How to derive the $\log$ of fraction of two Gaussian distribution

Question

I am designing a function $A(x)$ that map $x\to \{0,1\}$. The function $A$ can be expressed as $$A(x)=\begin{cases} 1 \text{ if $p_1(x) \ge p_2(x)$}\\ 0 \text{ otherwises} \end{cases} $$ where $p_1$ and $p_2$ are Gaussian distribution:

$p_1(x)=\frac{1}{\sqrt{2\pi}\sigma_1}\exp\left( {-\frac{(x-\mu_1)^2}{2\sigma_1^2}} \right)$;

$p_2(x)=\frac{1}{\sqrt{2\pi}\sigma_2}\exp\left( {-\frac{(x-\mu_2)^2}{2\sigma_2^2}} \right)$.

Could you help me to derive a short form of $A(x)$ ?

This is my solution. However, it maybe wrong

I derived the bellow eq. $$A(x)=\begin{cases} 1 \text{ if $\log \frac{p_1(x)}{p_2(x)}>0$}\\ 0 \text{ otherwises} \end{cases} $$ where $\log(e)=1$

We have $$\log \frac{p_1(x)}{p_2(x)}=\log \frac{\sigma_2}{\sigma_1}+\left( {\frac{x-\mu_2}{\sqrt{2}\sigma_2}} \right)^2-\left( {\frac{x-\mu_1}{\sqrt{2}\sigma_1}} \right)^2$$

Hence, my final solution is $$A(x)=\begin{cases} 1 \text{ if $\left (\log \frac{\sigma_2}{\sigma_1}+\left( {\frac{x-\mu_2}{\sqrt{2}\sigma_2}} \right)^2-\left( {\frac{x-\mu_1}{\sqrt{2}\sigma_1}} \right)^2 \right) \ge 0$}\\ 0 \text{ otherwises} \end{cases} $$

Thanks in advance

Answer 1

Your final condition is correct, but it doesn't show explicitly for which values of $x$ it is satisfied. This can be worked out as follows. From what you have it follows that $p_1(x)\ge p_2(x)$ is equivalent to

$$0\ge x^2(\sigma_2^2-\sigma_1^2)-2x(\mu_1\sigma_2^2-\mu_2\sigma_1^2)+\mu_1^2\sigma_2^2-\mu_2^2\sigma_1^2-2\sigma_1^2\sigma_2^2\log\left(\frac{\sigma_2}{\sigma_1}\right)\tag{1}$$

Let's first consider the case $\sigma_1=\sigma_2$. From $(1)$ it follows that in this case the condition on $x$ becomes

$$x\ge \frac{\mu_1^2-\mu_2^2}{2(\mu_1-\mu_2)} = \frac{\mu_1+\mu_2}{2}\tag{2}$$

where we assume $\mu_1\neq\mu_2$ because otherwise $p_1(x)=p_2(x)$, which makes no sense.

In the case $\sigma_1\neq\sigma_2$, we have to consider $(1)$ as a quadratic inequality in $x$, which is satisfied with equality for the following values of $x$:

$$\begin{align}x_1&=\frac{a}{2}-\sqrt{\frac{a^2}{4}-b}\\ x_2&=\frac{a}{2}+\sqrt{\frac{a^2}{4}-b}\end{align}\tag{3}$$

with

$$\begin{align}a&=\frac{2(\mu_1\sigma_2^2-\mu_2\sigma_1^2)}{\sigma_2^2-\sigma_1^2}\\ b&=\frac{\mu_1^2\sigma_2^2-\mu_2^2\sigma_1^2-2\sigma_1^2\sigma_2^2\log\left(\frac{\sigma_2}{\sigma_1}\right)}{\sigma_2^2-\sigma_1^2}\end{align}\tag{4}$$

Depending on the sign of $\sigma_2-\sigma_1$ we have two conditions equivalent to $p_1(x)\ge p_2(x)$:

$$\begin{align}\sigma_2>\sigma_1:\quad & x_1\le x\le x_2\\ \sigma_2<\sigma_1:\quad & x\le x_1\text{ or }x\ge x_2\end{align}\tag{5}$$

The inequalities in $(5)$ state explicit conditions on the variable $x$ that can be easily evaluated.

EDIT: As suggested by Dilip Sarwate in a comment it can be instructive to distinguish another special case: $\sigma_1\neq\sigma_2$ and $\mu_1=\mu_2=\mu$. In that case the limits $x_1$ and $x_2$ of the decision regions given by $(3)$ lie symmetrical to the common mean $\mu$, and $(3)$ simplifies to

$$x_{1,2}=\mu\pm\sqrt{\frac{2\sigma_1^2\sigma_2^2\log\left(\frac{\sigma_2}{\sigma_1}\right)}{\sigma_2^2-\sigma_1^2}}$$

Of course, the conditions $(5)$ remain valid.