# Relation between the Naive Bayes Classifier and GAM

Problem:

This problem is about establishing a connection between the Naive Bayes Classifier and GAM. Consider a classification problem with J classes. Let $$f_j (X), X ∈ ℝ^p$$, be a density function for class $$j = 1, ...,J$$. Show that for a given $$j∈ {1, 2, . . . , J − 1}$$

$$log\frac{P(Y=j|X}{P(Y=J|X}=\beta_{j0}+\sum_{k=1}^{p} h_{jk}(X_K)$$

for some constant $$\beta_{j0}$$ and funstions $$h_{jk}$$. Assume that $$f_j(X)=f_{j1}(X_1)...f_{jp}(X_p).$$

My guess:

The idea behind the Generalized Naive Bayes Classifier is to relax the conditional independence assumption by adjusting the Naive Effects. This is done by adding p functions, $$f_{j1}(X_1), . . . , f_{jp}(X_p)$$, to the Naive Effects, where $$f_j (x_j )$$ accounts for the marginal bias. I am not sure how can we connect this to show the above results. I appreciate your thoughts!