^{\text{T}}\mathit{p}+C,\end{array}$

  (12.111)

where C is a constant and Q = Q^T and invertible. We rewrite ...