Statistical Inference for Everyone: Technical Supplement

This document is the technical supplement, for instructors, for Statistical Inference for Everyone, the introductory statistical inference textbook from the perspective of "probability theory as logic".

Estimating the Ratio of Two Variances $\kappa\equiv \sigma_x^2/\sigma_y^2$

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From Estimating the Deviation we have \begin{eqnarray} p(\sigma_x|\bvec{x},I)&\propto& \frac{1}{\sigma_x^{n}}e^{-V_x/2\sigma_x^2} \end{eqnarray} For independent $\bvec{x}$ and $\bvec{y}$ we have \begin{eqnarray} p(\sigma_x,\sigma_y|\bvec{x},\bvec{y},I)&\propto& \frac{1}{\sigma_x^{n}}\frac{1}{\sigma_y^{m}} e^{-V_x/2\sigma_x^2}e^{-V_y/2\sigma_y^2} \end{eqnarray} Changing variables to $\kappa\equiv \sigma_x^2/\sigma_y^2$, we have the following definitions \begin{eqnarray} \kappa&\equiv& \sigma_x^2/\sigma_y^2 \\\\ \sigma_x&=&\sigma_y \kappa^{1/2} \\\\ \sigma_y&=&\sigma_x \kappa^{-1/2} \end{eqnarray} and then transform the posterior \begin{eqnarray} p(\kappa,\sigma_x|\bvec{x},\bvec{y},I)&=& p(\sigma_x,\sigma_y|\bvec{x},\bvec{y},I)\times \left|\frac{\partial(\sigma_x,\sigma_y)}{\partial(\kappa,\sigma_x)}\right| \\\\ &=& p(\sigma_x,\sigma_y|\bvec{x},\bvec{y},I)\times \left| \begin{array}{cc} \frac{\partial\sigma_x}{\partial \kappa} & \frac{\partial \sigma_x}{\partial \sigma_x}\\\\ \frac{\partial\sigma_y}{\partial \kappa} & \frac{\partial \sigma_y}{\partial \sigma_x} \\\\ \end{array} \right|\\\\ &=& p(\sigma_x,\sigma_y|\bvec{x},\bvec{y},I)\times \left| \begin{array}{cc} \frac{1}{2}\sigma_y \kappa^{-1/2} & 1 \\\\ -\frac{1}{2}\sigma_x \kappa^{-3/2} & 0 \end{array} \right|\\\\ &\propto&\frac{1}{\sigma_x^{n}}\frac{\kappa^{m/2}}{\sigma_x^{m}} e^{-V_x/2\sigma_x^2}e^{-V_y\kappa/2\sigma_x^2} \sigma_x \kappa^{-3/2} \end{eqnarray} Now we integrate out the nuisance parameter, $\sigma_x$, to get

\begin{eqnarray} p(\kappa|\bvec{x},\bvec{y},I)&=&\int d\sigma_x p(\kappa,\sigma_x|\bvec{x},\bvec{y},I) \\\\ &\propto& \kappa^{(m-3)/2} \int d\sigma_x \frac{1}{\sigma_x^{n+m-1}} e^{-(V_x+V_y\kappa)/2\sigma_x^2} \end{eqnarray}

from the gaussian integral trick we get

\begin{eqnarray} p(\kappa|\bvec{x},\bvec{y},I)&\propto&\kappa^{(m-3)/2} (V_x+V_y\kappa)^{(n+m-2)/2} \end{eqnarray}

A more common form is found with the substitutions \begin{eqnarray} \eta &\equiv&\kappa\times \frac{(V_y/f_y)}{(V_x/f_x)} \\\\ f_x&\equiv&n-1 \\\\ f_y&\equiv&m-1 \end{eqnarray}

from which it follows \begin{eqnarray} p(\eta|\bvec{x},\bvec{y},I)&\propto& \eta^{\frac{f_y}{2}-1} (f_x+f_y\eta)^{(f_x+f_y)/2} \end{eqnarray} which is the commonly used F distribution.

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