# Naive Bayes with Density estimation [duplicate]

The way I understand the Naive Bayes estimators is that the characteristic (or naive) assumption is that all Features are conditionally independent.

Now on top of this assumption, for continuos features it is also assumed that they are Gaussian distributed. My question is whether this is something inherent in the Naive Bayes estimator or why do we not just use a non-parametric density estimation to determine the one dimensional probability distributions of the features.

For example, suppose that you want to approximate a joint pdf over two variables (X,Y), i.e., $p(x,y)$. Now, if you cannot make any assumption about the pdf, then best you can do is use a multivariate KDE or some other estimator to approximate the true $p(x,y)$. If you can assume that x and y are conditionally independent (i.e., $p(x|y)=p(x)$), then you have $p(x,y)=p(x|y)p(y)=p(x)p(y)$. This means that you can separately estimate $p(x)$ and $p(y)$ by some estimator and just multiply them. If these two distributions are multimodal, then you probably need to use the KDE. However, if you can assume that each of the pdfs can be well approximated by a single Gaussian, then you can approximate your joint pdf $p(x,y)\approx \mathcal{G}(x| \mu_x, \sigma_x) \mathcal{G}(y| \mu_y, \sigma_y)$, where $\mathcal{G}(x| \mu_x, \sigma_x)$ and $\mathcal{G}(y| \mu_y, \sigma_y)$ are just 1D Guassians.