Naive Bayes Denominator clarification

Question

I came across an earlier post that was resolved and had a follow up to it but I couldn't comment because my reputation is under 50. Essentially I am interested in calculating the denominator in Naive Bayes.

Now I understand that the features in Naive Bayes are assumed to be independent so could we calculate $p(x) = p(x_{1})p(x_{2})...p(x_{n})$ or would we have to use this formula $$p(\mathbf{x}) = \sum_k p(C_k) \ p(\mathbf{x} \mid C_k)$$ with the conditional independence assumption that$$ p(\mathbf{x} \mid C_k) = \Pi_{i} \, p(x_i \mid C_k) $$

My question is would both ways of calculating give the same p(x)?

Link to the original question : https://datascience.stackexchange.com/posts/69699/edi

Edit** : Sorry I believe the features have conditional independence, rather than complete independence. Therefore it is incorrect to use $p(x) = p(x_{1})p(x_{2})...p(x_{n})$?

Lastly, I understand we don't actually need the denominator to find our probabilities but am asking out of curiosity.

Answer 1

The way to calculate $p(x)$ is indeed:

$$p(x) = \sum_k p(C_k) \ p(x| C_k)$$

Since in general one needs to calculate $p(C_k,x)$ (numerator) for every $k$, it's simple enough to sum all these $k$ values. It would be incorrect to use the product, indeed.

Lastly, I understand we don't actually need the denominator to find our probabilities but am asking out of curiosity.

Calculating the marginal $p(x)$ is not needed in order to find the most likely class $C_k$ because:

$$argmax_k(\{ p(C_k|x) \}) = argmax_k(\{ p(C_k,x) \})$$

However it's actually needed to find the posterior probability $p(C_k | x)$, that's why it's often useful to calculate the denominator $p(x)$ in order to obtain $p(C_k | x)$, especially if one wants to output the actual probabilities.