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!


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