# Reversed Naive Bayes - likelihood and parameter estimation

What happens if we flip the arrows in a Naive Bayes classifier? To clarify - from what I have found naive Bayes is defined for the following network structure:

I'm interested to understand what happens if instead from y->x the child will be y. As shown in the following:

I'm not entirely sure if we get the same results as a regular naive Bayes classifier (intuitively we don't), and if not how to estimate both the likelihood of the network and the parameters?

For further clarification: lets say there is a training set of size n; each data point consisting of (x,y) where x is a vector of size m of binary values and y is the class and is also binary. E.g. (0,1,0,1,1) where the last index is the class. I'm trying to figure out the likelihood of the network (where y is the child) and how to estimate the parameters in the case of y being the child.

• What do you mean by "flipping arrows"? Can you try being more precise?
– Tim
Dec 11, 2017 at 20:23
• edited, hope it is clearer now
– DML
Dec 11, 2017 at 21:02

There are undirected connections drawn between the $$X$$'s in the first graph and, strictly speaking, Naive Bayes assumes that there are only $$Y \rightarrow X$$ connections. So if the generative model is as you draw it then Naive Bayes will get things wrong by assuming those connections are absent.
In contrast, logistic regression - the second version - need not assume any distribution for $$X$$s at all (and if it does, it won't matter), so correlations induced by those connections are not problematic for estimation or for predicting $$Y$$. In that sense it's usually more practically appealing.