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I'd like to know if this is a sensible idea and if there exist any already formed methods to do this (I'm new to the data science area).

Essentially, I have used Naive Bayes to accurately classify three types of food, based on their nutritious value (fat, salt, sugar, protein, and carbohydrates as my features).

Now that I can accurately classify these foods, Is there a method which uses the Naive Bayes to reverse this approach, and find the extreme values these features can be to be still classified as a type of food?

E.g: The max fat food1 can be, to still be considered food1.

I realize that these values will change, as other nutrient variables are changed, but I wondered if an optimized set of equations could be obtained in 5 dimensions?

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What you are looking for is called "decision boundary", which is the set of (extreme) points laying on the boundary between classes.

Decision boundary of NB is a set of points $\boldsymbol{x}$ that satisfy at least one of these $K(K-1)/2$ conditions $$i\neq j \in [1, K]:{\Bbb P}(\boldsymbol{x}, C_i)={\Bbb P}(\boldsymbol{x}, C_j) \overset{\forall k}{\geq} {\Bbb P}(\boldsymbol{x}, C_k)$$ Generally, Naive Bayes does not learn an explicit decision boundary (specially when categorical features are involved), however, for example here is the derivation of NB boundary for continuous variables when ${\Bbb P}(\boldsymbol{x}|C_i)$ is chosen from exponential family (e.g. a Gaussian). Nonetheless, you could still select a set of random points, then assign a class to each, and finally plot them using something like t-SNE to get a sense of decision boundaries. For example, a tool like this interactive t-SNE plot would be helpful:

t-SNE explorer

Or a library like mlxtend.plotting that carries out a similar procedure:

On the contrary, models like multi-class logistic regression and SVM both have explicit decision boundaries, which are equations of learned parameters $\boldsymbol{w}$ such as $$w_0+w_1\text{fat}+w_2\text{salt}+w_3\text{sugar}=0$$ that you can access via common libraries.

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    $\begingroup$ This is great, thank you very much $\endgroup$ – Dan Savage May 3 at 20:34

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