Difference between Probabilistic kNN and Naive Bayes

Question

I'm trying to modify an standard kNN algorithm to obtain the probability of belonging to a class instead of just the usual classification. I haven't found much information about Probabilistic kNN, but as far as I understand, it works similar to kNN, with the difference that it calculates the percentage of examples of every class inside the given radius.

So I wonder, what's the difference then between Naive Bayes and Probabilistic kNN? I just can spot that Naive Bayes takes into consideration the prior possibility, while PkNN does not. Am I getting it wrong?

Thanks in advance!

Answer 1

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I would like to propose an opposite view that KNN is a kind of simplified Naive Bayes (NB) by viewing KNN as a mean of density estimation.

To perform density estimation, we attempt to estimate p(x) = k/NV, where k is the number of samples lying in a region R, N is the total sample number, and V is the volume of the region R. Usually, there are two ways to estimate it: (1) fixing V, calculate k, which is known as kernel density estimation or Parzen window; (2) fixing k, calculate V, which is the KNN-based density estimation. The latter one is much less famous than the former one due to its many drawbacks.

Yet, we can use KNN-based density estimation to connect KNN and NB. Given total N samples, Ni samples for class ci, we can write the NB in the form of KNN-based density estimation by considering a region contain x:

P(ci|x) = P(x|ci)P(ci)/P(x) = (ki/NiV)(Ni/N)/(k/NV) = ki/k,

where ki is the sample number of class ci lying in the region. The final form ki/k is actually the KNN classifier.

Answer 2

To be honest there is nearly no similarity.

Naive bayes assumes that each class is distributed according to a simple distribution, independent on feature basis. For contiuous case - It will fit a radial Normal distribution to your whole class (each of them) and then make a decision through argmax_y N(m_y, Sigma_y)

KNN on the other hand is not a probabilistic model. Modification that you are refering to is simply a "smooth" version of the original idea, where you return ratio of each class in the nearest neighbours set (and this is not really any "probabilistic kNN", it is just regular kNN which rough estimate of probability). This assumes nothing about data distribution (besides being localy smooth). In particular - it is a nonparametric model which, given enough training samples, will fit perfectly to any dataset. Naive Bayes will fit perfectly only to K gaussians (where K is number of classes).