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In chapter 3 of the book by Mitchell ("Generative and discriminative classifiers: Naive Bayes and logistic regression") he states that "accurately estimating P(X|Y) typically requires many more examples. To see why, consider the number of parameters we must estimate when Y is boolean and X is a vector of n boolean attributes. In this case, we need to estimate a set of parameters

$\theta_{ij} \equiv P(X=x_i|Y=y_i)$

where the index $i$ takes on $2^n$ possible values (one for each of the possible vector values of $X$), and $j$ takes on $2$ possible values. Therefore, we will need to estimate approximately $2^{n+1}$ parameters."

Is that meaning it is needed to find a parameter for each probability for each possible combination of $X$ and $Y$? Given I've done that, how to put all together?

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The author compares $P(Y)$ (not $P(Y|X)$) and $P(X|Y)$ in terms of estimation complexity, due to the quickly growing $2^n$ term wrt number of dimensions. This is because of the dependence modelling in the joint distribution. Naive Bayes uses conditional independence assumption and relaxes this to $2n$ terms.

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  • $\begingroup$ I'm not getting where that $2^n$ comes from. Is that due to the law of total probability? $\endgroup$
    – TonyRomero
    Feb 15, 2020 at 16:04
  • $\begingroup$ when X is boolean with dimension n, there are $2^n$ different possibilities for it. $\endgroup$
    – gunes
    Feb 15, 2020 at 20:56

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