1. Naive Bayes, Caret symbol..?

Input X contains 3 categorical features— X1, X2, X3. The joint distribution becomes:

P(X | Y) = P(X1 | Y) * P(X2 | X1, Y) * P(X3 | X1, X2, Y)

P(x1 ^ x2 ^ x3 | y) = P(x1 | y) * P(x2 | y ^ x1) * P(x3 | y ^ x1 ^ x2)
= P(x1 | y) * P(x2 | y ) * P(x3 | y ^ x2)

1) What is the caret symbol above means
2) How was the last equation derived from the one above?

2. Re: Naive Bayes, Caret symbol..?

In a situation like this, the "caret" symbol is "and". That is, the first, "P(X1^X2^X3|Y)", is "The probability that all three of X1, X2, and X3 are true given that Y is true" (or "that all three happen given that Y happens").

The last equation follows from the first if and only if X1, X2, and X3 are independent.

3. Re: Naive Bayes, Caret symbol..?

Thanks - so if you writing it on paper, instead of the "caret" symbol, what do you write for "and"?

Does P(x1 & x2 & x3 | y) work - or this P(x1 U x2 U x3 | y)?

4. Re: Naive Bayes, Caret symbol..?

Originally Posted by silicon
Thanks - so if you writing it on paper, instead of the "caret" symbol, what do you write for "and"?

Does P(x1 & x2 & x3 | y) work - or this P(x1 U x2 U x3 | y)?
If you type P(X_1\cap X_2\cap X_3|Y) between two dollar signs we get $P(X_1\cap X_2\cap X_3|Y)$.

5. Re: Naive Bayes, Caret symbol..?

I could not reproduce what you say - you mean
$P(X_1\cap X_2\cap X_3|Y)$
I typed the above and previewed it but did not get it?
Does the "cap" meant for upper case X?

Does it only work on this forum or Word?

6. Re: Naive Bayes, Caret symbol..?

Originally Posted by silicon
I could not reproduce what you say - you mean
$P(X_1\cap X_2\cap X_3|Y)$
I typed the above and previewed it but did not get it?
Does the "cap" meant for upper case X?

Does it only work on this forum or Word?
That is call LaTeX coding. It us used to type-set scientific publications.
The cap and cup are used for $\cap~\&~\cup$, intersection(and) & union(or).

$\mathcal{P}(X\cup Y)$ is read 'the probability of X or Y'.

$\mathcal{P}(X\cap Y)$ is read 'the probability of X and Y'.