# How to make a prediction with Bayes Classifier after computing MLE?

I'm trying to figure out the role of computing the MLE for classification/prediction purposes with the Bayes Classifier.

Let's say I'm given a set of data assumed to be Gaussian. I can then compute the MLE for parameters µ and ∑ (multivariate Gaussian) for each class. Now, how can I use those parameters to predict what class new data might be?

Do I just compute the two parameters directly and see if it matches to that of the MLE parameters I've computed or do plug in the new data in the Gaussian using the MLE parameters and see what the probability is?

I presume you've at least two different classes. Using MLE, you calculate $$\mu_i,\Sigma_i$$ and form the PDFs $$p_i(x)$$. Bayes classifier chooses the class with largest posterior, i.e. chooses the class with largest $$p_i(x)\pi_i$$, ignoring the denominator, where $$\pi_i$$ is the prior probability of an incoming sample belongs to class $$i$$. If equal priors is assumed, then you just choose the class with max $$p_i(x)$$.