Last night in bed I came upon what seems to me like a new and at times useful way to compute naïve Bayesian conditional probabilities.
But since I have never seen it before, and since as I get older I often
overlook things, I though I would share it with those on this list to see
(A) if my math contains one of my occasional bloopers, (B) if my math is
correct, if it is commonly known or used, and/or (C) if there is some reason
why I haven't seen it before (whereas I have seen the traditional
formulation for naïve Bayes many times). It seems that this new-to-me
formulation is more intuitively straight forward than the traditional,
somewhat ass-backward, formulation of the naïve Bayes formula. Presumably
there is some explanation for why I haven't seen the new-to-me formula
before that is better than merely that mathematicians enjoy doing things in
an ass-backward manner.
As you all know the Naïve Bayes formula for the conditional probability of H
given evidence E1, E2,...EN is
p(H|E1,E2,...EN) = p(H) * p(E1|H)/p(E1) * p(E2|H)/p(E2) *...*
p(EN|H)/p(EN)
Assume one has the following set of values: p(H), p(E1), p(E2),... p(EN)
It seems to me one can, based on the Naïve Bayes formula, calculate the
conditional probability of H with the just this set of values using the
following formula (i.e., the new-to-me equation for calculating naïve
Bayesian conditional probabilities):
p(H|E1,E2,...EN) = p(H) * p(H|E1)/p(H) * p(H|E2)/p(H) * ... *
p(H|EN)/p(H)
My reasoning is a follows:
Assume Ei is any evidence E1...EN
P(H|Ei)p(Ei) = p(H,Ei)
and
p(H,Ei)/P(H) = P(Ei|H)
thus
P(H|Ei)p(Ei)/P(H) = P(Ei|H)
That means the naïve bayes formula could be re-written, substituting
P(H|Ei)p(A)/P(H) for P(Ei|H), as
p(H|E1,E2,...EN) = p(H) * P(H|E1)p(E1)/P(H)/p(E1)
* P(H|E2)p(E2)/P(H)/p(E2) * ...
* p(H)*P(H|EN)p(EN)/P(H)/p(EN)
Crossing out the p(Ei)/p(Ei) components one gets
p(H|E1,E2,...EN) = p(H) * P(H|E1)/P(H) * P(H|E2)/P(H) * ... *
p(H)*P(H|EN)/P(H),
which was to be proved.
If you have any thoughts re my questions (A),(B),(C) above, I would
appreciate them, particularly, if this is wrong why, and if this is not
wrong, why is it not used more often.
Ed Porter
-------------------------------------------
agi
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