Decision boundary between two Gaussians of unequal variance This question is concerning a similar problem as mentioned in this question. The only difference is that in my case the variances are unequal.
To recap, consider a two class scenario. At the decision boundary, the posterior probability of classifying a data point into two classes will be equal i.e. $p(y=1|x) = p(y=2|x)$
Posterior definition
$p(y=1|x) = \frac{p(x|y=1) * P(y=1)}{p(x)}$
$p(y=2|x) = \frac{p(x|y=2) * P(y=2)}{p(x)}$
where likelihoods are Gaussian i.e.
$p(x|y=1) = \mathcal{N}(x|\mu_1, \sigma_1)$
$p(x|y=2) = \mathcal{N}(x|\mu_2, \sigma_2)$
So at the decision boundary, $p(y=1|x^*) = p(y=2|x^*)$ where $x^*$ is the threshold
$
\begin{align}
&p(y=1|x^*) = p(y=2|x^*) \\
&\Longrightarrow\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp(-\frac{(x^* - \mu_1)^2}{2\sigma_1^2}) * P(y=1) = \frac{1}{\sqrt{2\pi\sigma_2^2}}\exp(-\frac{(x^* - \mu_2)^2}{2\sigma_2^2})* P(y=2)\\
\end{align}
$
Taking log on both sides,
$
\begin{align}
\Rightarrow & \small-\frac{(x^* - \mu_1)^2}{2\sigma_1^2} -\log\sqrt{2\pi}\sigma_1 + \log P(y=1)= -\frac{(x^* - \mu_2)^2}{2\sigma_2^2} -\log\sqrt{2\pi}\sigma_2 + \log P(y=2)\\
\end{align}
$
To get the threshold, we would solve for $x^*$ in the above equation. But since the variances are unequal this will remain a quadratic equation and hence it is possible to get complex values for $x^*$.
If this is the case, then what does it mean to have a complex threshold?
Further context:
I'm using a 2 component Gaussian mixture model and planning to find the threshold to create a mask as specified in the paper snippet below. As such, I was expecting the threshold to be real in order to create the mask. Hence my confusion.


The solution x* will be two real values.
This sort of decision boundary is typical in mixture models, for instance $\sigma_1^2 = 1$ is a "typical" scenario, and $\sigma_2^2 = 100$ is an "outlier" scenario.
The low variance, highly peeked distribution is selected around its mean; elsewhere above and below this region attributed to the low variance distribution, the diffuse high variance distribution is selected.
West & Harrison discuss models involving "typical" and "outlier" observation variance $V$, in their notation $V_\text{typical}=1$ and $V_\text{outlier} = 100$.

*

*West, Mike, and Jeff Harrison. Bayesian forecasting and dynamic models. Springer Science & Business Media, 2006.

