0
$\begingroup$

Consider a binary classification task in 2D. Assume that the likelihood of both classes is Gaussian with uniform class prior, how should be the means $(\mu_1,\mu_2)$ and covariances $(\Sigma_1, \Sigma_2)$ of the discriminants $g_{y_1}(x)$ and $g_{y_1}(x)$ such that the decision surface is a spherical? For example, with equal isotropic covariance matrices, by equating both discriminants functions, we have a linear decision surface in the form $Ax + B =0$. My guess is to just take two differents means and covariance matrices in order to have a quadratic decision surface in the form $x'Ax + B'x +c = 0$. Can someone confirm/explain if wrong? Thanks

$\endgroup$
1
  • $\begingroup$ Read this answer for a discussion of the simpler case of two variables that are independent Gaussian under one hypothesis, and correlated jointly Gaussian under the other hypothesis. The decision boundary can be a parabola or a hyperbola. Your case is more complicated since the variables are correlated jointly Gaussian under both hypotheses. $\endgroup$ Jan 21, 2018 at 18:17

1 Answer 1

1
$\begingroup$

In the 2D case, you mean a circular decision boundary, not spherical, correct? This is the simplest way I can think of to accomplish that. Here's a plot I made showing the maximum of two bivariate gaussians: enter image description here

Both have mean (0, 0). The red one has covariance matrix:

$$\begin{pmatrix}1 & 0\\\ 0 & 1\end{pmatrix}$$

And the blue one has covariance matrix:

$$\begin{pmatrix}0.5 & 0\\\ 0 & 0.5\end{pmatrix}$$

Using these gaussians as your classes would give you a circular decision boundary. If the means were different, or the covariance matrices were full (rather than diagonal), you would end up with a more elliptical decision boundary.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.